Download CHAPTER 5 Probability: What Are the Chances?

CHAPTER 5
Probability: What Are
the Chances?
5.2
Probability Rules (Part 1)
The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
Probability Rules
Learning Objectives
After this section, you should be able to:
 DESCRIBE a probability model for a chance process.
 USE basic probability rules, including the complement rule and the
addition rule for mutually exclusive events.
 USE a two-way table or Venn diagram to MODEL a chance process
and CALCULATE probabilities involving two events.
 USE the general addition rule to CALCULATE probabilities.
The Practice of Statistics, 5th Edition
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Probability Models
In Section 5.1, we used simulation to imitate chance behavior.
Fortunately, we don’t have to always rely on simulations to determine
the probability of a particular outcome.
Descriptions of chance behavior contain two parts:
The sample space S of a chance process is the set of all possible
outcomes.
A probability model is a description of some chance process that
consists of two parts:
• a sample space S and
• a probability for each outcome.
The Practice of Statistics, 5th Edition
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Example: Building a probability model
Sample Space
36 Outcomes
The Practice of Statistics, 5th Edition
Since the dice are fair, each outcome is equally
likely.
Each outcome has probability 1/36.
4
Probability Models
Probability models allow us to find the probability of any collection of
outcomes.
An event is any collection of outcomes from some chance
process. That is, an event is a subset of the sample space. Events
are usually designated by capital letters, like A, B, C, and so on.
If A is any event, we write its probability as P(A).
In the dice-rolling example, suppose we define event A as “sum is 5.”
There are 4 outcomes that result in a sum of 5.
Since each outcome has probability 1/36, P(A) = 4/36.
Suppose event B is defined as “sum is not 5.” What is P(B)?
P(B) = 1 – 4/36 = 32/36
The Practice of Statistics, 5th Edition
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Flipping Coins
Imagine flipping a fair coin three times.
Problem: Give a probability model for this chance process.
Solution: There are eight possible outcomes when we flip the coin three times:
HHH HHT HTH HTT
TTT
TTH
THT
THH
Because the coin is fair, each of these eight outcomes will be equally likely and have probability
1/8.
The Practice of Statistics, 5th Edition
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Basic Rules of Probability
• The probability of any event is a number between 0 and 1.
• All possible outcomes together must have probabilities whose sum
is exactly 1.
• If all outcomes in the sample space are equally likely, the probability
that event A occurs can be found using the formula
P(A) =
number of outcomes corresponding to event A
total number of outcomes in sample space
• The probability that an event does not occur is 1 minus the
probability that the event does occur.
• If two events have no outcomes in common, the probability that one
or the other occurs is the sum of their individual probabilities.
Two events A and B are mutually exclusive (disjoint) if they
have no outcomes in common and so can never occur together—
that is, if P(A and B ) = 0.
The Practice of Statistics, 5th Edition
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Basic Rules of Probability
We can summarize the basic probability rules more concisely in
symbolic form.
Basic Probability Rules
•For any event A, 0 ≤ P(A) ≤ 1.
•If S is the sample space in a probability model,
P(S) = 1.
•In the case of equally likely outcomes,
number of outcomes corresponding to event A
P(A) =
total number of outcomes in sample space
•Complement rule: P(AC) = 1 – P(A)
•Addition rule for mutually exclusive events: If A and B are
mutually exclusive,
P(A or B) = P(A) + P(B).
The Practice of Statistics, 5th Edition
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AP® Statistics scores
Randomly select a student who took the 2013 AP® Statistics exam and record the student’s score.
Here is the probability model:
1
2
3
4
5
Score:
Probability: 0.235 0.188 0.249 0.202 0.126
Problem:
(a) Show that this is a legitimate probability model.
(b) Find the probability that the chosen student scored 3 or better.
Solution:
(a) All the probabilities are between 0 and 1 and the sum of the probabilities is 1, so this is a
legitimate probability model.
(b) There are two ways to find this probability:
By the addition rule, P(3 or better) = 0.249 + 0.202 + 0.126 = 0.577
By the complement rule and addition rule, P(3 or better) = 1 – P(2 or less) =
1 – (0.235 + 0.188) = 1 – 0.423 = 0.577.
The Practice of Statistics, 5th Edition
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Homework…..
Pg. 309: CYU #1-3
Pg. 314 – 315: #39, 41, 43, 45, 47
The Practice of Statistics, 5th Edition
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CHAPTER 5
Probability: What Are
the Chances?
5.2
Probability Rules (Part 2)
The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
Two-Way Tables and Probability
When finding probabilities involving two events, a two-way table can
display the sample space in a way that makes probability calculations
easier.
Consider the example on page 309. Suppose we choose a student at
random. Find the probability that the student (a) has pierced ears.
(b) is a male with pierced ears.
(c) is a male or has pierced
ears.
Define events A: is male and B: has pierced ears.
