Download Fundamental Counting Principle

Aim: Probability! Probability! How do I count the ways?
Do Now: Choose one item from each
category to make an entire meal.
Main Course
Drink
Dessert
Spaghetti
Milk
Ice Cream
Hamburger
Coke
Apple Pie
Hotdog
Aim: Fundamental Counting Principle
Chocolate
Cake
Course: Alg. 2 & Trig.
Dinner is Served!
Main Course
Drink
Dessert
Milk
Ice Cream
Apple Pie
Chocolate Cake
SMI
SMA
SMC
Coke
Ice Cream
Apple Pie
Chocolate Cake
SCI
SCA
SCC
Milk
Ice Cream
Apple Pie
Chocolate Cake
HMI
HMA
HMC
Coke
Ice Cream
Apple Pie
Chocolate Cake
HCI
HCA
HCC
Milk
Ice Cream
Apple Pie
Chocolate Cake
Ht M I
Ht M A
Ht M C
Coke
Ice Cream
Apple Pie
Chocolate Cake
Ht C I
Ht C A
Ht C C
Spaghetti
Hamburger
Hotdog
Aim: Fundamental Counting Principle
Sample Space
Course: Alg. 2 & Trig.
Three consecutive Events
Am I lost?
How many different
ways will get us from
MJ Petrides to Great
Adventure?
Outerbridge
Crossing
2
Great
Adventure
MJ Petrides
3
Tracing the different
routes we find there are
6 different routes.
Is there a shortcut
method for finding how
many different routes
there are?
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
The Fundamental Counting Principle
To find the total number of possible
outcomes in a sample space, multiply the
number of choices for each stage or event...
n( E )
in other words...
P ( E) 
n( S )
If event M can occur in m ways
and is followed by event N that
can occur in n ways, then the
event M followed by event N can
occur in m · n ways.
Main Ide a
Counting Principle
2 events: m · n
3 events: m · n · o
4 events: m · n · o · p
5 events: etc.
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
3
x 2 x 3
Main Course
= 18
Drink
Dessert
Milk
Ice Cream
Apple Pie
Chocolate Cake
SMI
SMA
SMC
Coke
Ice Cream
Apple Pie
Chocolate Cake
SCI
SCA
SCC
Milk
Ice Cream
Apple Pie
Chocolate Cake
HMI
HMA
HMC
Coke
Ice Cream
Apple Pie
Chocolate Cake
HCI
HCA
HCC
Milk
Ice Cream
Apple Pie
Chocolate Cake
Ht M I
Ht M A
Ht M C
Coke
Ice Cream
Apple Pie
Chocolate Cake
Ht C I
Ht C A
Ht C C
Spaghetti
Hamburger
Hotdog
Aim: Fundamental Counting Principle
Sample Space
Course: Alg. 2 & Trig.
Model Problem
Jamie has 3 skirts - 1 blue, 1 yellow, and 1 red. She has 4
blouses - 1 yellow, 1 white, 1 tan and 1 striped. How many
skirt-blouse outfits can she choose? What is the
probability she will chose the blue skirt and white blouse?
Skirt
3
blouse 4
yellow
white
tan
striped
yellow
white
tan
striped
by
bw
Blue
bt
bs
yy
yw
Yellow
yt
by
ry
yellow
rw
white
Red
rt
tan
ry
Aim: Fundamental
Counting Principle
striped
12 outcomes
in sample
space
n( E )
P ( E) 
n( S )
1
P ( b, w ) 
12
Course: Alg. 2 & Trig.
Regents Question
A four-digit serial number is to be created
from the digits 0 through 9. How many of
these serial numbers can be created if 0 can
not be the first digit, no digit may be
repeated, and the last digit must be 5?
1) 448
2) 2240
3) 504
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
possible
outcomes
8 
E1
8 
E2
Aim: Fundamental Counting Principle
4) 2,520
10 possible
outcomes to start
7  1 = 448
E3
E4
Course: Alg. 2 & Trig.
Types of Events
Compound Event - two or more activities
Ex. Rolling a pair of dice
What is the probability of rolling a pair of
dice and getting a total of four?
6
5
4
Die 1 3
2
1
7
6
5
4
3
2
8
7
6
5
4
3
1
2
9 10 11 12
8 9 10 11
7 8 9 10
6 7 8 9 P(4) = 3/36
5 6 7 8
= 1/12
4 5 6 7
3 4
Die 2
5
Aim: Fundamental Counting Principle
6
Course: Alg. 2 & Trig.
Types of Events
Mutually Exclusive Events – Two or more
events that can not happen at the same time
Ex. mutually exclusive?
rolling a 2 and a 3 on a die
Yes
rolling an even number or a multiple
of 3 on a die
No
Independent Event – when the outcome of
one event does not affect the outcome of a
second event.
Dependent Event - when the outcome of one
event affects the outcome of a second event.
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Probability of Two Independent Events
The probability of two independent events can be
found by multiplying the probability of the first
event by the probability of the second event.
(The Counting Principle w/Probabilities)
AND Probabilities
Ex: A die is tossed and a
spinner is spun.
What’s the probability of
throwing a 5 and spinning
red? P(5 and R)?
Faster than
drawing a
tree
diagram!!
P(A and B) = P(A) · P(B)
Event A
Event B
1
P (5) 
6
1
P ( red ) 
4
P(5 and Red) =
1
6
Aim: Fundamental Counting Principle

