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Chapter 6
Probability: Understanding Random Situations
page 6-1
Chapter 6
Probability
1
Chapter 6
Probability: Understanding Random Situations
page 6-2
Probability as Guide For Action
Select Target
Population
Statistics
Understand
Target Population
Probability
Act On
Target Population
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Chapter 6
Probability: Understanding Random Situations
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Basic Notions
 Random Experiment
Each observation is a ___________ ___________
from the target population
 Sample Space
A list of all possible outcomes
 Event
Any subset of the Sample Space
 Probability of an Event
A number between 0 and 1
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Chapter 6
Probability: Understanding Random Situations
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Example: I just purchased an iPod Shuffle, and
purchased three songs of my favorite.
Song 1:
Song 2:
Song 3:
A. What is the probability that next two songs are
(1) the same?
(2) different?
B. What is the probability of (1) and (2),
respectively, if I buy four songs?
C. If I want to make the probability of (1) for no
more than 5%, how many songs do I have to
buy?
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Chapter 6
Probability: Understanding Random Situations
page 6-5
Determining Probabilities
– Three Sources
1: Relative Frequency
 Law of Large Numbers
RE: Toss a Fair Coin n times. As n gets larger,
relative frequency gets closer to probability
Relative frequency
1
Probability = 0.5
0
0
50
100
150
200
Number n of times random experiment was run
Relative Frequency  Probability + / - SD
SD: From Table 6.3.1 on page 197:
n = 10
n = 100
n = 1000
Prob = 0.5
Prob =
0.25 / 0.75
Prob =
0.10 or 0.9
0.16
0.05
0.02
0.14
0.04
0.01
0.09
0.03
0.01
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Chapter 6
Probability: Understanding Random Situations
page 6-6
2 Theoretical Probability
•
The physical condition of the random experiment
Example: The Equally Likely Rule
If all N possible outcomes in the sample space are equally
likely, then the probability of any event A is:
Prob (A) = (# of outcomes in A) / N
 There are 35 defects in a production lot of 400.
Choose item at random.
Prob(defective) = 35/400 = 0.0875
 Flip a fair coin.
Prob(heads) = 1/2
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Chapter 6
Probability: Understanding Random Situations
page 6-7
3:Subjective Probability
 An Expert’s opinion
Example:
CFO: What do you think about the state of the economy
and the company profit?
Corporate economist forecasts:
Economy
Great
Good
Average
Poor
Company Profit
10
3
1
-8
Probability
0.10
0.25
0.40
0.25
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Chapter 6
Probability: Understanding Random Situations
page 6-8
Calculating Probabilities
A. Desired Probabilities:
1. Complementary Event
2. Union of Two Events
3. Intersection of Two Events
(A) Two Independent Events
(B) Two Mutually Exclusive Events
4. Conditional Probability
5. Sequential Events
B. Tools:
1. Venn Diagram
2. 2 by 2 table
3. Probability Trees
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Chapter 6
Probability: Understanding Random Situations
page 6-9
1. Complement of an Event, A
 The event, “not A”
 Venn diagram: A (in circle), “not A” (shaded)
not A
A
 Prob(not A) =
 Example:
If Prob (Succeed) = 0.7, then
Prob (Fail) =
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Chapter 6
Probability: Understanding Random Situations
page 6-10
2. Union of Two Events, A and B
 The event, “union” : either or both occur
 Venn diagram: Union “A or B” (shaded)
A
B
 P( A or B) = ___________________________
 Explaining the Formula When P(A and B) > 0
=
+
-
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Chapter 6
Probability: Understanding Random Situations
page 6-11
3. Intersection of Two Events, A and B
 The event, both A and B occur.
 P(A and B) > 0
 Venn diagram: Intersection “A and B” (shaded)
A
B
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Chapter 6
Probability: Understanding Random Situations
page 6-12
3 (A) Two Mutually Exclusive Events
 Two events A and B are Mutually Exclusive if they
cannot both happen:
 Venn Diagram
A
B
 Prob(A and B) = ______________________
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Chapter 6
Probability: Understanding Random Situations
page 6-13
3 (B) Independent Events
 Two Events, A and B are in Independent, if
information about one does not change the
probabilityof the other.
 Can a Venn Diagram show independent two events?
 Prob (A and B) = Prob (A)  Prob (B)
Two events are Dependent, if:
Prob (A and B) = Prob (A)  Prob (B)
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Chapter 6
Probability: Understanding Random Situations
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4. Conditional Probability
 Venn Diagram
A is now represented by A and B, since B must
happen
A
B
Probability of Event A occurring given Event B has
occurred.
Example:
A{ a customer purchases a Dryer, say}) given
B { S/he has purchases a Washer}

P  Aand B 
P  B given A   P  B | A  
P  A
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Chapter 6
Probability: Understanding Random Situations
page 6-15
Practice Exercises
Exercise 1: Two events A and B are mutually exclusive,
and P(A) = 0.3 P(B) = 0.4.
Find the probability that:
1. events A and B occur
2. event A occurs given that B occurs
3. event A or B occurs
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Chapter 6
Probability: Understanding Random Situations
page 6-16
Exercise 2: Two events A and B are independent.
P(A) = 0.3 and P(B) = 0.4.
Find the probability that:
1. events A and B occur
2. event A occurs given that B
occurs
3. event A or B occurs
4. event A occurs given that B did not occur
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Chapter 6
Probability: Understanding Random Situations
page 6-17
Exercise 3 : Event S = smoke Event H= heart disease
P(S) = 0.3 P(H) = 0.3 P(S and H) = 0.2
1. Define each event determined by two circles.
2. Find the probability that a selected person smokes
or has heart disease.
3.
Find the probability that a selected person has
heart disease given the person smokes.
4. Find the probability that a selected person does not
have heart disease given the person does not smoke.
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Chapter 6
Probability: Understanding Random Situations
page 6-18
Probability Trees
 Sequential: Event A occurring or not is first
observed. Then event B or not B is observed.
 P(A and B) = P(A) * P(B | A)
 Tree Diagram
Event B
Event A
P(A)
Yes
No
s
Ye
No
P(not A)
Yes
n
P(B give
P(“not B”
A)
given A)
”)
n “not A
e
v
i
g
B
(
P
No P(“no
t B” given
“not A”)
Prob(A and B)
Prob(A and “not B”)
Prob(“not A” and B)
Prob(“not A”
and “not B”)
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