Download The Fundamental Counting Principle and Permutations

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12.1
What you should learn
The Fundamental Counting
Principle and Permutations
GOAL 1
THE FUNDAMENTAL COUNTING PRINCIPLE
GOAL 1 Use the fundamental counting principle to
count the number of ways an
event can happen.
In many real-life problems you want to count the number of possibilities. For
instance, suppose you own a small deli. You offer 4 types of meat (ham, turkey, roast
beef, and pastrami) and 3 types of bread (white, wheat, and rye). How many choices
do your customers have for a meat sandwich?
GOAL 2 Use permutations
to count the number of ways
an event can happen, as
applied in Ex. 62.
One way to answer this question is to use a tree diagram, as shown below. From the
list on the right you can see that there are 12 choices.
Why you should learn it
RE
Turkey
Ham on white
Wheat
Ham on wheat
Rye
Ham on rye
White
Turkey on white
Wheat
Turkey on wheat
Rye
Turkey on rye
White
Roast beef on white
Wheat
Roast beef on wheat
Rye
Roast beef on rye
White
Pastrami on white
Wheat
Pastrami on wheat
Rye
Pastrami on rye
FE
䉲 To find the number of
ways a real-life event can
happen, such as the number
of ways skiers can finish in
an aerial competition
in Example 3.
AL LI
Ham
White
Roast beef
Pastrami
Another way to count the number of possible sandwiches is to use the fundamental
counting principle. Because you have 4 choices for meat and 3 choices for bread,
the total number of choices is 4 • 3 = 12.
F U N DA M E N TA L C O U N T I N G P R I N C I P L E
TWO EVENTS If one event can occur in m ways and another event can occur
in n ways, then the number of ways that both events can occur is m • n.
For instance, if one event can occur in 2 ways and another event can occur in
5 ways, then both events can occur in 2 • 5 = 10 ways.
THREE OR MORE EVENTS The fundamental counting principle can be extended
to three or more events. For example, if three events can occur in m, n, and p
ways, then the number of ways that all three events can occur is m • n • p.
For instance, if three events can occur in 2, 5, and 7 ways, then all three events
can occur in 2 • 5 • 7 = 70 ways.
12.1 The Fundamental Counting Principle and Permutations
701
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FOCUS ON
EXAMPLE 1
CAREERS
Using the Fundamental Counting Principle
CRIMINOLOGY Police use photographs of various facial features to help witnesses
identify suspects. One basic identification kit contains 195 hairlines, 99 eyes and
eyebrows, 89 noses, 105 mouths, and 74 chins and cheeks.
䉴 Source: Readers’ Digest: How In The World?
a. The developer of the identification kit claims that it can produce billions of
different faces. Is this claim correct?
b. A witness can clearly remember the hairline and the eyes and eyebrows of a
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suspect. How many different faces can be produced with this information?
POLICE
DETECTIVE
INT
A police detective is an
officer who collects facts
and evidence for criminal
cases. Part of a detective’s
duties may include helping
witnesses identify suspects.
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CAREER LINK
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SOLUTION
a. You can use the fundamental counting principle to find the total number of
different faces.
Number of faces = 195 • 99 • 89 • 105 • 74 = 13,349,986,650
䉴
The developer’s claim is correct since the kit can produce over 13 billion faces.
b. Because the witness clearly remembers the hairline and the eyes and eyebrows,
there is only 1 choice for each of these features. You can use the fundamental
counting principle to find the number of different faces.
Number of faces = 1 • 1 • 89 • 105 • 74 = 691,530
䉴
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License Plates
The number of faces that can be produced has been reduced to 691,530.
EXAMPLE 2
Using the Fundamental Counting Principle with Repetition
The standard configuration for a New York license
plate is 3 digits followed by 3 letters.
䉴 Source: New York State Department of Motor Vehicles
a. How many different license plates are possible
if digits and letters can be repeated?
b. How many different license plates are possible
if digits and letters cannot be repeated?
SOLUTION
a. There are 10 choices for each digit and 26 choices for each letter. You can use the
fundamental counting principle to find the number of different plates.
Number of plates = 10 • 10 • 10 • 26 • 26 • 26 = 17,576,000
䉴
The number of different license plates is 17,576,000.
b. If you cannot repeat digits there are still 10 choices for the first digit, but then
only 9 remaining choices for the second digit and only 8 remaining choices for
the third digit. Similarly, there are 26 choices for the first letter, 25 choices for
the second letter, and 24 choices for the third letter. You can use the fundamental
counting principle to find the number of different plates.
Number of plates = 10 • 9 • 8 • 26 • 25 • 24 = 11,232,000
䉴
702
The number of different license plates is 11,232,000.
Chapter 12 Probability and Statistics
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GOAL 2
USING PERMUTATIONS
An ordering of n objects is a permutation of the objects. For instance, there are six
permutations of the letters A, B, and C: ABC, ACB, BAC, BCA, CAB, CBA.
STUDENT HELP
Study Tip
Recall from Lesson 11.5
that n! is read as “n
factorial.” Also note that
0! = 1 and 1! = 1.
The fundamental counting principle can be used to determine the number of
permutations of n objects. For instance, you can find the number of ways you can
arrange the letters A, B, and C by multiplying. There are 3 choices for the first letter,
2 choices for the second letter, and 1 choice for the third letter, so there are
3 • 2 • 1 = 6 ways to arrange the letters.
In general, the number of permutations of n distinct objects is:
n! = n • (n º 1) • (n º 2) • . . . • 3 • 2 • 1
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Sports
EXAMPLE 3
Finding the Number of Permutations
Twelve skiers are competing in the final round of the Olympic freestyle skiing aerial
competition.
a. In how many different ways can the skiers finish the competition? (Assume there
are no ties.)
b. In how many different ways can 3 of the skiers finish first, second, and third to
win the gold, silver, and bronze medals?
SOLUTION
a. There are 12! different ways that the skiers can finish the competition.
12! = 12 • 11 • 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 479,001,600
b. Any of the 12 skiers can finish first, then any of the remaining 11 skiers can
finish second, and finally any of the remaining 10 skiers can finish third. So, the
number of ways that the skiers can win the medals is:
12 • 11 • 10 = 1320
..........
INT
STUDENT HELP
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ER T
KEYSTROKE HELP
Visit our Web site
www.mcdougallittell.com
to see keystrokes for
several models of
calculators.
STUDENT HELP
Derivations
For a derivation of the
formula for the permutation of n objects taken r
at a time, see p. 899.
Some calculators have special keys to evaluate
factorials. The solution to Example 3 is shown.
12!
12!/9!
The number in part (b) of Example 3 is called the
number of permutations of 12 objects taken 3 at a
479001600
1320
12!
(12 º 3)!
time, is denoted by 12P3, and is given by .
P E R M U TAT I O N S O F
n
O B J E C T S TA K E N
r
AT A T I M E
The number of permutations of r objects taken from a group of n distinct objects
is denoted by nPr and is given by:
nPr
n!
= (n º r)!
12.1 The Fundamental Counting Principle and Permutations
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College Visits
EXAMPLE 4
Finding Permutations of n Objects Taken r at a Time
You are considering 10 different colleges. Before you decide to apply to the colleges,
you want to visit some or all of them. In how many orders can you visit (a) 6 of the
colleges and (b) all 10 colleges?
SOLUTION
a. The number of permutations of 10 objects taken 6 at a time is:
10!
3,628,800
10!
= = = 151,200
10 P6 = 4!
24
(10 º 6)!
b. The number of permutations of 10 objects taken 10 at a time is:
10!
10!
= = 10! = 3,628,800
10 P10 = 0!
(10 º 10)!
..........
INT
STUDENT HELP
NE
ER T
KEYSTROKE HELP
Visit our Web site
www.mcdougallittell.com
to see keystrokes for
several models of
calculators.
Some calculators have special keys that are programmed
to evaluate nPr . The solution to Example 4 is shown.
So far you have been finding permutations of distinct
objects. If some of the objects are repeated, then some of
the permutations are not distinguishable. For instance, of
the six ways to order the letters M, O, and M—
MOM
OMM
MMO
MOM
OMM
MMO
10 nPr 6
10 nPr 10
151200
3628800
—only three are distinguishable without color: MOM, OMM, and MMO. In this
3!
2!
6
2
case, the number of permutations is = = 3, not 3! = 6.
P E R M U TAT I O N S W I T H R E P E T I T I O N
The number of distinguishable permutations of n objects where one object is
repeated q1 times, another is repeated q2 times, and so on is:
n!
q1! • q2! • . . . • qk!
EXAMPLE 5
Finding Permutations with Repetition
Find the number of distinguishable permutations of the letters in (a) OHIO
and (b) MISSISSIPPI.
SOLUTION
a. OHIO has 4 letters of which O is repeated 2 times. So, the number of
4!
24
distinguishable permutations is = = 12.
2!
2
b. MISSISSIPPI has 11 letters of which I is repeated 4 times, S is repeated 4 times,
and P is repeated 2 times. So, the number of distinguishable permutations is
39,916,800
11!
= = 34,650.
2 4 • 24 • 2
4! • 4! • 2!
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Chapter 12 Probability and Statistics
Page 5 of 7
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
1. What is a permutation of n objects?
2. Explain how the fundamental counting principle can be used to justify the
formula for the number of permutations of n distinct objects.
3. Rita found the number of distinguishable permutations of the letters in OHIO by
4!
evaluating the expression . Does this method give the same answer as
2! • 1! • 1!
in part (a) of Example 5? Explain.
Number of 3-digit numbers
with only even digits
=5•4•3
= 60
4. ERROR ANALYSIS Explain the error in
Skill Check
✓
calculating how many three-digit numbers
from 000 to 999 have only even digits.
Find the number of permutations of n distinct objects.
5. n = 2
6. n = 6
7. n = 1
8. n = 4
Find the number of permutations of n objects taken r at a time.
9. n = 6, r = 3
10. n = 5, r = 1
11. n = 3, r = 3
12. n = 10, r = 2
Find the number of permutations of n objects where one or more objects are
repeated the given number of times.
13. 7 objects with one object repeated 4 times
14. 5 objects with one object repeated 3 times and a second object repeated 2 times
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 956.
FUNDAMENTAL COUNTING PRINCIPLE Each event can occur in the given
number of ways. Find the number of ways all of the events can occur.
15. Event 1: 1 way, Event 2: 3 ways
16. Event 1: 3 ways, Event 2: 5 ways
17. Event 1: 2 ways, Event 2: 4 ways,
18. Event 1: 4 ways, Event 2: 6 ways,
Event 3: 5 ways
Event 3: 9 ways, Event 4: 7 ways
LICENSE PLATES For the given configuration, determine how many different
license plates are possible if (a) digits and letters can be repeated, and
(b) digits and letters cannot be repeated.
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 15–18,
55, 56
Example 2: Exs. 19–22,
57
Example 3: Exs. 23–30,
39–46, 59, 60
Example 4: Exs. 31–38,
61
Example 5: Exs. 47–54,
62, 63
19. 3 letters followed by 3 digits
20. 2 digits followed by 4 letters
21. 4 digits followed by 2 letters
22. 5 letters followed by 1 digit
FACTORIALS Evaluate the factorial.
23. 8!
24. 5!
25. 10!
26. 9!
27. 0!
28. 7!
29. 3!
30. 12!
PERMUTATIONS Find the number of permutations.
31. 3P3
32. 5P2
33. 2P1
35. 8P5
36. 9P4
37.
12P3
34. 7P6
38.
16P0
12.1 The Fundamental Counting Principle and Permutations
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FOCUS ON
APPLICATIONS
PERMUTATIONS WITHOUT REPETITION Find the number of distinguishable
permutations of the letters in the word.
39. HI
40. JET
41. IOWA
42. TEXAS
43. PENCIL
44. FLORIDA
45. MAGNETIC
46. GOLDFINCH
PERMUTATIONS WITH REPETITION Find the number of distinguishable
permutations of the letters in the word.
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COMPUTER
SECURITY On the
Internet there are three main
ways to secure a site: restrict
which addresses can access
the site, use public key
cryptography, or require a
user name and password.
47. DAD
48. PUPPY
49. OREGON
50. LETTER
51. ALGEBRA
52. ALABAMA
53. MISSOURI
54. CONNECTICUT
55.
STEREO You are going to set up a stereo system by purchasing separate
components. In your price range you find 5 different receivers, 8 different
compact disc players, and 12 different speaker systems. If you want one of each
of these components, how many different stereo systems are possible?
56.
PIZZA A pizza shop runs a special where you can buy a large pizza with one
cheese, one vegetable, and one meat for $9.00. You have a choice of 7 cheeses,
11 vegetables, and 6 meats. Additionally, you have a choice of 3 crusts and
2 sauces. How many different variations of the pizza special are possible?
57.
COMPUTER SECURITY To keep computer files secure, many programs
require the user to enter a password. The shortest allowable passwords are
typically six characters long and can contain both numbers and letters. How
many six-character passwords are possible if (a) characters can be repeated
and (b) characters cannot be repeated?
58. CRITICAL THINKING Simplify the formula for n Pr when r = 0. Explain
why this result makes sense.
59.
CLASS SEATING A particular classroom has 24 seats and 24 students.
Assuming the seats are not moved, how many different seating arrangements are
possible? Write your answer in scientific notation.
60.
RINGING BELLS “Ringing the changes” is a process where the bells in a
tower are rung in all possible permutations. Westminster Abbey has 10 bells in
its tower. In how many ways can its bells be rung?
INT
STUDENT HELP
NE
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61.
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with problem
solving in Ex. 61.
62.
PLAY AUDITIONS Auditions are being held
for the play shown. How many ways can the
roles be assigned if (a) 6 people audition and
(b) 9 people audition?
WINDOW DISPLAY A music store wants
to display 3 identical keyboards, 2 identical
trumpets, and 2 identical guitars in its store window.
How many distinguishable displays are possible?
63.
the
Drama
Club
is holding
Open Auditions
for parts in a one-act play
Parts available:
DOG SHOW In a dog show how many ways
Student A
Teacher
can 3 Chihuahuas, 5 Labradors, 4 poodles, and 3
beagles line up in front of the judges if the dogs
of the same breed are considered identical?
Student B
Librarian
Principal
Coach
64. CRITICAL THINKING Find the number of permutations of n objects taken n º 1
at a time for any positive integer n. Compare this answer with the number of
permutations of all n objects. Does this make sense? Explain.
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Chapter 12 Probability and Statistics
Page 7 of 7
Test
Preparation
★ Challenge
QUANTITATIVE COMPARISON In Exercises 65 and 66, choose the statement
that is true about the given quantities.
A
¡
B
¡
C
¡
D
¡
The quantity in column A is greater.
The quantity in column B is greater.
The two quantities are equal.
The relationship cannot be determined from the given information.
Column A
Column B
65.
5 P1
5!
66.
The number of permutations of
12 objects taken 7 at a time
The number of permutations
of 12 objects, one of which
is repeated 5 times
67. CIRCULAR PERMUTATIONS You have learned that
A
n! represents the number of ways that n objects can be
placed in a linear order, where it matters which object is
placed first. Now consider circular permutations, where
objects are placed in a circle so it does not matter which
object is placed first. Find a formula for the number of
permutations of n objects placed in clockwise order
around a circle when only the relative order of the objects
matters. Explain how you derived your formula.