1
4

1
24
Course: Alg. 2 & Trig.
Independent Event
Not So Independent!
There are 4 red, 3 pink, 2 green and 1 blue
chips in a bag. What is P(pink)? 3/10
What is the probability of picking a pink and
then reaching in and picking a second pink
without replacing the first picked pink?
Event A
Event B ONLY IF THE
FIRST PINK CHIP
P(pink)
P(pink) WAS NOT
RETURNED TO
2/9
3/10
THE BAG.
3 2 6
1
P(pink and pink) =
 

10 9 90 15
The selection of the second event was affected
by the selection
ofCounting
the first.
Aim: Fundamental
Principle
Course: Alg. 2 & Trig.
Dependent Event
Model Problem
Find the probability of
choosing two pink chips
without replacement.
Event A
Event B
Dependent Events
P(pink)
P(pink)
P(pink, pink) =
3/10
2/9
3/10 • 2/9 = 6/90
or 1/15
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Counting Principle w/Probabilities
Model Problem
Find the probability of
choosing blue and then a
red chip without
replacement.
Event A
Event B
P(blue)
P(red)
1/10
4/9
Dependent Events
P(blue, red) =
1/10 • 4/9 = 4/90
or 2/45
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Counting Principle w/Probabilities
Dependent Events
Two events are dependent events if the
occurrence of one of them has an effect on
the probability of the other.
AND Probabilities with Dependent Events
If A and B are dependent events, then
P(A and B) = P(A) · P(B given that A has
occurred)
extends to multiple dependent events
You are dealt three cards from a 52-card deck.
Find the probability of getting 3 hearts.
P(1st heart) = 13/52
P(2nd heart) = 12/51
P(3rd heart) = 11/50
Principle
Alg. 2 & Trig.
P(hearts)Aim:
= Fundamental
13/52 ·Counting
12/51
· 11/50 Course:
= 1716/162600
 0.0129
Probability of Dependent Events
1. Calculate the probability of the first event.
2. Calculate the probability of the second
event, etc. ... but NOTE:
The sample space for the probability of the
subsequent event is reduced because of the previous
events.
3. Multiply the the probabilities.
Ex. A bag contains 3 marbles, 2 black and one white.
Select one marble and then, without replacing it in the bag,
select a second marble. What is the probability of
selecting first a black and then a white marble?
Event A
Event B
2
3
1
P(A) =
P(B) = 2
2 1 2 1
P(Black 1st, White 2nd =   
3 2 6 3
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Key words - without replacement
Model Problem
From a deck of 10 cards (5 ten-point
cards, 3 twenty-point cards, and 2
fifty-point cards), Ronnie can only
pick 2 cards. In order to win the
game, he must pick the 2 fifty-point
cards. What is the probability that he
will win?
10
10
10
10
10
20
Aim: Fundamental Counting Principle
20
20
10
20
10
50
20
10
10
50
50
Course: Alg. 2 & Trig.
Model Problem
From a deck of 10 cards Ronnie can
only pick 2 cards. In order to win the
game, he must pick the 2 fifty-point
cards. What is the probability that he
will win?
10
10
10
10
10
20
20
20
10
20
10