(a)
(b) Each
(c)
We want
student
to find
is equally
P(male likely
or
and
pierced
pierced
to be ears),
chosen.
ears),
that
that
103
is,is,
students
P(A
P(A
orand
B).have
There
B).
pierced
are
ears.
in
So,
theP(pierced
classofand
ears)
103
=
individuals
P(B)
103/178.
with
pierced
ears.There
Look90atmales
the intersection
the
“Male”
row=and
“Yes”
column.
are 19 males
with pierced
ears. So,
P(A
and B)
= 19/178.
However,
19 males
have pierced
ears
– don’t
count
them twice!
P(A or B) = (19 + 71 + 84)/178. So, P(A or B) = 174/178
The Practice of Statistics, 5th Edition
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Who owns a home?
What is the relationship between educational achievement and home ownership? A random
sample of 500 U.S. adults was selected. Each member of the sample was identified as a high
school graduate (or not) and as a home owner (or not). The two-way table displays the data.
High
school
graduate
Homeowner
Not a homeowner
Total
221
89
310
Not a
high
Total
school
graduate
119
340
71
160
190
500
Problem: Suppose we choose a member of the sample at random. Find the probability that the
member:
(a) is a high school graduate.
(b) is a high school graduate and owns a home.
(c) is a high school graduate or owns a home.
We will define event G as being a high school graduate and event H as being a homeowner.
The Practice of Statistics, 5th Edition
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High
school
graduate
Homeowner
Not a homeowner
Total
221
89
310
Not a
high
Total
school
graduate
119
340
71
160
190
500
Problem: Suppose we choose a member of the sample at
random. Find the probability that the member:
(a) is a high school graduate.
(b) is a high school graduate and owns a home.
(c) is a high school graduate or owns a home.
Solution:
(a) Because 310 of the 500 members of the sample graduated from high school, P(G) = 310/500.
(b) Because 221 of the 500 members of the sample graduated from high school and own a home,
P(G and H) = 221/500.
(c) Because there are 221 + 89 + 119 = 429 people who graduated from high school or own a
home, P(G or H) is 429/500.
The Practice of Statistics, 5th Edition
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General Addition Rule for Two Events
We can’t use the addition rule for mutually exclusive events unless the
events have no outcomes in common.
General Addition Rule for Two Events
If A and B are any two events resulting from some chance
process, then
P(A or B) = P(A) + P(B) – P(A and B)
The Practice of Statistics, 5th Edition
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Venn Diagrams and Probability
Because Venn diagrams have uses in other branches of mathematics,
some standard vocabulary and notation have been developed.
The complement AC contains exactly the outcomes that are not in A.
The events A and B are mutually exclusive (disjoint) because they do not
overlap. That is, they have no outcomes in common.
The Practice of Statistics, 5th Edition
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Venn Diagrams and Probability
The intersection of events A and B (A ∩ B) is the set of all outcomes
in both events A and B.
The union of events A and B (A ∪ B) is the set of all outcomes in either
event A or B.
Hint: To keep the symbols straight, remember ∪ for union and ∩ for intersection.
The Practice of Statistics, 5th Edition
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Venn Diagrams and Probability
Recall the example on gender and pierced ears. We can use a Venn
diagram to display the information and determine probabilities.
Define events A: is male and B: has pierced ears.
The Practice of Statistics, 5th Edition
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Who owns a home?
Here is the two-way table summarizing the relationship between educational status and home
ownership from the previous example. Our events of interest were G: is a high school graduate
and H: is a homeowner.
High
school
graduate
Homeowner
Not a homeowner
Total
221
89
310
Not a
high
Total
school
graduate
119
340
71
160
190
500
The four distinct regions in the Venn diagram below correspond to the four cells in the two-way
table as follows:
Region in Venn
diagram
In the intersection of
two circles
Inside circle G,
outside circle H
Inside circle H,
outside circle G
Outside both circles
The Practice of Statistics, 5th Edition
In words
In symbols
Graduate and owns home
G H
Graduate and doesn’t own home
G  HC
Not a graduate but owns home
GC  H
Not a graduate and doesn’t own
home
GC  HC
Count
221
89
119
71
19
Draw a Venn Diagram based on earlier information…
89
221
119
71
Are all values in the two-way table accounted in the Venn Diagram?
If we add all these numbers together, do we get the total number of
people in Sample space?
The Practice of Statistics, 5th Edition
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Homework….
Pg. 311 CYU: #1-3
Pg. 315 – 317: 49, 51, 53, 55, 57 – 60
The Practice of Statistics, 5th Edition
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Probability Rules
Section Summary
In this section, we learned how to…
 DESCRIBE a probability model for a chance process.
 USE basic probability rules, including the complement rule and the
addition rule for mutually exclusive events.
 USE a two-way table or Venn diagram to MODEL a chance process
and CALCULATE probabilities involving two events.
 USE the general addition rule to CALCULATE probabilities.
The Practice of Statistics, 5th Edition
22