EXTRA CHALLENGE
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E
B
C
D
C
D
B
E
A
This is the same
permutation.
MIXED REVIEW
SPECIAL PRODUCTS Find the product. (Review 6.3 for 12.2)
68. (x + 9)(x º 9)
69. (x2 + 2)2
70. (2x º 1)3
71. (4x + 5)(4x º 5)
72. (2y + 3x)2
73. (8y º x)2
GRAPHING Graph the equation of the parabola. (Review 10.2)
74. y2 = 8x
75. x2 = º10y
76. y2 = º4x
77. x2 = 26y
78. x + 14y2 = 0
79. y º 2x2 = 0
FINDING SUMS Find the sum of the infinite geometric series if there is one.
(Review 11.4)
∑ 3冉12冊
81.
∑ 2冉75冊
84.
‡
80.
n=0
‡
83.
nº1
n=1
∑ º4冉14冊
82.
∑ º5冉18冊
85.
‡
n
n=0
‡
n=0
n
∑ 2冉23冊
‡
n
n=0
‡
∑
n=1
n
1
(0.3)n º 1
2
CONNECTION Ohm’s law states that the resistance R (in ohms) of a
conductor varies directly with the potential difference V (in volts) between two
points and inversely with the current I (in amperes). The constant of variation is
1. What is the resistance of a light bulb if there is a current of 0.80 ampere when
the potential difference across the bulb is 120 volts? (Review 9.1)
86. SCIENCE
12.1 The Fundamental Counting Principle and Permutations
707
Page 1 of 8
12.2
What you should learn
GOAL 1 Use combinations
to count the number of ways
an event can happen, as
applied in Ex. 55.
GOAL 2 Use the binomial
theorem to expand a binomial
that is raised to a power.
Combinations and
the Binomial Theorem
GOAL 1
USING COMBINATIONS
In Lesson 12.1 you learned that order is important for some counting problems.
For other counting problems, order is not important. For instance, in most card
games the order in which your cards are dealt is not important. After your cards
are dealt, reordering them does not change your card hand. These unordered
groupings are called combinations. A combination is a selection of r objects
from a group of n objects where the order is not important.
n
䉲 To solve real-life
problems, such as finding
the number of different
combinations of plays
you can attend
in Example 3.
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The number of combinations of r objects taken from a group of n distinct
objects is denoted by nCr and is given by:
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C O M B I N AT I O N S O F
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O B J E C T S TA K E N
r
Why you should learn it
n Cr
AT A T I M E
n!
ᎏ
=ᎏ
(n º r)! • r!
For instance, the number of combinations of 2 objects taken from a group of
5!
5 objects is 5C2 = ᎏᎏ = 10.
3! • 2!
EXAMPLE 1
Finding Combinations
A standard deck of 52 playing cards has 4 suits with
13 different cards in each suit as shown.
a. If the order in which the cards are dealt is not
important, how many different 5-card hands
are possible?
b. In how many of these hands are all five cards
of the same suit?
SOLUTION
a. The number of ways to choose 5 cards from a
deck of 52 cards is:
52C5
52 !
47! • 5!
ᎏ
=ᎏ
Standard 52-Card Deck
K♠
Q♠
J♠
10 ♠
9♠
8♠
7♠
6♠
5♠
4♠
3♠
2♠
A♠
K♣
Q♣
J♣
10 ♣
9♣
8♣
7♣
6♣
5♣
4♣
3♣
2♣
A♣
K♦
Q♦
J♦
10 ♦
9♦
8♦
7♦
6♦
5♦
4♦
3♦
2♦
A♦
K♥
Q♥
J♥
10 ♥
9♥
8♥
7♥
6♥
5♥
4♥
3♥
2♥
A♥
52 • 51 • 50 • 49 • 48 • 47!
47! • 5!
= ᎏᎏᎏ
= 2,598,960
b. For all five cards to be the same suit, you need to choose 1 of the 4 suits and then
5 of the 13 cards in the suit. So, the number of possible hands is:
4C1
708
4!
13!
3! • 1! 8! • 5!
4 • 3!
3! • 1!
13 • 12 • 11 • 10 • 9 • 8!
8! • 5!
• 13C5 = ᎏᎏ • ᎏᎏ = ᎏᎏ • ᎏᎏᎏ = 5148
Chapter 12 Probability and Statistics
Page 2 of 8
When finding the number of ways both an event A and an event B can occur, you
need to multiply (as you did in part (b) of Example 1). When finding the number
of ways that an event A or an event B can occur, you add instead.
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Menu Choices
EXAMPLE 2
Deciding to Multiply or Add
A restaurant serves omelets that can be ordered
with any of the ingredients shown.
a. Suppose you want exactly 2 vegetarian
ingredients and 1 meat ingredient in your
omelet. How many different types of
omelets can you order?
b. Suppose you can afford at most
3 ingredients in your omelet. How many
different types of omelets can you order?
Omelets $3.00
(plus $.50 for each ingredient)
Meat
Vegetarian
green pepper
ham
red pepper
bacon
onion
sausage
mushroom
steak
tomato
cheese
SOLUTION
a. You can choose 2 of 6 vegetarian ingredients and 1 of 4 meat ingredients. So, the
number of possible omelets is:
6C2
6!
4!
4! • 2! 3! • 1!
• 4C1 = ᎏᎏ • ᎏᎏ = 15 • 4 = 60
b. You can order an omelet with 0, 1, 2, or 3 ingredients. Because there are 10 items
to choose from, the number of possible omelets is:
10C0
..........
INT
STUDENT HELP
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KEYSTROKE HELP
Visit our Web site
www.mcdougallittell.com
to see keystrokes for
several models of
calculators.
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+ 10C1 + 10C2 + 10C3 = 1 + 10 + 45 + 120 = 176
Theater
Some calculators have special keys to evaluate
combinations. The solution to Example 2 is shown.
Counting problems that involve phrases like “at least”
or “at most” are sometimes easier to solve by
subtracting possibilities you do not want from the
total number of possibilities.
EXAMPLE 3
(6 nCr 2)(4 nCr 1)
60
10 nCr 0+10 nCr 1
+10 nCr 2+10 nCr 3
176
Subtracting Instead of Adding
A theater is staging a series of 12 different plays. You want to attend at least 3 of the
plays. How many different combinations of plays can you attend?
SOLUTION
You want to attend 3 plays, or 4 plays, or 5 plays, and so on. So, the number of
combinations of plays you can attend is 12C3 + 12C4 + 12C5 + . . . + 12C12.
Instead of adding these combinations, it is easier to use the following reasoning. For
each of the 12 plays, you can choose to attend or not attend the play, so there are 212
total combinations. If you attend at least 3 plays you do not attend only 0, 1, or 2
plays. So, the number of ways you can attend at least 3 plays is:
212 º (12C0 + 12C1 + 12C2) = 4096 º (1 + 12 + 66) = 4017
12.2 Combinations and the Binomial Theorem
709
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FOCUS ON
PEOPLE
GOAL 2
USING THE BINOMIAL THEOREM
If you arrange the values of nCr in a triangular pattern in which each row corresponds
to a value of n, you get what is called Pascal’s triangle. It is named after the
famous French mathematician Blaise Pascal (1623–1662).
1
0 C0
1C0
2C0
3 C0
4C0
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BLAISE PASCAL
developed his
arithmetic triangle in 1653.
The following year he and
fellow mathematician
Pierre Fermat outlined the
foundations of probability
theory.
5C0
2 C1
3C1
4C1
5C1
1
2C2
3C2
4C2
5C2
1
1C1
4C3
5 C3
1
3 C3
1
4C4
5C4
1
5 C5
1
2
3
4
1
3
6
1
4
1
5 10 10 5
1
Pascal’s triangle has many interesting patterns and properties. For instance, each
number other than 1 is the sum of the two numbers directly above it.
ACTIVITY
Developing
Concepts
1
Investigating Pascal’s Triangle
Expand each expression. Write the terms of each expanded expression so
that the powers of a decrease.
a. (a + b)2
b. (a + b)3
c. (a + b)4
2
Describe the relationship between the coefficients in parts (a), (b), and (c)
of Step 1 and the rows of Pascal’s triangle.
3
Describe any patterns in the exponents of a and the exponents of b.
In the activity you may have discovered the following result, which is called the
binomial theorem. This theorem describes the coefficients in the expansion of the
binomial a + b raised to the nth power.
THE BINOMIAL THEOREM
The binomial expansion of (a + b)n for any positive integer n is:
(a + b)n = C anb0 + C an º 1b1 + C an º 2b2 + . . . + C a0bn
n 0
n
=∑
r=0
EXAMPLE 4
STUDENT HELP
Study Tip
You can calculate
combinations using
either Pascal’s triangle
or the formula on p. 708.
n 1
n 2
nºr r
b
nCr a
Expanding a Power of a Simple Binomial Sum
Expand (x + 2)4.
SOLUTION
(x + 2)4 = 4C0 x420 + 4C1x321 + 4C2 x 222 + 4C3x123 + 4C4x024
= (1)(x4)(1) + (4)(x3)(2) + (6)(x 2)(4) + (4)(x)(8) + (1)(1)(16)
= x4 + 8x3 + 24x 2 + 32x + 16
710
n n
Chapter 12 Probability and Statistics
Page 4 of 8
Expanding a Power of a Binomial Sum
EXAMPLE 5
Expand (u + v2)3.
SOLUTION
(u + v 2)3 = 3C0u3(v 2)0 + 3C1u2(v 2)1 + 3C2u1(v 2)2 + 3C3u0(v 2)3
= u3 + 3u2v 2 + 3uv4 + v6
..........
To expand a power of a binomial difference, you can rewrite the binomial as a sum.
The resulting expansion will have terms whose signs alternate between + and º.
Expanding a Power of a Simple Binomial Difference
EXAMPLE 6
Expand (x º y)5.
SOLUTION
(x º y)5 = [x + (ºy)]5
= 5C0 x 5(ºy)0 + 5C1x4(ºy)1 + 5C2 x 3(ºy)2 + 5C3 x 2(ºy)3 +
1
4
0
5
5C4 x (ºy) + 5C5 x (ºy)
= x 5 º 5x4y + 10x 3y2 º 10x 2y3 + 5xy4 º y5
Expanding a Power of a Binomial Difference
EXAMPLE 7
Expand (5 º 2a)4.
SOLUTION
(5 º 2a)4 = [5 + (º2a)]4
= 4C054(º2a)0 + 4C153(º2a)1 + 4C252(º2a)2 + 4C351(º2a)3 +
0
4
4C45 (º2a)
= (1)(625)(1) + (4)(125)(º2a) + (6)(25)(4a2) + (4)(5)(º8a3) +
(1)(1)(16a4)
= 625 º 1000a + 600a2 º 160a3 + 16a4
EXAMPLE 8
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.
Finding a Coefficient in an Expansion
Find the coefficient of x4 in the expansion of (2x º 3)12.
SOLUTION From the binomial theorem you know the following:
12
(2x º 3)12 = ∑
r=0
12 º r
(º3)r
12Cr(2x)
The term that has x4 is 12C8(2x)4(º3)8 = (495)(16x4)(6561) = 51,963,120x4.
䉴
The coefficient is 51,963,120.
12.2 Combinations and the Binomial Theorem
711
Page 5 of 8
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
1. Explain the difference between a permutation and a combination.
2. Describe a situation in which to find the total number of possibilities you would
(a) add two combinations and (b) multiply two combinations.
3. Write the expansions for (x + y)4 and (x º y)4.
How are they similar? How are they different?
10C6
4. ERROR ANALYSIS What error was made in
Skill Check
✓
10 • 9 • 8 • 7
= ᎏᎏᎏ
6 • 5 • 4 • 3 • 2 • 1
=7
the calculation of 10C6? Explain.
Find the number of combinations of n objects taken r at a time.
5. n = 8, r = 2
6. n = 6, r = 5
7. n = 5, r = 1
8. n = 9, r = 9
Expand the power of the binomial.
9. (x + y)3
10. (x + 1)4
11. (2x + 4)3
12. (2x + 3y)5
13. (x º y)5
14. (x º 2)3
15. (3x º 1)4
16. (4x º 4y)3
17. Complete this equation:
? x2y3 + 405xy4 + 㛭㛭㛭
? y5
(x + 3y)5 = x5 + 15x4y + 90x3y2 + 㛭㛭㛭
PRACTICE AND APPLICATIONS
STUDENT HELP
COMBINATIONS Find the number of combinations.
Extra Practice
to help you master
skills is on p. 956.
18. 10C2
19. 8C5
20. 5C2
21. 8C6
22. 12C4
23. 12C12
24. 14C6
25. 11C3
CARD HANDS In Exercises 26–30, find the number of possible 5-card hands
that contain the cards specified.
26. 5 face cards (either kings, queens, or jacks)
27. 4 aces and 1 other card
28. 1 ace and 4 other cards (none of which are aces)
29. 2 aces and 3 kings
30. 4 of one kind (kings, queens, and so on) and 1 of a different kind
31. PASCAL’S TRIANGLE Copy Pascal’s triangle on page 710 and add the rows for
n = 6 and n = 7 to it.
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 18–30,
47, 48
Example 2: Exs. 49–52
Example 3: Exs. 53, 54
Examples 4–7: Exs. 31–43
Example 8: Exs. 44–46
712
PASCAL’S TRIANGLE Use the rows of Pascal’s triangle from Exercise 31 to
write the binomial expansion.
32. (x + 4)6
33. (x º 3y)6
34. (x 2 + y)7
35. (2x º y3)7
BINOMIAL THEOREM Use the binomial theorem to write the binomial expansion.
36. (x º 2)3
37. (x + 4)5
38. (x + 3y)4
39. (2x º y)6
40. (x 3 + 3)5
41. (3x 2 º 3)4
42. (2x º y 2)7
43. (x 3 + y 2)3
Chapter 12 Probability and Statistics
Page 6 of 8
FOCUS ON
APPLICATIONS
44. Find the coefficient of x5 in the expansion of (x º 3)7.
45. Find the coefficient of x4 in the expansion of (x + 2)8.
46. Find the coefficient of x6 in the expansion of (x2 + 4)10.
RE
FE
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AL I
NOVELS Your English teacher has asked you to select 3 novels from a list
of 10 to read as an independent project. In how many ways can you choose
which books to read?
48.
GAMES Your friend is having a party and has 15 games to choose from.
There is enough time to play 4 games. In how many ways can you choose which
games to play?
49.
CARS You are buying a new car. There are 7 different colors to choose
from and 10 different types of optional equipment you can buy. You can
choose only 1 color for your car and can afford only 2 of the options. How
many combinations are there for your car?
CARS A 1998
survey showed that
of 7 basic car colors, white is
the most popular color for
full-size cars with 18.8% of
the vote. Green came in
second with 16.4% of the
vote.
INT
47.
50.
NE
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ART CONTEST There are 6 artists each presenting 5 works of art in an art
contest. The 4 works judged best will be displayed in a local gallery. In how many
ways can these 4 works all be chosen from the same artist’s collection?
APPLICATION LINK
www.mcdougallittell.com
51. LOGICAL REASONING Look back at Example 2. Suppose you can afford at
most 7 ingredients. How many different types of omelets can you order?