50
20
10
50
10
50
Dependent
Event A
Event B
P(50, 50) =
P(50) =
P(50) =
2/10 • 1/9 =
2/90 = 1/45
2/10
1/9
Counting Principle
Aim: Fundamental Counting Principle w/Probabilities
Course: Alg. 2 & Trig.
Regents Question
Penny has 3 boxes, each containing 10 colored balls.
The first box contains 1 red ball and 9 white balls,
the second box contains 3 red balls and 7 white balls,
and the third box contains 7 red balls and 3 white
balls. Penny pulls 1 ball out of each box.
Box 1
Box 2
Box 3
A. What is the probability that Penny pulled
3 red balls?
P(r,r,r) = 1/10 • 3/10 • 7/10 = 21/1000
B. If Penny pulled 3 white balls and did not
replace them, what is the probability that
she will Aim:
now
pull 3 red balls? Course: Alg. 2 & Trig.
Fundamental Counting Principle
P(r,r,r) = 1/9 • 3/9 • 7/9 = 21/729
Venn Diagrams
Find the probability of rolling a die and getting a
number that is both odd and greater than 2.
>2
odd
1
3
5
In logic, a
sentence p and
q, written p 
q, is true only
when p is true
and q is true.
P(odd) = 3/6
2
1
3
5
4
6
P(> 2) = 4/6
3
5
4
6
Aim: Fundamental Counting Principle
P(odd  > 2)
= 2/6
Course: Alg. 2 & Trig.
Probability of A and B
Probability of (A  B)
n( A and B )
P ( A and B ) 
(A and B are
n( S )
separate events)
n( A  B )
P(A  B) 
n( S )
Example: Find the probability of rolling a die and
getting a number that is both odd and greater than 2.
P ( odd ) 
n( E ) 3
 ,
n( S ) 6
n( E ) 4
P (  2) 
 ,
n( S ) 6
{1, 3 ,5 }
{ 3 ,4 ,5 ,6 }
2
P ( odd   2 ) 
6
{3, 5}
In probability, an outcome is in event (A and B) only
when the outcome
is inCounting
event
A and the
Aim: Fundamental
Principle
Course:outcome
Alg. 2 & Trig. is
also in event B.
Mutually Exclusive Events
Mutually exclusive – two events A & B are
mutually exclusive if they can not occur at
the same time. That is, A and B are
mutually exclusive when A  B = 
An outcome for A or B is in one or the other.
If the events are mutually exclusive then
P(A or B) = P(A) + P(B)
If one card is randomly selected from a
deck of cards, what is the probability of
selecting a king or a queen? mutually
yes
exclusive?
P ( king or queen)  P ( k )  P (q )
4
4
8
2




Aim: Fundamental Counting Principle
& Trig.
52 52 Course:
52Alg. 213
‘Or’ Probabilities Not Mutually Exclusive
From a standard deck you randomly select one
card. What is the probability of selecting a
mutually
diamond or a face card?
no
exclusive?
13
P (diamonds ) 
52
12
P ( facecards ) 
52
13
13 12
12 325 22
  
P(or fcd) = 52
52 52
52 52
52 52
common elements A  B
n(A  B) = 3
{K, Q, J}
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Mutually Exclusive or Not
1. A card is drawn from a standard deck of
52. Find P(king or queen) mutually exclusive
4
4
P(king) =
P(queen) =
52
52
4
4
8
2



P(king or queen) =
52 52 52 13
2. A card is drawn from a standard deck of
52. Find P(king or face card) not mutually exclusive
4
12
P(face) =
P(king) =
52
52
4 12 4
8
2