52.
AMUSEMENT PARKS An amusement park has 20 different rides. You want to
ride at least 15 of them. How many different combinations of rides can you go on?
53.
FISH From the list of different species
of fish shown, an aquarium enthusiast is
interested in knowing how compatible any
group of 3 or more different species are.
How many different combinations are there
to consider?
54.
CONCERTS A summer concert series
has 12 different performing artists. You
decide to attend at least 4 of the concerts.
How many different combinations of
concerts can you attend?
On Sale This Month
Freshwater Tropical Fish
Neon Tetras
Black Mollies
Tiger Barbs
Zebra fish
Red Platys
Bala Sharks
Angelfish
Lyretails
Blue Gouramis
Catfish
CRITICAL THINKING Decide whether the problem requires combinations or
permutations to find the answer. Then solve the problem.
55.
MARCHING BAND Eight members of a school marching band are
auditioning for 3 drum major positions. In how many ways can students be
chosen to be drum majors?
56.
YEARBOOK Your school yearbook has an editor-in-chief and an assistant
editor-in-chief. The staff of the yearbook has 15 students. In how many ways can
students be chosen for these 2 positions?
57.
RELAY RACES A relay race has 4 runners who run different parts of the
race. There are 16 students on your track team. In how many ways can your
coach select students to compete in the race?
58.
COLLEGE COURSES You must take 6 elective classes to meet your
graduation requirements for college. There are 12 classes that you are interested
in. In how many ways can you select your elective classes?
12.2 Combinations and the Binomial Theorem
713
Page 7 of 8
59. CRITICAL THINKING Write an equation that relates nPr and nCr .
STACKING CUBES In Exercises 60º63,
use the diagram shown which illustrates
the different ways to stack four cubes.
60. Sketch the different ways to stack
three cubes.
61. Sketch the different ways to stack
five cubes.
62. How does the number of ways to
stack three, four, and five cubes
relate to Pascal’s triangle?
63. In how many different ways can
you stack ten cubes?
1 column
2 columns
3 columns
4 columns
PASCAL’S TRIANGLE In Exercises 64–66, use the diagram of Pascal’s triangle
shown.
1
64. What is the sum of the numbers in row n
of Pascal’s triangle? Explain.
1
65. What is the sum of the numbers in rows 0
1
through 20 of Pascal’s triangle?
formed by the sums of the numbers along
the diagonal segments of Pascal’s triangle.
Test
Preparation
1
1
4
Row 1
1
2
3
1
66. LOGICAL REASONING Describe the pattern
Row 0
3
6
Row 2
Row 3
1
4
1
Row 4
1
67. MULTI-STEP PROBLEM A group of 20 high school students is volunteering to
help elderly members of their community. Each student will be assigned a job
based on requests received for help. There are 8 requests for raking leaves,
7 requests for running errands, and 5 requests for washing windows.
a. One way to count the number of possible job assignments is to find the
number of permutations of 8 L’s (for “leaves”), 7 E’s (for “errands”), and
5 W’s (for “windows”). Use this method to write the number of possible
job assignments first as an expression involving factorials and then as a
simple number.
b. Another way to count the number of possible job assignments is to first
choose the 8 students who will rake leaves, then choose the 7 students who
will run errands from the students who remain, and then choose the 5 students
who will wash windows from the students who still remain. Use this method
to write the number of possible job assignments first as an expression
involving factorials and then as a simple number.
c.
Writing How do the answers to parts (a) and (b) compare to each other?
Explain why this makes sense.
★ Challenge
714
COMBINATORIAL IDENTITIES Verify the identity.
68. nC0 = 1
69. nCn = 1
70. nC1 = nP1
71. nCr = nCn º r
72. nCr • rCm = nCm • n º mCr º m
73. n + 1Cr = nCr + nCr º 1
Chapter 12 Probability and Statistics
Page 8 of 8
MIXED REVIEW
FINDING AREA Find the area of the figure. (Skills Review, p. 914)
74. Circle with radius 18 centimeters
75. Rectangle with sides 9.5 inches and 11.3 inches
76. Triangle with base 13 feet and height 9 feet
77. Trapezoid with bases 10 meters and 13 meters, and height 27 meters
GRAPHING Graph the equation of the hyperbola. (Review 10.5)
y2
x2
78. ᎏᎏ º ᎏᎏ = 1
25
14 4
y2
x2
79. ᎏᎏ º ᎏᎏ = 1
10 0
36
49y2
80. x2 º ᎏᎏ = 1
16
y2
x2
81. ᎏᎏ º ᎏᎏ = 9
4
9
82. 64y2 º x2 = 64
83. 9x2 º 4y2 = 144
WRITING RULES Decide whether the sequence is arithmetic or geometric.
Then write a rule for the nth term. (Review 11.2, 11.3)
84. 3, 9, 27, 81, 243, . . .
85. 3, 10, 17, 24, 31, . . .
86. 2, 10, 50, 250, 1250, . . .
87. 1, º2, 4, º8, 16, . . .
88. 8, 6, 4, 2, 0, . . .
89. º10, º5, 0, 5, 10, . . .
90.
POTTERY A potter has 70 pounds of clay and 40 hours to make soup bowls
and dinner plates to sell at a craft fair. A soup bowl uses 3 pounds of clay and a
dinner plate uses 4 pounds of clay. It takes 3 hours to make a soup bowl and
1 hour to make a dinner plate. If the profit on a soup bowl is $25 and the profit
on a dinner plate is $20, how many bowls and plates should the potter make in
order to maximize profit? (Review 3.4)
QUIZ 1
Self-Test for Lessons 12.1 and 12.2
Find the number of distinguishable permutations of the letters in the word.
(Lesson 12.1)
1. POP
2. JUNE
3. IDAHO
4. KANSAS
5. WYOMING
6. THURSDAY
7. SEPTEMBER
8. CALIFORNIA
Write the binomial expansion. (Lesson 12.2)
9. (x + y)6
13. (x2 + 3y)4
10. (x + 2)4
11. (x º 2y)5
12. (3x º 4y)3
14. (4x2 º 2)6
15. (x3 º y3)3
16. (2x4 + 5y2)5
17. Find the coefficient of x3 in the expansion of (x + 3)5. (Lesson 12.2)
18. Find the coefficient of y4 in the expansion of (5 º y2)3. (Lesson 12.2)
19.
RESTAURANTS You are eating dinner at a restaurant. The restaurant offers
6 appetizers, 12 main dishes, 6 side orders, and 8 desserts. If you order one of
each of these, how many different dinners can you order? (Lesson 12.1)
20.
FLOWERS You are buying a flower arrangement. The florist has 12 types of
flowers and 6 types of vases. If you can afford exactly 3 types of flowers and need
only 1 vase, how many different arrangements can you buy? (Lesson 12.2)
12.2 Combinations and the Binomial Theorem
715
Page 1 of 7
12.3
What you should learn
An Introduction to Probability
GOAL 1
THEORETICAL AND EXPERIMENTAL PROBABILITY
GOAL 1 Find theoretical
and experimental
probabilities.
The probability of an event is a number between 0 and 1 that indicates the
likelihood the event will occur. An event that is certain to occur has a probability of 1.
An event that cannot occur has a probability of 0. An event that is equally likely to
Find geometric
probabilities, as applied in
Example 5.
occur or not occur has a probability of ᎏᎏ.
GOAL 2
Why you should learn it
RE
FE
䉲 To solve real-life
problems, such as finding the
probability that an archer hits
the center of a target
in Ex. 46.
AL LI
1
2
1
Pⴝ0
Event will
not occur.
Pⴝ2
Event is equally likely
to occur or not occur.
Pⴝ1
Event is certain
to occur.
There are two types of probability: theoretical and experimental. Theoretical
probability is defined below and experimental probability is defined on page 717.
T H E T H E O R E T I C A L P R O BA B I L I T Y O F A N E V E N T
When all outcomes are equally likely, the
theoretical probability that an event A will
occur is:
all possible outcomes
event A
number of outcomes in A
P(A) = ᎏᎏᎏᎏ
total number of outcomes
outcomes
4
P(A) = ᎏᎏ
9
The theoretical probability of an event is often
simply called the probability of the event.
EXAMPLE 1
Finding Probabilities of Events
You roll a six-sided die whose sides are numbered from 1 through 6. Find the
probability of (a) rolling a 4, (b) rolling an odd number, and (c) rolling a number
less than 7.
SOLUTION
a. Only one outcome corresponds to rolling a 4.
number of ways to roll a 4
number of ways to roll the die
1
6
P(rolling a 4) = ᎏᎏᎏᎏ = ᎏᎏ
b. Three outcomes correspond to rolling an odd number: rolling a 1, 3, or 5.
number of ways to roll an odd number
number of ways to roll the die
3
6
1
2
P(rolling an odd number) = ᎏᎏᎏᎏ = ᎏᎏ = ᎏᎏ
c. All six outcomes correspond to rolling a number less than 7.
number of ways to roll less than 7
number of ways to roll the die
6
6
P(rolling less than 7) = ᎏᎏᎏᎏ = ᎏᎏ = 1
716
Chapter 12 Probability and Statistics
Page 2 of 7
You can express a probability as a fraction, a decimal, or a percent. For instance, in
part (b) of Example 1 the probability of rolling an odd number can be written as
1
ᎏᎏ, 0.5, or 50%.
2
RE
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Music
EXAMPLE 2
Probabilities Involving Permutations or Combinations
You put a CD that has 8 songs in your CD player. You set the player to play the songs
at random. The player plays all 8 songs without repeating any song.
a. What is the probability that the songs are played in the same order they are listed
on the CD?
b. You have 4 favorite songs on the CD. What is the probability that 2 of your
favorite songs are played first, in any order?
SOLUTION
a. There are 8! different permutations of the 8 songs. Of these, only 1 is the order
STUDENT HELP
in which the songs are listed on the CD. So, the probability is:
Skills Review
For help with
converting decimals,
fractions, and
percents, see p. 906.
1
8!
1
40,320
P(playing 8 in order) = ᎏᎏ = ᎏᎏ ≈ 0.0000248
b. There are 8C2 different combinations of 2 songs. Of these, 4C2 contain 2 of your
favorite songs. So, the probability is:
4C 2
6
28
3
14
P(playing 2 favorites first) = ᎏᎏ = ᎏᎏ = ᎏᎏ ≈ 0.214
8C 2
..........
Sometimes it is not possible or convenient to find the theoretical probability of an
event. In such cases you may be able to calculate an experimental probability by
performing an experiment, conducting a survey, or looking at the history of the event.
Internet
EXAMPLE 3
Finding Experimental Probabilities
In 1998 a survey asked Internet users for their
ages. The results are shown in the bar graph.
Find the experimental probability that a
randomly selected Internet user is (a) at most
20 years old, and (b) at least 41 years old.
䉴 Source: GVU’s WWW User Surveys™
SOLUTION The number of people surveyed
was 1636 + 6617 + 3693 + 491 + 6 = 12,443.
a. Of the people surveyed, 1636 are at most
20 years old. So, the probability is:
Internet Users
Age (years)
RE
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Under 21
21–40
41–60
61–80
Over 80
1636
6617
3693
491
6
0
2000
4000
6000
Number of users
1636
12,443
P(user is at most 20) = ᎏᎏ ≈ 0.131
b. Of the people surveyed, 3693 + 491 + 6 = 4190 are at least 41 years old.
So, the probability is:
4 1 90
12,443
P(user is at least 41) = ᎏᎏ ≈ 0.337
12.3 An Introduction to Probability
717
Page 3 of 7
GOAL 2
GEOMETRIC PROBABILITY
Some probabilities are found by calculating a ratio of two lengths, areas, or volumes.
Such probabilities are called geometric probabilities.
EXAMPLE 4
Using Area to Find Probability
You throw a dart at the board shown. Your dart is equally
likely to hit any point inside the square board. Are you
more likely to get 10 points or 0 points?
3 in.
SOLUTION
10
The two probabilities are as follows.
STUDENT HELP
2
area of smallest circle
P(10 points) = ᎏᎏᎏ
area of entire board
Skills Review
For help with area,
see p. 914.
5
0
π • 32
9π
π
324
36
18
area outside largest circle
P(0 points) = ᎏᎏᎏ
area of entire board
= ᎏᎏ
= ᎏᎏ = ᎏᎏ ≈ 0.0873
2
182 º (π • 92)
18
324 º 81π
324
4ºπ
4
= ᎏᎏ
= ᎏᎏ = ᎏᎏ ≈ 0.215
2
䉴
RE
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Entertainment
You are more likely to get 0 points.
EXAMPLE 5
Using Length to Find Probability
You have recorded a 2 hour movie at the beginning of a videocassette that has
6 hours of recording time. Starting at a random location on the videocassette, your
brother records a 30 minute television show. What is the probability that your
brother’s television show accidentally records over part of your movie?
SOLUTION
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.
You can think of the videocassette as a number line from 0 to 6. The movie can be
represented as a line segment 2 units long and the television show as a line segment
0.5 unit long. Because you know the movie starts at the beginning of the
videocassette, the number line is as shown.
movie
0
1
2
3
4
5
6
If the 30 minute, or half hour, television show is to fit on the tape, it must start somewhere between 0 and 5.5. If it records over part of the movie, it must start somewhere
between 0 and 2. So, the probability of recording over part of the movie is:
length where show will record over movie
length where show will fit on tape
P(recording over movie) = ᎏᎏᎏᎏᎏ
2º0
5.5 º 0
2
5.5
4
11
= ᎏᎏ = ᎏᎏ = ᎏᎏ ≈ 0.364
718
Chapter 12 Probability and Statistics
Page 4 of 7
GUIDED PRACTICE
Vocabulary Check
Concept Check
✓
1. Complete this statement: A probability that involves length, area, or volume is
✓
2. P(A) = 0.2 and P(B) = 0.6. Which event is more likely to occur? Explain.
? probability.
called a(n) 㛭㛭㛭
3. Explain the difference between theoretical probability and experimental
Skill Check
✓
probability. Give an example of each.
A jar contains 2 red marbles, 3 blue marbles, and 1 green marble. Find the
probability of randomly drawing the given type of marble.
4. a red marble
5. a green marble
6. a blue or a green marble
7. a red or a blue marble
Find the probability that a dart thrown at the given target will hit the shaded
region. Assume the dart is equally likely to hit any point inside the target. The
targets and regions within are either squares, circles, or triangles.
9.
10.
3
3
3兹2
6
POPULATION The bar graph
shown gives the resident population
(in thousands) of the United States in
1997. For a randomly selected person
in the United States, find the
probability of the given event.
INT
11.
NE
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6
United States Population
Population
(thousands)
8.
120,000
83,608
55,446
34,076
0
24 and 25 to 44 45 to 64 65 and
under
older
Age (years)
DATA UPDATE of Statistical Abstract of the
United States data at www.mcdougallittell.com
a. The person is 24 years old or under.
94,507
60,000
b. The person is at least 45 years old.
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 956.
CHOOSING NUMBERS You have an equally likely chance of choosing any
integer from 1 through 20. Find the probability of the given event.
12. An odd number is chosen.
13. A number less than 7 is chosen.
14. A perfect square is chosen.
15. A prime number is chosen.
16. A multiple of 3 is chosen.
17. A factor of 240 is chosen.
CHOOSING CARDS A card is drawn randomly from a standard 52-card deck.
Find the probability of drawing the given card.
STUDENT HELP
18. the ace of hearts
19. any ace
Look Back
For help with a standard
52-card deck, see p. 708.
20. a diamond
21. a red card
22. a card other than 10
23. a face card (a king, queen, or jack)
12.3 An Introduction to Probability
719
Page 5 of 7
ROLLING A DIE The results of rolling a six-sided die 120 times are shown.
STUDENT HELP
HOMEWORK HELP
Use the table to find the experimental probability of each event. Also find
the theoretical probability. How do the probabilities compare?
Example 1: Exs. 12–23
Example 2: Exs. 35–40
Example 3: Exs. 24–29,
41–43
Example 4: Exs. 30–34,
46, 47
Example 5: Exs. 44, 45
Results from Rolling a Die 120 Times
Roll
Number of
occurrences
1
2
3
4
5
6
15
18
20
17
24
26
24. rolling a 6
25. rolling a 3 or 4
26. rolling an odd number
27. rolling an even number
28. rolling a number greater than 2
29. rolling anything but a 1
GEOMETRY CONNECTION Find the probability that
a dart thrown at the square target shown will hit
the given region. Assume the dart is equally likely
to hit any point inside the target.
2 in.
30. the red center
31. the white border
32. the red center or the white border
33. the four rings or the red center
24 in.
34. the yellow or green ring
SPEECHES In Exercises 35 and 36, use the following information.
Your English teacher is drawing names to see who will give the first speech. There
are 26 students in the class and 4 speeches will be given each day.
35. What is the probability that you will give your speech first?
36. What is the probability that you will give your speech on the first day?
WORD GAMES In Exercises 37 and 38, use the following information.
You and a friend are playing a word game
Distribution of Letters
that involves lettered tiles. The distribution
A: 9
H: 2
O: 8
V: 2
of letters is shown at the right. At the start
B:
2
I:
9
P:
2
W:
2
of the game you choose 7 letters.
C: 2
J: 1
Q: 1
X: 1
37. What is the probability that you will
D: 4
K: 1
R: 6
Y: 2
choose three vowels and four
consonants? (Count “Y” as a vowel.)
E: 12
L: 4
S: 4
Z: 1
F:
2
M:
2
T:
6
Blank:
2
38. What is the probability that you will
G: 3
N: 6
U: 4
choose the letters A, B, C, D, E, F, and
G in order?
LOTTERIES In Exercises 39 and 40, find the probability of winning the
lottery according to the given rules. Assume numbers are selected at random.
39. You must correctly select 6 out of 51 numbers. The order of the numbers is not
important.
40. You must correctly select 3 numbers, each from 0 to 9. The order of the numbers
is important.
720
Chapter 12 Probability and Statistics
Page 6 of 7
FOCUS ON
APPLICATIONS
41.
MEDIA CONCERN In a 1998 survey, parents were asked what media influence
on their children most concerned them. The results are shown in the bar graph. Find
the experimental probability that a randomly selected parent is most concerned about
the given topic. 䉴 Source: Annenberg Public Policy Center
INT
NE
ER T
APPLICATION LINK
400
180
156
43
41
es
s
tk
no
Do
Vi
d
M
ag
az
in
w
Ne
ie
s
es
ov
ga
m
rn
eo
M
et
ic
te
In
ev
isi
Te
l
a. Television
50
23
w
46
0
no
200
on
The largest one-day
stock-market loss in Wall
Street history occurred on
October 19, 1987. That day
the Dow Jones fell 508.32
points, or 22.6%. The net
change for the year,
however, was a
growth of 20.9%.
672
600
us
STOCK MARKET
800
M
RE
FE
L
AL I
Number of parents
Media Influence
b. Video games
The table shows how many years the stock market
gained or lost a given percent over a recent 17 year period based on the Dow
Jones Industrial Average. Find the experimental probability of the given event.
42. HISTORY
www.mcdougallittell.com
Range
Years
CONNECTION
Lost more
than 0%
Gained
0% to 9%
4
Gained
Gained
Gained more
10% to 19% 20% to 29% than 29%
2
4
a. The stock market has a loss.
3
b. The stock market gains at least 10%.
CONNECTION The table
shows how people in the United States
got to work in 1990. For a randomly
selected person in the United States,
find the probability that the person
chose the given type of transportation.
43. STATISTICS
4
Means of transportation
Automobile
Public transportation
Number
99,592,932
6,069,589
Motorcycle
237,404
䉴 Source: The World Almanac
Bicycle
466,856
a. Used public transportation
Other
5,297,468
b. Drove to work (either in an
None (work at home)
3,406,025
automobile or on a motorcycle)
44.
VIDEOCASSETTES Look back at Example 5. Suppose you recorded your
movie starting 1 hour into the videocassette. What is the probability that your
brother’s television show accidentally records over part of your movie?
45.
CABLE INSTALLATION You set up an appointment to have cable television
installed between 12:00 P.M. and 4:00 P.M. The installer will wait 15 minutes if
no one is home. Your cousin asks for a favor that would take you away from your
home from 1:30 P.M. to 2:00 P.M. If you do the favor, what is the probability that
you will miss the cable installer?
46.
KYUDO Kyudo is a form of Japanese archery. The
most common target is shown. Find the probability that
an arrow shot at the target will hit the center circle.
Assume the arrow is equally likely to hit any point
inside the target.
12.3 An Introduction to Probability
3.6 cm
1.5 cm
3.3 cm
3.0 cm
3.0 cm
3.6 cm
721
Page 7 of 7
47.
Test
Preparation
CONTACT LENSES You have just stepped
into the tub to take a shower when one of your
contact lenses falls out. (You have not yet turned
on the shower.) Assuming that the lens is equally
likely to land anywhere on or inside of the tub,
what is the probability that it landed in the drain?
drain
2 in.
26 in.
50 in.
48. MULTIPLE CHOICE On a multiple choice question, you know that the answer is
not B or D, but you are not sure about answers A, C, or E. What is the probability
that you will get the right answer if you guess?
A
¡
1
ᎏᎏ
5
B
¡
C
¡
4
ᎏᎏ
5
D
¡
1
ᎏᎏ
3
E
¡
2
ᎏᎏ
3
3
ᎏᎏ
5
49. MULTIPLE CHOICE A dart thrown at the circular target
shown is equally likely to hit any point inside the target.
What is the probability that it hits the region outside the
triangle?
★ Challenge
A
¡
D
¡
0.5
0.47
B
¡
E
¡
C
¡
0.75
5
0.32
0.68
50. PROBABILITY Find the probability that the graph of y = x2 º 4x + c intersects
the x-axis if c is a randomly chosen integer from 1 to 6.
MIXED REVIEW
DETERMINANTS Evaluate the determinant of the matrix. (Review 4.3)
51.
54.
冋 册
2 7
5 9
冋 册
1 2 3
2 3 1
3 1 2
52.
冋
冋
6
0
1 º3
册
0 1
53.
5
7
2 1 º4
55. º3 4
册
56.
冋 册
3 8
º2 1
冋
º1 º2
2
4
3 º4
2
4
6
册
MULTIPLYING Multiply the rational expressions. Simplify the result.
(Review 9.4)
6xy2 10y4
• ᎏᎏ
57. ᎏ
5x3y 9xy
x2 + 3x + 2
x2 º 3x
58. ᎏᎏ
• ᎏᎏ
2
2
x ºxº6 x ºxº2
x2 º 4x º 21
25x2 º 16
59. ᎏᎏ • ᎏᎏ
5x º 4
5x3 º 31x2 º 28x
4x2 º 12x
60. ᎏᎏ
• (x 2 + 3x + 9)
27 º x3
WRITING TERMS Write the first five terms of the sequence. (Review 11.5)
61. a0 = 3
an = an º 1 + 7
64. a0 = 1
a1 = 1
an = an º 1 + an º 2
67.
722
62. a0 = º1
an = 3 • a n º 1
65. a0 = º2
a1 = 0
an = an º 1 º an º 2
63. a0 = 2
an = (an º 1)3
66. a0 = 1
a1 = º2
an = an º 1 • an º 2
TRADE SHOWS You are attending a trade show that has booths from
20 different vendors. You hope to visit at least 5 of the booths. How many
combinations of booths can you visit? (Review 12.2 for 12.4)
Chapter 12 Probability and Statistics
Page 1 of 6
12.4
What you should learn
GOAL 1 Find probabilities
of unions and intersections of
two events.
GOAL 2 Use complements
to find the probability of an
event, as applied in
Example 5.
Probability of Compound Events
GOAL 1
PROBABILITIES OF UNIONS AND INTERSECTIONS
When you consider all the outcomes for either of two events A and B, you form the
union of A and B. When you consider only the outcomes shared by both A and B, you
form the intersection of A and B. The union or intersection of two events is called a
compound event.
A
A
B
A
B
B
Why you should learn it
RE
FE
䉲 To solve real-life
problems, such as finding the
probability that friends will be
in the same college dormitory
in Ex. 49.
AL LI
Union of A and B
Intersection of A and B
is empty.
Intersection of A and B
To find P(A or B) you must consider what outcomes, if any, are in the intersection
of A and B. If there are none, then A and B are mutually exclusive events and
P(A or B) = P(A) + P(B). If A and B are not mutually exclusive, then the outcomes
in the intersection of A and B are counted twice when P(A) and P(B) are added. So,
P(A and B) must be subtracted once from the sum.
P R O BA B I L I T Y O F C O M P O U N D E V E N T S
If A and B are two events, then the probability of A or B is:
P(A or B) = P(A) + P(B) º P(A and B)
If A and B are mutually exclusive, then the probability of A or B is:
P(A or B) = P(A) + P(B)
EXAMPLE 1
Probability of Mutually Exclusive Events
A card is randomly selected from a standard
deck of 52 cards. What is the probability that
it is an ace or a face card?
A
B
K♠
A♠
K♣
SOLUTION
A♣
Let event A be selecting an ace, and let
event B be selecting a face card. Event A has
4 outcomes and event B has 12 outcomes.
Because A and B are mutually exclusive, the
probability is:
4
52
12
52
16
52
4
13
P(A or B) = P(A) + P(B) = ᎏᎏ + ᎏᎏ = ᎏᎏ = ᎏᎏ ≈ 0.308
724
Chapter 12 Probability and Statistics
Q♠
Q♣
J♠
J♣
Page 2 of 6
EXAMPLE 2
Probability of a Compound Event
A card is randomly selected from a standard deck
of 52 cards. What is the probability that the card
is a heart or a face card?
A
B
K♠ Q♠ J♠
SOLUTION
Let event A be selecting a heart, and let event B
be selecting a face card. Event A has 13 outcomes
and event B has 12 outcomes. Of these, three
outcomes are common to A and B. So, the
probability of selecting a heart or a face card is:
P(A or B) = P(A) + P(B) º P(A and B)
13
52
12
52
3
52
= ᎏᎏ + ᎏᎏ º ᎏᎏ
RE
FE
Business
Write general formula.
Substitute known probabilities.
22
52
Combine terms.
= ᎏᎏ
11
26
Simplify.
≈ 0.423
Use a calculator.
= ᎏᎏ
L
AL I
K♣ Q♣ J♣
EXAMPLE 3
Using Intersection to Find Probability
Last year a company paid overtime wages or hired
temporary help during 9 months. Overtime wages
were paid during 7 months and temporary help
was hired during 4 months. At the end of the year,
an auditor examines the accounting records and
randomly selects one month to check the
company’s payroll. What is the probability that the
auditor will select a month in which the company
paid overtime wages and hired temporary help?
A
Overtime
wages
B
Temporary
help
SOLUTION
Let event A represent paying overtime wages during a month, and let event B
represent hiring temporary help during a month. From the given information you
know that:
7
12
4
12
9
12
P(A) = ᎏᎏ, P(B) = ᎏᎏ, and P(A or B) = ᎏᎏ
The probability that the auditor will select a month in which the company paid
overtime wages and hired temporary help is P(A and B).
P(A or B) = P(A) + P(B) º P(A and B)
9
7
4
ᎏᎏ = ᎏᎏ + ᎏᎏ º P(A and B)
12
12
12
7
12
4
12
2
12
1
6
9
12
P(A and B) = ᎏᎏ + ᎏᎏ º ᎏᎏ
P(A and B) = ᎏᎏ = ᎏᎏ ≈ 0.167
Write general formula.
Substitute known probabilities.
Solve for P (A and B).
Simplify.
12.4 Probability of Compound Events
725
Page 3 of 6
GOAL 2
USING COMPLEMENTS TO FIND PROBABILITY
The event A§, called the complement of event A, consists of all outcomes that are
not in A. The notation A§ is read as “A prime.”
P R O BA B I L I T Y O F T H E C O M P L E M E N T O F A N E V E N T
The probability of the complement of A is P(A§) = 1 º P(A).
EXAMPLE 4
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.
Probabilities of Complements
When two six-sided dice are tossed, there are
36 possible outcomes as shown. Find the
probability of the given event.
a. The sum is not 8.
b. The sum is greater than or equal to 4.
SOLUTION
a. P(sum is not 8) = 1 º P(sum is 8)
b. P(sum ≥ 4) = 1 º P(sum < 4)
5
36
3
36
= 1 º ᎏᎏ
= 1 º ᎏᎏ
= ᎏᎏ
31
36
= ᎏᎏ
33
36
≈ 0.861
= ᎏᎏ
FOCUS ON
APPLICATIONS
11
12
≈ 0.917
EXAMPLE 5
Using a Complement in Real Life
HOME ELECTRONICS Four houses in a neighborhood have the same model of
garage door opener. Each opener has 4096 possible transmitter codes. What is the
probability that at least two of the four houses have the same code?
SOLUTION
RE
FE
L
AL I
HOME
ELECTRONICS
One type of garage door
opener has 12 switches that
can be set in one of two
positions (off or on) to create
a code. So for this type of
garage door opener, there
are 212, or 4096, possible
transmitter codes.
726
The total number of ways to assign codes to the four openers is 4096 4. The number
of ways to assign different codes to the four openers is 4096 • 4095 • 4094 • 4093.
So, the probability that at least two of the four openers have the same code is:
P(at least 2 are the same) = 1 º P(none are the same)
Chapter 12 Probability and Statistics
4096 • 4095 • 4094 • 4093
4096
= 1 º ᎏᎏᎏ
4
≈ 1 º 0.99854
= 0.00146
Page 4 of 6
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
Skill Check
✓
1. Describe what it means for two events to be mutually exclusive.
2. Write a formula for computing P(A or B) that applies to any events A and B.
How can you simplify this formula when A and B are mutually exclusive?
3. Are the events A and A§ mutually exclusive? Explain.
Events A and B are mutually exclusive. Find P (A or B).
4. P(A) = 0.2, P(B) = 0.3
5. P(A) = 0.5, P(B) = 0.5
3
1
6. P(A) = ᎏᎏ, P(B) = ᎏᎏ
8
8
1
1
7. P(A) = ᎏᎏ, P(B) = ᎏᎏ
3
4
Find P (A or B).
2
3
9. P(A) = ᎏᎏ, P(B) = ᎏᎏ,
5
5
1
P(A and B) = ᎏᎏ
5
8. P(A) = 0.5, P(B) = 0.4,
P(A and B) = 0.3
Find P (A and B).