P(king or face) =
52 52 52 52 13
In #1 the 2 events have no common elements. They
are mutually exclusive. In #2 a card can be both face
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
and king. They
are not mutually exclusive.
Probability of A or B
What is the probability of spinning a
number greater than 8 or an odd number?
12
Count the number of
successes for
1
11
2
10
3
9
4
9, 10, 11, 12
n>8
8
5
7
6
4
n - odd 1, 3, 5, 7, 9, 11 6
not mutually exclusive
10
P (greater than 8 or odd)  P ( 8)  P (odd) =
12
4
6
2
8 2
P ( 8 or odd) 




12 12 12 12 3
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Probability of (A or B)
If A and B are not mutually exclusive events,
then P(A or B) = P(A) + P(B) - P(A and B)
P(A  B) = P(A) + P(B) - P(A  B)
P(A  B) = n(A) + n(B) - n(A  B)
n(S)
n(S)
n(S)
Example: Find the probability of rolling a die and
getting a number that is odd or greater than 2.
n( E ) 3
n( E ) 4
P (odd) 

P ( 2) 

n( S ) 6
n( S ) 6
successes
{1,3,5}
{3,4,5,6}
n(odd) n( 2) n(odd  >2)
P (odd  >2) 


n( S )
n( S )
n( S )
3 Counting
4 Principle
2 5 Course: Alg. 2 & Trig.
Aim:
Fundamental
P (odd  >2)    
6 6 6 6
Probability Rules
1. The probability of an impossible event is 0.
2. The probability of an event that is certain
to occur is 1.
3. The probability of an event E must be
greater than or equal to 0 and less that or equal
to 1.
4. P(A and B) = n(A  B)
n(S)
5. P(A or B) = P(A) + P(B) - P(A  B)
6. P(Not A) = 1 - P(A)
7. The probability of any even is equal to
the sum of the probabilities of the
singleton outcomes in the event.
8. The sum of the probabilities of all possible
Fundamental Counting
Course: space
Alg. 2 & Trig.
singletonAim:
outcomes
forPrinciple
any sample
must
always equal 1.
Model Problems
In drawing a card from the deck at random,
find the probability that the card is:
P(A and B) = P(A) · P(B)
A. A red king
A red king must be red and a king
26 4
1
2
 g 
P(red and king) =
52 52 52 26
mutually exclusive
P(A  B) = P(A) + P(B) - P(A  B)
B. A 10 or an ace
10’s and aces have no common outcomes
4 _ 0 = 8
4
+
P(10’s or aces) =
52
52
52
52
not mutually exclusive
C. A jack or a club P(A  B) = P(A) + P(B) - P(A  B)
There are 4 jacks and 13 clubs, but one of the
cards is both (jack of clubs)
Aim: Fundamental Counting
Principle13
16
Course:1Alg. 2 & Trig.
_
4
=
+
P(jacks or clubs) =
52
52
52
52
Model Problems
Based on the table below, if one person is
P(A 
B) = P(A)
P(B)US
- P(A
 B)
randomly
selected
from+ the
military,
find the probability that this person is in the
Army or is a woman. not mutually exclusive
Active Duty US Military Personnel, in 000’s
Air
Army Marines Navy Total
Force
Male
290
400
160
320
1170
Female
70
70
10
50
200
Total
360
470
170
370
1370
P(Army  Female) = P(A) + P(F) - P(A  F)
470 200
70
60




1370 1370 1370 137
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Model Problems
Five more men than women are riding a bus
as passengers. The probability that a man
will be the first passenger to leave the bus is
2/3. How many passengers on the bus are
men, and how many are women?
x = number of women 5
x + 5 = number of men 10
2x + 5 = number of passengers
P(man) = Number of men
Number of passengers
2 = x+5
4x + 10 = 3x + 15
3
2x + 5
x=5
x + 5 = 10
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Model Problem
A special family has had nine girls in a row. Find
the probability of this occurrence.
Having a girl is an independent event
with P(1 girl) = 1/2
Probability of two Independent Events
P(A and B) = P(A) · P(B)
extends to multiple independent events
1 1 1 1 1 1 1 1 1
P (nine girls in a row) = g g g g g g g g
2 2 2 2 2 2 2 2 2
9
1
1
  