5
7
11. P(A) = ᎏᎏ, P(B) = ᎏᎏ,
16
16
9
P(A or B) = ᎏᎏ
16
10. P(A) = 0.7, P(B) = 0.2,
P(A or B) = 0.8
Find P (A§).
12. P(A) = 0.5
13. P(A) = 0.75
1
14. P(A) = ᎏᎏ
3
4
15. P(A) = ᎏᎏ
7
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 957.
FINDING PROBABILITIES Find the indicated probability. State whether A and B
are mutually exclusive.
16. P(A) = 0.4
P(B) = 0.35
P(A or B) = 0.5
?
P(A and B) = 㛭㛭㛭
13
19. P(A) = ᎏᎏ
17
?
P(B) = 㛭㛭㛭
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 16–24,
29–34, 42, 43
Example 2: Exs. 16–24,
29–34, 44, 45
Example 3: Exs. 16–24,
29–34, 46, 47
Example 4: Exs. 25–28,
35–40
Example 5: Exs. 48, 49
14
17
6
P(A and B) = ᎏᎏ
17
P(A or B) = ᎏᎏ
22. P(A) = 5%
P(B) = 29%
?
P(A or B) = 㛭㛭㛭
P(A and B) = 0%
17. P(A) = 0.6
18. P(A) = 0.25
?
P(B) = 㛭㛭㛭
P(A or B) = 0.70
P(A and B) = 0
P(B) = 0.2
?
P(A or B) = 㛭㛭㛭
P(A and B) = 0.1
1
20. P(A) = ᎏᎏ
3
1
P(B) = ᎏᎏ
4
3
21. P(A) = ᎏᎏ
4
1
P(B) = ᎏᎏ
3
7
12
P(A or B) = ᎏᎏ
?
P(A or B) = 㛭㛭㛭
?
P(A and B) = 㛭㛭㛭
P(A and B) = ᎏᎏ
23. P(A) = 30%
1
4
24. P(A) = 16%
?
P(B) = 㛭㛭㛭
P(A or B) = 50%
P(A and B) = 10%
FINDING PROBABILITIES OF COMPLEMENTS Find P (A§).
3
26. P(A) = 0
27. P(A) = ᎏᎏ
25. P(A) = 0.34
4
P(B) = 24%
P(A or B) = 32%
?
P(A and B) = 㛭㛭㛭
28. P(A) = 1
12.4 Probability of Compound Events
727
Page 5 of 6
STUDENT HELP
Look Back
For help with a standard
52-card deck in
Exs. 29–34, see p. 708.
Look Back
For help with simulations
on a graphing calculator
in Ex. 41, see p. 723.
CHOOSING CARDS A card is randomly drawn from a standard 52-card deck.
Find the probability of the given event. (A face card is a king, queen, or jack.)
29. a queen and a heart
30. a queen or a heart
31. a heart or a diamond
32. a five or a six
33. a five and a six
34. a three or a face card
USING COMPLEMENTS Two six-sided dice are rolled. Find the probability of
the given event. (Refer to Example 4 for a diagram of all possible outcomes.)
35. The sum is not 3.
36. The sum is greater than or equal to 5.
37. The sum is neither 3 nor 7.
38. The sum is less than or equal to 10.
39. The sum is greater than 2.
40. The sum is less than 8 or greater than 11.
41.
EXPERIMENTAL PROBABILITIES Simulate rolling two dice 120 times.
Use separate lists for the results of each die, and a third list for the sum.
Record the frequency of each sum in a table. Find the experimental probabilities
of the events in Exercises 35–40. How do your experimental results compare
with the theoretical results?
FOCUS ON
CAREERS
42.
COMPANY MOVE An employee of a large national company is promoted
to management and will be moved within six months. The employee is told that
there is a 33% probability of being moved to Denver, Colorado, and a 50%
probability of being moved to Dallas, Texas. What is the probability that the
employee will be moved to Dallas or Denver?
43.
CLASS ELECTIONS You and your best friend are among several candidates
running for class president. You estimate that there is a 40% chance you will win
the election and a 35% chance your best friend will win. What is the probability
that either you or your best friend wins the election?
44.
PARAKEETS A pet store contains 35 light green parakeets (14 females and
21 males) and 44 sky blue parakeets (28 females and 16 males). You randomly
choose one of the parakeets. What is the probability that it is a female or a sky
blue parakeet?
45.
HONORS BANQUET Of 162 students honored at an academic awards
banquet, 48 won awards for mathematics and 78 won awards for English.
Fourteen of these students won awards for both mathematics and English.
One of the 162 students is chosen at random to be interviewed for a newspaper
article. What is the probability that the student interviewed won an award for
English or mathematics?
CONNECTION A tree in a forest is not growing properly. A botanist
determines that there is an 85% probability the tree has a disease or is being
damaged by insects, a 45% probability it has a disease, and a 50% probability
it is being damaged by insects. What is the probability that the tree both has a
disease and is being damaged by insects?
46. SCIENCE
RE
FE
L
AL I
BOTANIST
INT
A botanist studies
plants and their environment.
Some botanists specialize
in the causes and cures of
plant illnesses, as discussed
in Ex. 46.
NE
ER T
CAREER LINK
www.mcdougallittell.com
728
47.
RAIN A weather forecaster says that the probability it will rain on Saturday
or Sunday is 50%, the probability it will rain on Saturday is 20%, and the
probability it will rain on Sunday is 40%. What is the probability that it will
rain on both Saturday and Sunday?
48.
POTLUCK DINNER The organizer of a potluck dinner sends 5 people a list
of 8 different recipes and asks each person to bring one of the items on the list. If
all 5 people randomly choose a recipe from the list, what is the probability that at
least 2 will bring the same thing?
Chapter 12 Probability and Statistics
Page 6 of 6
49.
CAMPUS HOUSING Four high school friends will all be attending the
same university next year. There are 14 dormitories on campus. Find the
probability that at least 2 of the friends will be in the same dormitory.
50. CRITICAL THINKING What is the complement of A§? Explain.
Test
Preparation
51.
MULTI-STEP PROBLEM Follow the steps below to explore a famous
probability problem called the birthday problem.
a. Suppose that 5 people are chosen at random. Find the probability that at least
two share the same birthday (Assume that there are 365 possible birthdays).
INT
STUDENT HELP
NE
ER T
KEYSTROKE HELP
Visit our Web site
www.mcdougallittell.com
to see keystrokes for
several models of
calculators.
b. Suppose that 10 people are chosen at random. Find the probability that at least
two of the people share the same birthday.
c. Generalize the results from parts (a) and (b) by writing a formula that gives
the probability P(n) that in a group of n people at least two people share the
same birthday. (Hint: Use n Pr notation.)
d. LOGICAL REASONING Enter the formula for P(n) from part (c) into a
graphing calculator put in sequence mode. Use the Table feature to make a
table of values for P(n). How large must a group be if the probability that at
least two of the people share the same birthday exceeds 50%?
★ Challenge
ODDS The odds in favor of an event occurring are the ratio of the probability
that the event will occur to the probability that the event will not occur. The
reciprocal of this ratio represents the odds against the event occurring.
52. Five marbles in a jar are green. The odds against choosing a green marble are
“4 to 1.” How many marbles are in the jar?
53. If a jar contains 4 red marbles and 7 blue marbles, what are the odds in favor of
choosing a red marble? What are the odds against choosing a red marble?
54. Write a formula that converts the odds in favor of an event to the probability of
the event.
EXTRA CHALLENGE
www.mcdougallittell.com
55. Write a formula that converts the probability of an event to the odds in favor of
the event.
MIXED REVIEW
SOLVING EXPONENTIAL EQUATIONS Solve the equation. (Review 8.6 for 12.5)
56. 72x = 4916
57. 9 x = 3x + 1
58. 24x + 8 = 3219
59. 5x = 21
60. 103x º 1 º 13 = 8
61. 72 = 91eº0.023x + 50
WRITING EQUATIONS Write the standard form of the equation of the circle that
passes through the given point and whose center is the origin. (Review 10.3)
62. (0, 7)
63. (3, 4)
64. (º1, 6)
65. (8, º2)
66. (4, 4)
67. (3, 10)
68. (º4, º2)
69. (16, 0)
LICENSE PLATES For the given configuration, determine how many different
license plates are possible if (a) digits and letters can be repeated, and (b) digits
and letters cannot be repeated. (Review 12.1 for 12.5)
70. 3 letters followed by 4 digits
71. 4 letters followed by 3 digits
12.4 Probability of Compound Events
729
Page 1 of 8
12.5
What you should learn
GOAL 1 Find the probability
of independent events.
GOAL 2 Find the probability
of dependent events, as
applied in Ex. 33.
Why you should learn it
PROBABILITIES OF INDEPENDENT EVENTS
Two events are independent if the occurrence of one has no effect on the
occurrence of the other. For instance, if a coin is tossed twice, the outcome of the
first toss (heads or tails) has no effect on the outcome of the second toss.
P R O BA B I L I T Y O F I N D E P E N D E N T E V E N T S
If A and B are independent events, then the probability that both A and B occur
is P(A and B) = P(A) • P(B).
EXAMPLE 1
Probability of Two Independent Events
$
7 2$ $
0
3
0 05 0
0
$
4
0
0
$
3
9 5
0 0
0
$
T
36
$
4 $
0 2 9$
5
00 5
0
24 24
..........
A
A
$
7
5
0
on the first spin, and let event B be going bankrupt
on the second spin. The two events are independent.
So, the probability is:
8 2
1
P(A and B) = P(A) • P(B) = ᎏᎏ • ᎏᎏ = ᎏᎏ ≈ 0.028
A
A
B
$ A $
$ 1 N 5 $
5 5 K 0 3
5 0 UR 0 5
0 P0 0
T
$
$ 3
$ 1 0
0
5 00
0
SOLUTION Let event A be getting more than $500
B
A
6
You are playing a game that involves spinning the
money wheel shown. During your turn you get to
spin the wheel twice. What is the probability that
you get more than $500 on your first spin and then
go bankrupt on your second spin?
B
$ A $
$ 2 N 3 $
1 $
K 0
3
5 00 UR 0 0 0 5
0 P0 0 0
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FE
䉲 To solve real-life
problems, such as finding
the probability that the
Florida Marlins win three
games in a row in
Example 2.
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Probability of Independent and
Dependent Events
A
A
B
The formula given above for the probability of two independent events can be
extended to the probability of three or more independent events.
EXAMPLE 2
Probability of Three Independent Events
BASEBALL During the 1997 baseball season, the Florida Marlins won 5 out of 7
home games and 3 out of 7 away games against the San Francisco Giants. During the
1997 National League Division Series with the Giants, the Marlins played the first
two games at home and the third game away. The Marlins won all three games.
Estimate the probability of this happening. 䉴 Source: The Florida Marlins
SOLUTION Let events A, B, and C be winning the first, second, and third games.
The three events are independent and have experimental probabilities based on the
regular season games. So, the probability of winning the first three games is:
5 5 3
7 7 7
75
343
P(A and B and C) = P(A) • P(B) • P(C) = ᎏᎏ • ᎏᎏ • ᎏᎏ = ᎏᎏ ≈ 0.219
730
Chapter 12 Probability and Statistics
A
Page 2 of 8
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Trading Cards
EXAMPLE 3
Using a Complement to Find a Probability
You collect hockey trading cards. For one team there are
25 different cards in the set, and you have all of them
except for the starting goalie card. To try and get this card,
you buy 8 packs of 5 cards each. All cards in a pack are
different and each of the cards is equally likely to be in a
given pack. Find the probability that you will get at least
one starting goalie card.
SOLUTION
In one pack the probability of not getting the starting goalie card is:
24C5
P(no starting goalie) = ᎏ
25C5
Buying packs of cards are independent events, so the probability of getting at least
one starting goalie card in the 8 packs is:
P(at least one starting goalie) = 1 º P(no starting goalie in any pack)
冉 冊
C 8
25 C 5
24 5
ᎏ
=1º ᎏ
≈ 0.832
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Manufacturing
EXAMPLE 4
Solving a Probability Equation
A computer chip manufacturer has found that only 1 out of 1000 of its chips is
defective. You are ordering a shipment of chips for the computer store where you
work. How many chips can you order before the probability that at least one chip is
defective reaches 50%?
SOLUTION
STUDENT HELP
Look Back
For help with solving
exponential equations,
see p. 501.
Let n be the number of chips you order. From the given information you know that
999
P(chip is not defective) = ᎏᎏ = 0.999. Use this probability and the fact that each
1000
chip ordered represents an independent event to find the value of n.
P(at least one chip is defective) = 0.5
1 º P(no chips are defective) = 0.5
1 º (0.999)n = 0.5
º(0.999)n = º0.5
n
(0.999) = 0.5
log 0.5
log 0.999
䉴
Write given assumption.
Use complement.
Substitute known probability.
Subtract 1 from each side.
Divide each side by –1.
n = ᎏᎏ
Solve for n.
n ≈ 693
Use a calculator.
If you order 693 chips, you have a 50% chance of getting a defective chip.
Therefore, you can order 692 chips before the probability that at least one chip
is defective reaches 50%.
12.5 Probability of Independent and Dependent Events
731
Page 3 of 8
GOAL 2
PROBABILITIES OF DEPENDENT EVENTS
Two events A and B are dependent events if the occurrence of one affects the
occurrence of the other. The probability that B will occur given that A has occurred
is called the conditional probability of B given A and is written P(B|A).
P R O BA B I L I T Y O F D E P E N D E N T E V E N T S
If A and B are dependent events, then the probability that both A and B occur is
P(A and B) = P(A) • P(B|A).
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Endangered Species
EXAMPLE 5
Finding Conditional Probabilities
The table shows the number of endangered and threatened animal species in the
United States as of November 30, 1998. Find (a) the probability that a listed animal
is a reptile and (b) the probability that an endangered animal is a reptile.
䉴 Source: United States Fish and Wildlife Service
Mammals
Birds
Reptiles
Amphibians
Other
Endangered
59
75
14
9
198
Threatened
8
15
21
7
69
SOLUTION
number of reptiles
35
a. P(reptile) = ᎏᎏᎏ = ᎏᎏ ≈ 0.0737
475
total number of animals
number of endangered reptiles
b. P(reptile|endangered) = ᎏᎏᎏᎏ
total number of endangered animals
14
355
= ᎏᎏ ≈ 0.0394
EXAMPLE 6
STUDENT HELP
Look Back
For help with a standard
52-card deck, see p. 708.
Comparing Dependent and Independent Events
You randomly select two cards from a standard 52-card deck. What is the probability
that the first card is not a face card (a king, queen, or jack) and the second card is a
face card if (a) you replace the first card before selecting the second, and (b) you do
not replace the first card?
SOLUTION
a. If you replace the first card before selecting the second card, then A and B are
independent events. So, the probability is:
40 12
52 52
30
169
P(A and B) = P(A) • P(B) = ᎏᎏ • ᎏᎏ = ᎏᎏ ≈ 0.178
b. If you do not replace the first card before selecting the second card, then A and B
are dependent events. So, the probability is:
40 12
52 51
40
221
P(A and B) = P(A) • P(B|A) = ᎏᎏ • ᎏᎏ = ᎏᎏ ≈ 0.181
732
Chapter 12 Probability and Statistics
Page 4 of 8
The formula for finding probabilities of dependent events can be extended to three or
more events, as shown in Example 7.
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Dining Out
Probability of Three Dependent Events
EXAMPLE 7
You and two friends go to a restaurant and order a sandwich. The menu has 10
types of sandwiches and each of you is equally likely to order any type. What is
the probability that each of you orders a different type?
SOLUTION
STUDENT HELP
Study Tip
You can also use the
fundamental counting
principle to find the
probability in Example 7.
Let event A be that you order a sandwich, event B be that one friend orders a different
type, and event C be that your other friend orders a third type. These events are
dependent. So, the probability that each of you orders a different type is:
P(A and B and C) = P(A) • P(B|A) • P(C|A and B)
10 9 8
1 0 10 10
= ᎏᎏ • ᎏᎏ • ᎏᎏ
P(all different)
no. of different orders
= ᎏᎏᎏ
18
25
no. of possible orders
= ᎏᎏ = 0.72
10 • 9 • 8
= ᎏᎏ = 0.72
10 • 10 • 10
Using a Tree Diagram to Find Conditional Probabilities
EXAMPLE 8
HEALTH The American Diabetes Association estimates that 5.9% of Americans
have diabetes. Suppose that a medical lab has developed a simple diagnostic test for
diabetes that is 98% accurate for people who have the disease and 95% accurate for
people who do not have it. If the medical lab gives the test to a randomly selected
person, what is the probability that the diagnosis is correct?
SOLUTION
A probability tree diagram, where the probabilities are given along the branches,
can help you see the different ways to obtain a correct diagnosis. Notice that the
probabilities for all branches from the same point must sum to 1.
0.059
Event A:
Person has
diabetes.
0.98
Event C:
Correct diagnosis
0.02
Event D:
Incorrect diagnosis
Event B:
Person does not
have diabetes.
0.95
Event C :
Correct diagnosis
0.05
Event D:
Incorrect diagnosis
Total
Population
0.941
So, the probability that the diagnosis is correct is:
P(C) = P(A and C) + P(B and C)
Follow branches leading to C.
= P(A) • P(C|A) + P(B) • P(C|B)
Use formula for dependent events.
= (0.059)(0.98) + (0.941)(0.95)
Substitute.
≈ 0.952
Use a calculator.
12.5 Probability of Independent and Dependent Events
733
Page 5 of 8
GUIDED PRACTICE
Vocabulary Check
Concept Check
✓
1. Explain the difference between dependent events and independent events, and
✓
2. If event A is drawing a queen from a deck of cards and event B is drawing a king
give an example of each.
from the remaining cards, are events A and B dependent or independent?
3. If event A is rolling a two on a six-sided die and event B is rolling a four on a
Skill Check
✓
different six-sided die, are events A and B dependent or independent?
Events A and B are independent. Find the indicated probability.
4. P(A) = 0.3
P(B) = 0.9
?
P(A and B) = 㛭㛭㛭
?
5. P(A) = 㛭㛭㛭
P(B) = 0.3
P(A and B) = 0.06
6. P(A) = 0.75
?
P(B) = 㛭㛭㛭
P(A and B) = 0.15
Events A and B are dependent. Find the indicated probability.
7. P(A) = 0.1
P(B|A) = 0.8
?
P(A and B) = 㛭㛭㛭
?
8. P(A) = 㛭㛭㛭
P(B|A) = 0.5
P(A and B) = 0.25
9. P(A) = 0.9
?
P(B|A) = 㛭㛭㛭
P(A and B) = 0.54
READING LIST In Exercises 10 and 11, use the following information.
Three friends are taking an English class that has a summer reading list. Each student
is required to read one book from the list, which contains 3 biographies, 10 classics,
and 5 historical novels.
10. Find the probability that the first friend chooses a biography, the second friend
chooses a classic, and the third friend chooses a historical novel.
11. Find the probability that the three friends each choose a different classic.
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 957.
SPINNING A WHEEL You are playing a game
that involves spinning the wheel shown. Find
the probability of spinning the given colors.
12. red, then blue
13. red, then green
14. yellow, then red
15. green, then yellow
16. blue, then yellow,
17. green, then red,
then green
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 12–15
Example 2: Exs. 16, 17,
24, 25
Example 3: Exs. 26, 27
Example 4: Exs. 28, 29
Example 5: Exs. 30–32
Example 6: Exs. 18–21
Example 7: Exs. 22, 23,
33, 34
Example 8: Exs. 35, 36
734
then blue
DRAWING CARDS Find the probability of drawing the given cards from a
standard 52-card deck (a) with replacement and (b) without replacement.
18. a heart, then a diamond
19. a jack, then a king
20. a 2, then a face card (K, Q, or J)
21. a face card (K, Q, or J), then a 2
22. an ace, then a 2, then a 3
23. a heart, then a diamond, then another heart
24.
GAMES You are playing a game that involves drawing three numbers from
a hat. There are 25 pieces of paper numbered 1 to 25 in the hat. Each number is
replaced after it is drawn. What is the probability that each number is greater
than 20 or less than 4?
Chapter 12 Probability and Statistics
Page 6 of 8
25.
LAWN CARE The owner of a one-man lawn mowing business owns three
old and unreliable riding mowers. As long as one of the three is working he can
stay productive. From past experience, one of the mowers is unusable 12 percent
of the time, one 6 percent of the time, and one 20 percent of the time. Find the
probability that all three mowers are unusable on a given day.
26.
TRADING CARDS You collect movie trading cards, which have different
scenes from a movie. For one movie there are 90 different cards in the set, and
you have all of them except the final scene. To try and get this card, you buy
10 packs of 8 cards each. All cards in a pack are different and each of the cards is
equally likely to be in a given pack. Find the probability that you will get the
final scene.
FREE THROWS Chris Mullin of the Indiana Pacers led the National
27.
Basketball Association in free-throw percentage during the 1997–1998 season.
He made 93.9% of his free-throw attempts. If he attempted 10 free throws in a
game, what is the probability that he missed at least one? 䉴 Source: NBA
MANUFACTURING Look back at Example 4. Suppose the computer chip
28.
manufacturer has improved quality control so that only 1 out of 10,000 of its
chips is defective. Now how many chips can you order before the probability that
at least one chip is defective reaches 50%?
INT
STUDENT HELP
NE
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HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with problem
solving in Ex. 29.
FOCUS ON
CAREERS
LOTTERY To win a state lottery, a player must correctly match six different
29.
numbers from 1 to 42. If a computer randomly assigns six numbers per ticket,
how many tickets would a person have to buy to have a 1% chance of winning?
STATISTICS
CONNECTION
In Exercises 30 and 31, use the following information.
The table, based on a Gallup Poll, shows the number of voters (in 1000’s) by party
affiliation who were expected to vote for Bill Clinton and Bob Dole in the 1996
Presidential election. 䉴 Source: The Gallup Organization
Clinton
Dole
Democrat
Republican
Independent
31,378
3,340
12,685
2,092
28,386
8,721
30. Find the probability that a randomly selected person voted for Clinton.
31. Find the probability that a randomly selected Democrat voted for Clinton.
TEACHERS In the United States during the 1993–1994 school year,
32.
TEACHER
INT
In addition to
teaching, teachers plan daily
lessons and activities,
assign and correct homework, and prepare and grade
exams. They also monitor
homerooms, study halls, and
cafeterias, meet with
parents, and supervise
extracurricular activities.
NE
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CAREER LINK
www.mcdougallittell.com
INT
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39.6% of all male teachers and 26.1% of all female teachers had twenty years
or more of full-time teaching experience. That year 694,000 males and
1,867,000 females were teachers. What is the probability that a randomly chosen
teacher in the United States that year was a female with twenty years or more of
full-time teaching experience?
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DATA UPDATE of Statistical Abstract of the United States data at www.mcdougallittell.com
33.
COSTUMES You and four of your friends go to the same store at different
times to buy costumes for a costume party. There are 20 different costumes at
the store, and the store has at least five duplicates of each costume. Find the
probability that all five of you choose different costumes.
34.
AIRPLANE MEALS On a long flight an airline usually serves a meal. If
there are 2 choices for the meal, what is the probability that all 6 people in the
first row choose the same meal assuming choices are made independently?
12.5 Probability of Independent and Dependent Events
735
Page 7 of 8
Test
Preparation
35.
RETIREMENT PLAN At a particular company 64% of the employees are
forty years old or over. Of those employees, 83% are enrolled in the company’s
retirement plan. Only 61% of the employees under forty years old are enrolled in
the plan. Make a probability tree diagram and use it to find the probability that a
randomly selected employee is enrolled in the company’s retirement plan.
36.
FOCUS TESTING A company is focus testing a new type of fruit drink. The
focus group is 47% male. Of the males in the group, 40% said they would buy
the fruit drink, and of the females, 54% said they would buy the fruit drink.
Make a probability tree diagram and use it to find the probability that a randomly
selected person would buy the fruit drink.
QUANTITATIVE COMPARISON In Exercises 37 and 38, choose the statement
that is true about the given quantities. Assume a standard 52-card deck is used.
A
¡
B
¡
C
¡
D
¡
The quantity in column A is greater.
The quantity in column B is greater.
The two quantities are equal.
The relationship cannot be determined from the given information.
Column A
★ Challenge
Column B
37.
The probability of drawing at least
one heart when drawing 4 times with
replacement
The probability of drawing at least
one heart when drawing 4 times without
replacement
38.
The probability of not drawing any
card twice when drawing 9 times
with replacement
The probability of drawing any card
2 or more times when drawing 9
times with replacement
39. CONDITIONAL PROBABILITY Using the data from Example 8, find the
conditional probability that a randomly selected person has diabetes given that
the person is diagnosed incorrectly.
MIXED REVIEW
HISTOGRAMS Using the given intervals, make a frequency distribution of the
data set. Then draw a histogram of the data set. (Review 7.7 for 12.6)
40. Use five intervals beginning with 51–60.
56, 68, 73, 79, 82, 82, 83, 85, 85, 87, 88, 89, 90, 90, 91, 93, 95, 100
41. Use four intervals beginning with 0.01–0.25.
0.3, 0.3, 0.4, 0.4, 0.5, 0.6, 0.6, 0.6, 0.7, 0.9, 0.9, 0.9, 0.9, 0.9
SOLVING EQUATIONS Solve the equation. (Review 8.8)
5
42. ᎏ
=4
1 + 2eºx
10
43. ᎏ
=8
1 + 6eºx
9
44. ᎏᎏ
=6
1 + 3eº3x
12
45. ᎏ
=4
1 + eº2x
1
1
46. ᎏᎏ
= ᎏᎏ
2
1 + 3eº5x
70
2
47. ᎏᎏ
= ᎏᎏ
9
1 + 12eº10x
BINOMIAL THEOREM Expand the power of the binomial. (Review 12.2 for 12.6)
48. (x + 1)5
736
49. (2x º 1)7
Chapter 12 Probability and Statistics
50. (x º 3y)4
51. (x º 1)6
Page 8 of 8
QUIZ 2
Self-Test for Lessons 12.3–12.5
A jar contains 6 blue marbles, 12 green marbles, and 7 yellow marbles. Find the
probability of randomly drawing the given marble. (Lesson 12.3)
1. a green marble
2. a blue marble
3. a green or a blue marble
Find the probability that a dart thrown at the circular target
shown will hit the given region. Assume the dart is equally
likely to hit any point inside the target. (Lesson 12.3)
4. the center circle
2
5. outside the square
6
6. the area inside the square but outside the center circle
Find the indicated probability. (Lesson 12.4)
7. P(A) = 0.7
8. P(A) = 0.5
P(B) = 0.2
?
P(A or B) = 㛭㛭㛭
P(A and B) = 0.1
?
P(A§) = 㛭㛭㛭
FRUIT You and four friends are in line at lunch and are each selecting a
piece of fruit to eat. If there are 5 types of fruit available, what is the probability
that you each select a different type? (Lesson 12.5)
INT
10.
9. P(A) = 0.25
P(B) = 0.4
P(A or B) = 0.9
?
P(A and B) = 㛭㛭㛭
APPLICATION LINK
www.mcdougallittell.com
Probability Theory
THEN
NE
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IN 1654 Blaise Pascal and Pierre Fermat solved the first probability problem, which
asked how to divide the stakes of an interrupted game of chance between two players
of equal ability. Suppose Player A has 2 points, Player B has 1 point, and the game
is won by the first player to score 4 out of a possible 7 points.
1. If the stakes are divided based on which player is closest to winning, what
fraction of the stakes should each player receive?
2. If the stakes are divided based on the number of points each player has so far,
what fraction of the stakes should each player receive?
3. What is the probability Player A will win the game? What is the probability
Player B will win the game? Based on these probabilities, what fraction of the
stakes should each player receive?
NOW
TODAY probability theory is used to decide more than just games of chance.
Actuaries, for example, use probability theory to design financial plans, to calculate
insurance rates, and to price corporate securities offerings.
Actuary is named
the best job in America.
Dice are in use.
JOB RANKINGS
c. 2000 B.C.
1820
c. A . D . 1450
Playing cards
are in use.
Probability theory is
extended to astronomy.
1. ACTUARY
2. SOFTWARE ENGINEER
3. COMPUTER SYSTEMS
ANALYST
1995
4. ACCOUNTANT
5. PARALEGAL ASSISTANT
6. MATHEMATICIAN
12.5 Probability of Independent and Dependent Events
737
Page 1 of 6
12.6
What you should learn
GOAL 1 Find binomial
probabilities and analyze
binomial distributions.
Binomial Distributions
GOAL 1
There are many probability experiments in which the results of each trial can be
reduced to two outcomes. If such an experiment satisfies the following conditions,
then it is called a binomial experiment.
GOAL 2 Test a hypothesis,
as applied in Example 4.
•
•
•
Why you should learn it
䉲 To solve real-life problems, such as determining
whether a computer manufacturer’s claim is correct
in Ex. 46.
AL LI
There are n independent trials.
Each trial has only two possible outcomes: success and failure.
The probability of success is the same for each trial. This probability is
denoted by p. The probability of failure is given by 1 º p.
F I N D I N G A B I N O M I A L P R O BA B I L I T Y
For a binomial experiment consisting of n trials, the probability of exactly
k successes is
FE
RE
FINDING BINOMIAL PROBABILITIES
P(k successes) = nCk pk (1 º p)n º k
where the probability of success on each trial is p.
EXAMPLE 1
Finding a Binomial Probability
UFOS According to a survey taken by USA Today,
about 37% of adults believe that Unidentified Flying
Objects (UFOs) really exist. Suppose you randomly survey
6 adults. What is the probability that exactly 2 of them
believe that UFOs really exist?
SOLUTION
Let p = 0.37 be the probability that a randomly selected
adult believes that UFOs really exist. By surveying 6 adults,
you are conducting n = 6 independent trials. The probability
of getting exactly k = 2 successes is:
Do UFOs Exist?
Yes
No
or
37%
Do not
know
63%
P(k = 2) = 6C2(0.37)2(1 º 0.37)6 º 2
6!
4! • 2!