512
 2
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Model Problem
If the probability that South Florida will be hit by a
hurricane in any single year is 5/19
a) What is the probability that S. Florida will be hit
by a hurricane in three consecutive years?
3
5 5 of
5 event
125the
 5 (A)
 plus
•
The
probability
P (hurricane - 3) =
  
 0.018
probability of19"not
1:
19 A”
19 or ~A,
19  equals
6859
P(A) + P(~A) = 1; P(A) = 1 – P(~A);
b) What is the probability that S. Florida will not be
P(~A) = 1 – P(A)
hit by a hurricane in the next ten years?
P (no hurricane) = 1  P (hurricane)
5 14
 1

 0.737
19 19
10
10
 14 
P (no hurricane 10 yrs) =     0.737   0.047
Aim: Fundamental Counting
Principle
Course: Alg. 2 & Trig.
 19

Model Problem
Three people are randomly selected, one
person at a time, from 5 freshman, two
sophomores, and four juniors. Find the
probability that the first two people selected
are freshmen and the third is a junior.
P(1st selection is freshman)
= 5/11
P(2nd selection is freshman) = 4/10
P(3rd selection is junior)
= 4/9
P(F, F, J) = 5/11 · 4/10 · 4/9 = 8/99
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Model Problems
A sack contains red marbles and green marbles. If
one marble is drawn at random, the probability that
it is red is 3/4. Five red marbles are removed from
the sack. Now, if one marble is drawn, the
probability that it is red is 2/3. How many red and
how many green marbles were in the sack at the
start?
x = original red marbles 5
y = original number of green marbles 15
3
x__
2
x-5
=
=
4
x+y
3
x+y-5
3x + 3y = 4x
3y = x
2x + 2y - 10 = 3x - 15
2y + 5 = x
3y = 2y + 5
y= 5
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
3y = x = 15
Model Problem
Several players start playing a game with a
full deck of 52 cards. Each player draws
two cards at random, one at a time, from
the full deck. Find the probability that a
player does not draw a pair.
1st card 2nd card
1
48
48
g

1
51
51
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Model Problem
Find the probability of rolling a die and getting a
number that is odd or greater than 2.
>2
odd
1
3
5
3
5
P(odd) = 3/6
2
4
6
P(> 2) = 4/6
A  B = {1, 3, 4, 5, 6}
1
3
5
4
6
n(A  B) = 5
n(U) = 6
n( A  B ) 5
P (> 2 or odd) 

n(U )
6
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
Model Problem
In a group of 50 students, 23 take math, 11 take
psychology, and 7 take both. If one student is
selected at random, find the probability that the
student takes math or psychology
P(A  B) = P(A) + P(B) - P(A  B)
23
P( M ) 
50
11
7
P ( Psy ) 
P ( M  Psy ) 
50
50
23 11 7 27
P ( M  Psy ) 



50 50 50 50
M
Psy
23
16
7
Aim: Fundamental Counting Principle
4
Course: Alg. 2 & Trig.
Determine the number of outcomes:
4 coins are tossed
A die is rolled and a coin is tossed
A tennis club has 15 members: 8 women
and seven men. How many different teams
may be formed consisting of one woman and
one man on each team?
A state issues license plates consisting of
letters and numbers. There are 26 letters,
and the letters may be repeated on a plate;
there are 10 digits, and the digits may be
repeated. The how many possible license
plates the state may issue when a license
consists of: 2 letters, followed by 3 numbers,
2 numbers
followed
byPrinciple
3 letters.
Aim: Fundamental
Counting
Course: Alg. 2 & Trig.
Model Problem
One bag contains 3 red and 4 white balls. A
2nd bag contains 6 yellow and 3 green balls.
One ball is drawn from each bag. Find the
probability of choosing a red and yellow
ball.
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.
The Product Rule
Aim: Fundamental Counting Principle
Course: Alg. 2 & Trig.