= ᎏᎏ(0.37)2(0.63)4
≈ 0.323
䉴
The probability that exactly 2 of the people surveyed believe that UFOs really
exist is about 32%.
..........
A binomial distribution shows the probabilities of all possible numbers of
successes in a binomial experiment, as illustrated in Example 2 on the next page.
12.6 Binomial Distributions
739
Page 2 of 6
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UFOs
EXAMPLE 2
Constructing a Binomial Distribution
Draw a histogram of the binomial distribution for the survey in Example 1. Then find
the probability that at most 2 of the people surveyed believe that UFOs really exist.
SOLUTION
P(k = 0) = 6C0(0.37)0(0.63)6 ≈ 0.063
STUDENT HELP
Probability
P(k = 1) = 6C1(0.37)1(0.63)5 ≈ 0.220
Study Tip
You can check your
calculations for a
binomial distribution by
adding all the
probabilities. The sum
should always be 1.
P(k = 2) = 6C2(0.37)2(0.63)4 ≈ 0.323
P(k = 3) = 6C3(0.37)3(0.63)3 ≈ 0.253
0.3
0.2
0.1
P(k = 4) = 6C4(0.37)4(0.63)2 ≈ 0.112
0
P(k = 5) = 6C5(0.37)5(0.63)1 ≈ 0.026
0 1 2 3 4 5 6
Number of believers
in UFOs
P(k = 6) = 6C6(0.37)6(0.63)0 ≈ 0.003
The probability of getting at most k = 2 successes is:
P(k ≤ 2) = P(2) + P(1) + P(0) ≈ 0.323 + 0.220 + 0.063 = 0.606
䉴
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Egg Incubation
The probability that at most 2 of the people surveyed believe that UFOs really
exist is about 61%.
EXAMPLE 3
Interpreting a Binomial Distribution
For a science project you are incubating 12 chicken eggs. The probability that a
chick is female is 0.5. Draw a histogram of the binomial distribution based on the
probability that exactly k of the chicks are female. Then find the most likely number
of female chicks.
SOLUTION Begin by calculating each binomial probability
Probability
using the formula P(k) = nCk(0.5)k(0.5)n º k = nCk(0.5)n as
shown in the table. Then draw the histogram.
0.2
0.1
0
0
2
4
6
8
10
12
Number of female chicks
The most likely number of female chicks is the value of k for
which P(k) is greatest. This probability is greatest for k = 6.
So, the most likely number of female chicks is 6.
..........
k
P(k)
0
1
2
3
4
5
6
7
8
9
10
11
12
0.000
0.003
0.016
0.054
0.121
0.193
0.226
0.193
0.121
0.054
0.016
0.003
0.000
The distribution in Example 3 is symmetric, which means that the left half of the
histogram is a mirror image of the right half. A distribution that is not symmetric is
called skewed. The distribution in Example 2 is an example of a skewed distribution.
740
Chapter 12 Probability and Statistics
Page 3 of 6
GOAL 2
TESTING HYPOTHESES
Example 1 presented the claim that 37% of adults believe that UFOs really exist.
If you wanted to test this claim, you could use a procedure from statistics called
hypothesis testing. Here are the basic steps.
HYPOTHESIS TESTING
STUDENT HELP
Study Tip
Different standards are
used by different
statisticians, but “small”
in Step 3 typically means
that the probability is less
than 0.1, 0.05, or 0.01.
STEP 1
State the hypothesis you are testing. The hypothesis should make a
statement about some statistical measure (mean, standard deviation,
or proportion) of a population.
STEP 2
Collect data from a random sample of the population and compute
the statistical measure of the sample.
STEP 3
Assume that the hypothesis is true and calculate the resulting
probability of obtaining the sample statistical measure or a more
extreme sample statistical measure. If this probability is small, you
should reject the hypothesis.
Testing a Hypothesis
EXAMPLE 4
UFOS To test the claim that 37% of adults believe UFOs really exist, you conduct a
survey of 10 randomly selected adults. In your survey only 2 of the adults believe that
UFOs really exist. Should you reject the claim? Explain.
SOLUTION
1
Assume the claim 37% of adults believe that UFOs really exist is true.
2
Form the binomial distribution for the probability of selecting exactly k people
out of 10 who believe that UFOs really exist.
FOCUS ON
RE
FE
INT
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APPLICATION LINK
www.mcdougallittell.com
7
0.000
6
0.001
5
0.028
4
0.006
2
0.085
3
0.173
0.246
1
8
9
10
Number of believers in UFOs
UFOS In 1938 a
radio adaptation of
The War of the Worlds
caused thousands of people
to believe that UFOs had
landed. The War of the
Worlds was made into a
feature film in 1953.
0
0.239
0.1
0.058
0.2
0
L
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0.153
0.3
0.010
Probability
APPLICATIONS
3
Calculate the probability that you could randomly select 2 or fewer adults out of
10 who believe that UFOs really exist.
P(k ≤ 2) = 0.010 + 0.058 + 0.153 = 0.221
䉴
If it is true that 37% of adults believe that UFOs really exist, then there is a 22%
probability of finding only 2 or fewer believers in a random sample of 10.
With a probability this large, you should not reject the claim.
12.6 Binomial Distributions
741
Page 4 of 6
GUIDED PRACTICE
Vocabulary Check
Concept Check
✓
1. Explain the difference between a binomial experiment and
✓
2. You draw ten cards from a standard 52-card deck without
a binomial distribution.
replacement. Is this a binomial experiment? Explain.
3. Consider the binomial distribution shown. Is the distribution
Skill Check
✓
skewed or symmetric? Explain.
Calculate the probability of k successes for a binomial
experiment consisting of n trials with probability p of
success on each trial.
4. k = 7, n = 12, p = 0.7
k
P (k)
0
0.0625
1
0.25
2
0.375
3
0.25
4
0.0625
Ex. 3
5. k ≤ 3, n = 14, p = 0.45
A binomial experiment consists of n trials with probability p of success on each
trial. Draw a histogram of the binomial distribution that shows the probability
of exactly k successes. Then find the most likely number of successes.
6. n = 6, p = 0.5
9.
7. n = 8, p = 0.33
8. n = 10, p = 0.25
CLASS RINGS You read an article that claims only 30% of graduating
seniors will buy a class ring. To test this claim you survey 15 randomly selected
seniors in your school and find that 4 are planning to buy class rings. Should you
reject the claim? Explain. 䉴 Source: America by the Numbers
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 957.
CALCULATING PROBABILITIES Calculate the probability of tossing a coin
20 times and getting the given number of heads.
10. 1
11. 3
12. 5
13. 9
14. 10
15. 11
16. 15
17. 17
BINOMIAL PROBABILITIES Calculate the probability of randomly guessing the
given number of correct answers on a 30-question multiple-choice exam that
has choices A, B, C, and D for each question.
18. 0
19. 2
20. 5
21. 10
22. 15
23. 20
24. 25
25. 30
BINOMIAL DISTRIBUTIONS Calculate the probability of k successes for a binomial
experiment consisting of n trials with probability p of success on each trial.
STUDENT HELP
26. k ≥ 3, n = 5, p = 0.2
27. k ≤ 2, n = 6, p = 0.5
HOMEWORK HELP
28. k ≤ 1, n = 9, p = 0.15
29. k ≤ 5, n = 12, p = 0.64
Example 1: Exs. 10–25,
36, 37
Example 2: Exs. 26–29,
38–41
Example 3: Exs. 30–35,
42–45
Example 4: Exs. 46–48
742
HISTOGRAMS A binomial experiment consists of n trials with probability p
of success on each trial. Draw a histogram of the binomial distribution that
shows the probability of exactly k successes. Then find the most likely number
of successes.
30. n = 2, p = 0.4
31. n = 4, p = 0.7
32. n = 5, p = 0.17
33. n = 8, p = 0.92
34. n = 9, p = 0.125
35. n = 12, p = 0.033
Chapter 12 Probability and Statistics
Page 5 of 6
FOCUS ON
PEOPLE
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DR. CHARLES
DREW was a
leading expert in blood
collection and storage,
discussed in Exs. 42 and 43.
Although he died in 1950 at
the age of 45, his contributions to medicine continue
to save lives.
36.
COURT CASE In a particular court case, legal analysts determine from
the evidence presented that there is about a 90% probability a juror will vote
that the defendant is guilty. Find the probability that all 12 jurors vote that the
defendant is guilty.
37.
VIDEOCASSETTES A product-quality researcher runs a study on a particular
brand of videocassette tape and discovers that on a random basis one out of 400
tapes is defective. Find the probability that you buy 20 of these tapes and exactly
2 are defective.
PUPPIES In Exercises 38 and 39, use the following information.
A dog gives birth to a litter of 8 puppies. Assume that the probability of a puppy
being female is 0.5.
38. Draw a histogram of the binomial distribution for the number of female puppies
in the litter.
39. What is the probability of having at least 5 female puppies?
EGGS In Exercises 40 and 41, use the following information.
A study states 34% of people prefer their eggs scrambled. You randomly select
7 people who had eggs for breakfast 䉴 Source: America by the Numbers
40. Draw a histogram of the binomial distribution for the number of people who
prefer their eggs scrambled.
41. What is the probability that at most 3 of these people prefer scrambled eggs?
BLOOD DRIVE In Exercises 42 and 43, use the following information.
A hospital is having a blood drive. The hospital is desperately in need of type
Oº blood, which 7% of the people in the United States have. Each hour at the
hospital, an average of 12 people give blood. 䉴 Source: American Red Cross
42. Draw a histogram of the binomial distribution for the number of people each
hour who give type Oº blood.
43. What is the most likely number of people each hour who give type Oº blood?
WORK HOURS In Exercises 44 and 45, use the following information.
A recent survey found that 25% of women wish they had more flexible work hours.
A small company has 24 female employees. 䉴 Source: America by the Numbers
44. Draw a histogram of the binomial distribution for the number of female
employees who want more flexible work hours.
45. At the small company, what is the most likely number of female employees who
want more flexible work hours?
INT
STUDENT HELP
NE
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HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with problem
solving in Ex. 47.
46.
PERSONAL COMPUTERS A manufacturer of personal computers claims
that under normal work use only 5% of its computers will fail to operate at some
point in a month. A small business firm uses 30 of the manufacturer’s computers
under normal work use and has 2 failures in a month. Would you reject the
manufacturer’s claim? Explain.
47.
STUDENT ATHLETES In an article about athletes at a particular college,
a journalist claims that 75% of the athletes who were given scholarships would
still be attending the college if they had not been given scholarships. Curious,
a professor polls a random sample of 10 students with sports scholarships.
Three of the students said they had plans to attend the college with or without a
scholarship. Would you reject the journalist’s claim? Explain.
12.6 Binomial Distributions
743
Page 6 of 6
48.
Test
Preparation
BOTTLED JUICES A company that makes bottled juices has created a new
brand of apple juice. The company claims 80% of people prefer the new apple
juice over a competitor’s apple juice. A taste test is conducted to test this claim.
Of 20 people sampled, 12 preferred the new apple juice. Would you reject the
company’s claim? Explain.
49. MULTIPLE CHOICE A baseball player’s batting average is .310. What is the
probability that the player will get 3 or more hits in a game in which the player
has 5 official at-bats?
A
¡
B
¡
0.500
C
¡
0.142
0.035
D
¡
0.177
E
¡
0.600
50. MULTIPLE CHOICE Based on sales figures it is assumed that 60% of the
students in a particular high school prefer Beverage A to Beverage B. In a
survey of 10 randomly selected students, 3 preferred Beverage A. What is the
probability of obtaining this sample result or a more extreme sample result?
★ Challenge
A
¡
B
¡
0.947
C
¡
0.043
0.055
D
¡
0.300
E
¡
0.200
51. FINDING A FORMULA The mean of a binomial distribution is the sum of the
products of the numbers of successes and the corresponding probabilities.
Mathematically, the mean can be represented by:
n
∑ k • P(k)
k=0
EXTRA CHALLENGE
www.mcdougallittell.com
Find the mean of the binomial distribution in Example 2. Then divide the mean
by n. What significance does this number have relative to the probability of
success on any trial? Use this result to find a simpler formula for the mean of a
binomial distribution.
MIXED REVIEW
MEASURES OF DISPERSION Find the range and standard deviation of the
data set. (Review 7.7 for 12.7)
52. 8, 9, 9, 9, 10
53. 16, 18, 19, 21, 25, 27, 27
54. 1.2, 1.3, 1.4, 1.7, 1.8
55. 81, 87, 88, 91, 99, 100
SOLVING SYSTEMS Find the points of intersection, if any, of the graphs in the
system. (Review 10.7)
56. x 2 + y2 = 1
57. x 2 + y2 = 74
x º y = º2
y=x
59. 5x 2 + y2 = 10
2
2
60. x 2 º y2 = 49
y=7
x +y =9
58. x 2 + 4y2 = 1
y=x+1
61. 2x 2 º 3y2 = 6
y = 3x + 1
WRITING RULES Write a recursive rule for the sequence. The sequence may be
arithmetic, geometric, or neither. (Review 11.5)
62. 3, 5, 9, 15, . . .
63. 4, 40, 400, 4000, . . .
64. 80, 60, 40, 20, . . .
65. 1, 3, 3, 9, . . .
66. 160, 40, 10, 2.5, . . .
67. 1, 2, 3, 5, . . .
The radius r of a sphere can be approximated by
苶 where V is the volume of the sphere. Graph the model. Then
r = 0.62兹V
determine the volume of a sphere with a radius of 3 inches. (Lesson 7.5)
68. GEOMETRY
CONNECTION
3
744
Chapter 12 Probability and Statistics
Page 1 of 7
E X P L O R I N G DATA
A N D S TAT I S T I C S
n ⫽ 15
p ⫽ 0.4
GOAL 2
RE
FE
䉲 To solve real-life
problems, such as finding
the probability that certain
numbers of patients
are nearsighted in
Exs. 53–55.
AL LI
In both cases the binomial distribution can be approximated by a smooth,
symmetrical, bell-shaped curve called a normal curve. Areas under this curve
represent probabilities from normal distributions.
CONCEPT
SUMMARY
AREAS UNDER A NORMAL CURVE
The mean x
苶 and standard deviation ␴ of a normal distribution determine the
following areas.
•
•
•
•
The total area under the curve is 1.
68% of the area lies within 1 standard deviation of the mean.
95% of the area lies within 2 standard deviations of the mean.
99.7% of the area lies within 3 standard deviations of the mean.
From the second bulleted statement above
and the symmetry of a normal curve, you
can deduce that 34% of the area lies within
1 standard deviation to the left of the mean,
and 34% of the area lies within 1 standard
deviation to the right of the mean. The
diagram shows other partial areas
(expressed as decimals rather than percents)
based on the properties of a normal curve.
You can interpret these areas as probabilities.
In a normal distribution, the probability that
a randomly chosen x-value is between a and
b is given by the area under the normal curve
between a and b. For instance, the probability
that a randomly selected x-value is between 1
and 2 standard deviations to the right of the
mean is:
P(x苶 + ß ≤ x ≤ x苶 + 2ß) = 0.135
746
Chapter 12 Probability and Statistics
0.34
0.34
0.135
0.0235
0.0015
0.135
0.0235
0.0015
x
x⫹
σ
x⫹
2σ
x⫹
3σ
Why you should learn it
n ⫽ 20
p ⫽ 0.3
2σ
x⫺
σ
Use normal
distributions to approximate
binomial distributions, as
applied in Example 3.
As statisticians began to study binomial distributions consisting of n trials with
probability P of success on each trial, they discovered that when np and n(1 º p)
are both greater than or equal to five, the distributions all resemble one another.
Here are two examples.
3σ
GOAL 1 Calculate
probabilities using normal
distributions.
USING NORMAL DISTRIBUTIONS
x⫺
What you should learn
GOAL 1
x⫺
12.7
Normal Distributions
P (a ≤ x ≤ b)
a
b
Page 2 of 7
Many real-life distributions are normal or approximately normal.
RE
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Shopping
EXAMPLE 1
Using a Normal Distribution
A survey shows that the time spent by shoppers in supermarkets is normally
distributed with a mean of 45 minutes and a standard deviation of 12 minutes.
a. What percent of the shoppers at a supermarket will spend between 33 and
57 minutes in the supermarket?
b. What is the probability that a randomly chosen shopper will spend between
45 and 69 minutes in the supermarket?
SOLUTION
a. The given times of 33 minutes and
and the time of 69 minutes is two
standard deviations to the right of the
mean, as shown below. So, the
probability that a randomly chosen
shopper will spend between 45 and
69 minutes in the supermarket is 0.475.
68%
P (45 ≤ x ≤ 69) ⫽ 0.34 ⫹ 0.135 ⫽ 0.475
9
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Health Statistics
9
21 33 45 57 69 81
Shopping time (minutes)
EXAMPLE 2
21 33 45 57 69 81
Shopping time (minutes)
Using a Normal Distribution (Compound Event)
According to a survey by the National Center for Health Statistics, the heights of
adult men in the United States are normally distributed with a mean of 69 inches and
a standard deviation of 2.75 inches. If you randomly choose 3 adult men, what is the
probability that all three are 71.75 inches or taller?
SOLUTION
A height of 71.75 inches is one standard deviation
to the right of the mean, as shown. The probability
of randomly selecting a man who is this height or
taller is:
P(x ≥ 71.75) = 0.135 + 0.0235 + 0.0015
0.16
.75
63
.5
66
.25
69
71
.75
74
.5
77
.25
Study Tip
When calculating
probabilities using a
normal distribution, refer
to the diagram below the
property box on p. 746.
b. The time of 45 minutes is the mean
57 minutes represent one standard
deviation on either side of the mean,
as shown below. So, 68% of the
shoppers will spend between 33 and
57 minutes in the supermarket.
60
STUDENT HELP
= 0.16
Height (inches)
Randomly choosing men are independent events, so the probability that all three
randomly chosen men are 71.75 inches or taller is:
P(all are 71.75 inches or taller) = (0.16)3
≈ 0.00410
12.7 Normal Distributions
747
Page 3 of 7
GOAL 2
APPROXIMATING BINOMIAL DISTRIBUTIONS
When n is large it can be tedious to compute binomial probabilities using the formula
P(k) = nCk pk(1 º p)n º k. In such cases you may be able to use a normal distribution
to approximate a binomial distribution.
N O R M A L A P P R OX I M AT I O N O F A B I N O M I A L D I S T R I B U T I O N
Consider the binomial distribution consisting of n trials with probability p
of success on each trial. If np ≥ 5 and n(1 º p) ≥ 5, then the binomial
distribution can be approximated by a normal distribution with a mean of
Æ = np
x
and a standard deviation of
ß = 兹n
苶p
苶 (1
苶苶
º苶)
p苶.
EXAMPLE 3
Finding a Binomial Probability
SURVEYS According to a survey conducted by the Harris Poll, 29% of adults in the
United States say that they or someone in their family plays soccer regularly. You are
conducting a random survey of 238 adults. What is the probability that you will find
at most 55 adults who come from a family in which someone plays soccer regularly?
SOLUTION
FOCUS ON
CAREERS
To answer the question using the binomial probability formula, you would have to
calculate the following:
P(x ≤ 55) = P(0) + P(1) + P(2) + . . . + P(55)
This would be tedious. Instead you can approximate the answer with a normal
distribution having a mean of
苶x = np = 238(0.29) ≈ 69
and a standard deviation of
ß = 兹n
苶p苶(1
苶苶
º苶)
p苶 = 兹2苶3苶8苶(0
苶.2
苶9苶)(
苶0苶.7
苶1苶)苶 ≈ 7.
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MARKET
RESEARCHER
INT
A market researcher gathers
data about the market
potential of a product or
service. The data collected
are used to identify
opportunities to improve a
company’s success in the
marketplace.
NE
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CAREER LINK
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748
For this normal distribution, 55 is two
standard deviations to the left of the mean.
So, the probability that you will find at most
55 people from families in which someone
plays soccer regularly is:
P(x ≤ 55) ≈ 0.0015 + 0.0235
0.025
48 55 62 69 76 83 90
People from “soccer families”
= 0.025
..........
If you had instead used the binomial probability formula in Example 3, you would
have found the actual binomial probability to be approximately 0.0249. So, you can
see that the normal approximation is a very good approximation.
Chapter 12 Probability and Statistics
Page 4 of 7
GUIDED PRACTICE
Vocabulary Check
Concept Check
✓
? can be used to approximate a binomial
1. Complete this statement: A(n) 㛭㛭㛭
✓
2. A normal curve is symmetric about what x-value?
distribution when np and n(1 º p) are both greater than or equal to 5.
3. What percent of the area under a normal curve lies within 1 standard deviation of
Skill Check
✓
the mean? within 2 standard deviations of the mean? within 3 standard deviations
of the mean?
A normal distribution has a mean of 10 and a standard deviation of 1. Find the
probability that a randomly selected x-value is in the given interval.
4. between 8 and 12
5. between 7 and 13
6. between 8 and 11
7. at most 10
8. at least 12
9. at most 9
Find the mean and standard deviation of a normal distribution that
approximates a binomial distribution consisting of n trials with probability p
of success on each trial.
10. n = 10, p = 0.5
11. n = 17, p = 0.3
12. n = 28, p = 0.2
13. n = 20, p = 0.25
14. n = 12, p = 0.42
15. n = 30, p = 0.17
COLORBLINDNESS In Exercises 16–18, use the fact that approximately 2%
of people are colorblind, and consider a class of 460 students.
16. What is the probability that 15 or fewer students are colorblind?
17. What is the probability that 12 or more students are colorblind?
18. What is the probability that between 6 and 18 students are colorblind?
PRACTICE AND APPLICATIONS
19.
20.
2σ
x⫹
x⫹
σ
22.
2σ
21.
x⫹
HOMEWORK HELP
x⫺
x⫺
STUDENT HELP
Example 1: Exs. 19–28,
38–43
Example 2: Exs. 29–31,
44–49
Example 3: Exs. 32–37,
50–55
σ
3σ
x
2σ
x⫺
σ
Extra Practice
to help you master
skills is on p. 957.
USING A NORMAL CURVE Give the percent of the area under a normal curve
represented by the shaded region.
x⫺
STUDENT HELP
12.7 Normal Distributions
749
Page 5 of 7
NORMAL DISTRIBUTIONS A normal distribution has a mean of 22 and a
standard deviation of 3. Find the probability that a randomly selected x-value is
in the given interval.
23. between 19 and 25
24. between 13 and 22
25. between 16 and 31
26. at most 25
27. at least 19
28. at most 28
FINDING PROBABILITIES A normal distribution has a mean of 64 and a
standard deviation of 7. Find the given probability.
29. three randomly selected x-values are all 71 or greater
30. four randomly selected x-values are all 50 or less
31. two randomly selected x-values are both between 57 and 78
APPROXIMATING BINOMIAL DISTRIBUTIONS Find the mean and standard
deviation of a normal distribution that approximates a binomial distribution
consisting of n trials with probability p of success on each trial.
32. n = 18, p = 0.7
33. n = 50, p = 0.1
34. n = 32, p = 0.8
35. n = 49, p = 0.12
36. n = 24, p = 0.67
37. n = 140, p = 0.06
DRIVE-THROUGH In Exercises 38–40, use the following information.
A certain bank is busiest during the Friday evening rush hours from 3:00 P.M.
until 6:00 P.M. During these hours the waiting time for drive-through customers is
normally distributed with a mean of 8 minutes and a standard deviation of 2 minutes.
38. What percent of drive-through customers will wait for 10 minutes or longer
during the Friday evening rush hours?
39. What is the probability that a customer will wait between 4 and 12 minutes
during the Friday evening rush hours?
40. What is the probability that a customer will wait 2 minutes or less during the
Friday evening rush hours?
LIGHT BULBS In Exercises 41–43, use the following information.
A company produces light bulbs having a life expectancy that is normally distributed
with a mean of 1000 hours and a standard deviation of 50 hours.
FOCUS ON
APPLICATIONS
41. What percent of the bulbs will last for 1000 hours or more?
42. What is the probability that a randomly chosen bulb will burn out in 900 hours
or less?
43. What is the probability that a randomly chosen bulb will last between 850 and
1050 hours?
BIOLOGY
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LIGHT BULBS
Four compact
fluorescent light bulbs use
the same energy as one
incandescent light bulb.
Compact fluorescent light
bulbs also last longer, with
an average life expectancy
of 10,000 hours.
750
CONNECTION
In Exercises 44–46, use the following information.
According to a survey by the National Center for Health Statistics, the heights of
adult women in the United States are normally distributed with a mean of 64 inches
and a standard deviation of 2.7 inches.
44. What is the probability that three randomly selected women are all 58.6 inches
or shorter?
45. What is the probability that five randomly selected women are all between the
heights of 61.3 and 66.7 inches?
46. What is the probability that four randomly selected women are all 72.1 inches
or taller?
Chapter 12 Probability and Statistics
Page 6 of 7
NE
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HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with problem
solving in Exs. 47–49.
SAT SCORES In Exercises 47–49, use the following information.
In 1998 scores on the mathematics section of the SAT (Scholastic Aptitude Test)
were normally distributed with a mean of 512 and a standard deviation of 112. Scores
on the English section of the SAT were normally distributed with a mean of 505 and
DATA UPDATE of SAT data at www.mcdougallittell.com
a standard deviation of 111.
INT
INT
STUDENT HELP
NE
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47. What is the probability that a randomly chosen student who took the SAT in
1998 scored at least 736 on the mathematics section and at least 727 on the
English section? Assume the scores are independent.
48. What is the probability that five randomly chosen students who took the SAT in
1998 all scored at most 394 on the English section?
49. What is the probability that two randomly chosen students who took the SAT in
1998 both scored between 400 and 624 on the mathematics section?
LEFT-HANDEDNESS In Exercises 50–52, use the fact that approximately 9%
of people are left-handed, and consider a high school with 1221 students.
50. What is the probability that at least 140 students are left-handed?
51. What is the probability that at most 100 students are left-handed?
52. What is the probability that between 80 and 130 students are left-handed?
MYOPIA In Exercises 53–55, use the fact that myopia, or nearsightedness,
is a condition that affects approximately 25% of the adult population in the
United States, and consider a random sample of 192 people.
53. What is the probability that 42 or more people are nearsighted?
54. What is the probability that between 36 and 60 people are nearsighted?
55. What is the probability that 66 or fewer people are nearsighted?
Test
Preparation
56. MULTI-STEP PROBLEM In 1998 Ben took both the SAT (Scholastic Aptitude
Test) and the ACT (American College Test). On the mathematics section of the
SAT, he earned a score of 624. On the mathematics section of the ACT, he earned
a score of 31. For the SAT the mean was 512 and the standard deviation was 112.
For the ACT the mean was 21 and the standard deviation was 5.
a. What percent of students did Ben outscore on the math section of the SAT?
b. What percent of students did Ben outscore on the math section of the ACT?
c. On which exam did Ben score better?
d.
★ Challenge
Writing Explain how you could translate ACT scores such as 15, 20, 25,
and 30 into equivalent SAT scores if you know the mean and standard
deviation of each exam.
57.
NORMAL CURVE A normal curve is defined by an equation whose general
form is as follows:
x º x苶
冉ᎏ
␴ᎏ冊
1
ºᎏᎏ
1
2
␴兹2
苶π
苶
y=ᎏe
EXTRA CHALLENGE
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2
Use a graphing calculator to draw a histogram of a binomial distribution
consisting of n = 20 trials with probability p = 0.5 of success. Also graph the
normal curve that approximates this binomial distribution. Then graph other
binomial distributions and normal curves in which you change p but leave n
constant. When is the normal curve a good approximation of a binomial
distribution and when is it a poor approximation? Why?
12.7 Normal Distributions
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Page 7 of 7
MIXED REVIEW
EVALUATING EXPRESSIONS Evaluate the expression without using a
calculator. (Review 7.1 for 13.1)
58. 93/2
3
61. 兹1
苶2苶5苶
59. 2563/4
60. 49º1/2
62. 兹8
苶1苶
63. (兹6
苶2苶5苶)2
4
IDENTIFYING PARTS Write the equation of the ellipse in standard form (if not
already). Then identify the vertices, co-vertices, and foci of the ellipse. (Review 10.4)
y2
x2
64. ᎏᎏ + ᎏᎏ = 1
4
16
y2
x2
65. ᎏᎏ + ᎏᎏ = 1
14 4
169
y2
x2
66. ᎏᎏ + ᎏᎏ = 1
5
10
y2
x2
67. ᎏᎏ + ᎏᎏ = 1
6
21
68. 4x2 + 9y2 = 36
69. 10x2 + 7y2 = 70
USING COMPLEMENTS Two six-sided dice are rolled. Find the probability of
the given event. (Review 12.4)
70. The sum is not 4.
71. The sum is less than or equal to 10.
72. The sum is not 3 or 12.
73. The sum is greater than 3.
QUIZ 3
Self-Test for Lessons 12.6 and 12.7
Calculate the probability of rolling a six-sided die 50 times and getting the
given number of ones. (Lesson 12.6)
1. 0
2. 1
3. 8
4. 17
5. 25
6. 33
7. 42
8. 50
A binomial experiment consists of n trials with probability p of success on each
trial. Draw a histogram of the binomial distribution that shows the probability
of exactly k successes. Then find the most likely number of successes.
(Lesson 12.6)
9. n = 4, p = 0.3
12. n = 10, p = 0.33
10. n = 7, p = 0.5
11. n = 8, p = 0.6
13. n = 12, p = 0.48
14. n = 15, p = 0.21
A normal distribution has a mean of 62 and a standard deviation of 4. Find the
probability that a randomly selected x-value is in the given interval.
(Lesson 12.7)
752
15. between 58 and 66
16. between 62 and 74
17. between 50 and 70
18. 62 or greater
19. 58 or less
20. 50 or less
21.
WELL-BEING A survey that asked people in the United States about their
feelings of personal well-being found that 85% are generally happy. To test this
finding you survey 26 people at random and find that 19 consider themselves
generally happy. Would you reject the survey’s findings? Explain. (Lesson 12.6)
22.
DRINKING WATER Approximately 64% of people in the United States
think that the nation’s water supply is safe to drink. A town has 625 people. What
is the probability that 400 or more people think that the nation’s water supply is
safe to drink? (Lesson 12.7)
Chapter 12 Probability and Statistics