Chapter 15
Probability Rules!
Copyright © 2010 Pearson Education, Inc.
The General Addition Rule
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When two events A and B are disjoint, we can
use the addition rule for disjoint events from
Chapter 14:
P(A  B) = P(A) + P(B)
However, when our events are not disjoint, this
earlier addition rule will double count the
probability of both A and B occurring. Thus, we
need the General Addition Rule.
Let’s look at a picture…
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Slide 15 - 3
The General Addition Rule (cont.)
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General Addition Rule:
 For any two events A and B,
P(A  B) = P(A) + P(B) – P(A  B)
The following Venn diagram shows a situation in
which we would use the general addition rule:
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Slide 15 - 4
The General Addition Rule (cont.)
A survey of college students found that 56% live on
campus, 62% are on a campus meal program,
and 42% do both.
What is the probability that:
 a student either lives or eats on campus?
 a student lives off campus and doesn’t have a
meal program?
 a student lives in a dorm but doesn’t have a meal
program?
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Slide 15 - 5
It Depends…
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Back in Chapter 3, we looked at contingency
tables and talked about conditional distributions.
When we want the probability of an event from a
conditional distribution, we write P(B|A) and
pronounce it “the probability of B given A.”
A probability that takes into account a given
condition is called a conditional probability.
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Slide 15 - 6
It Depends… (cont.)
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To find the probability of the event B given the
event A, we restrict our attention to the outcomes
in A. We then find the fraction of those outcomes
B that also occurred.
P(B|A) P(A  B)
P(A)
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Note: P(A) cannot equal 0, since we know that A
has occurred.
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Slide 15 - 7
It Depends… (cont.)
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P(B|A) P(A  B)
P(A)
Our survey found that 56% live on campus, 62%
have a meal program, 42% do both.
While dining in a campus facility open only to
students with meal programs you meet someone
new. What is the probability they live on campus?
Copyright © 2010 Pearson Education, Inc.
Slide 15 - 8
The General Multiplication Rule
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When two events A and B are independent, we
can use the multiplication rule for independent
events from Chapter 14:
P(A  B) = P(A) x P(B)
However, when our events are not independent,
this earlier multiplication rule does not work.
Thus, we need the General Multiplication Rule.
Copyright © 2010 Pearson Education, Inc.
Slide 15 - 9
The General Multiplication Rule (cont.)
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We encountered the general multiplication rule in
the form of conditional probability.
Rearranging the equation in the definition for
conditional probability, we get the General
Multiplication Rule:
 For any two events A and B,
P(A  B) = P(A)  P(B|A)
or
P(A  B) = P(B)  P(A|B)
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Slide 15 - 10
Independence
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Independence of two events means that the
outcome of one event does not influence the
probability of the other.
With our new notation for conditional
probabilities, we can now formalize this definition:
 Events A and B are independent whenever
P(B|A) = P(B). (Equivalently, events A and B
are independent whenever P(A|B) = P(A).)
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Slide 15 - 11
Independence
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Events A and B are independent
whenever P(B|A) = P(B).
Are living on campus and having a meal plan
independent? Are they disjoint?
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Slide 15 - 12
Independent ≠ Disjoint
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Disjoint events cannot be independent! Well, why not?
 Since we know that disjoint events have no outcomes
in common, knowing that one occurred means the
other didn’t.
 Thus, the probability of the second occurring changed
based on our knowledge that the first occurred.
 It follows, then, that the two events are not
independent.
A common error is to treat disjoint events as if they were
independent, and apply the Multiplication Rule for
independent events—don’t make that mistake.
Copyright © 2010 Pearson Education, Inc.
Slide 15 - 13
Depending on Independence
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It’s much easier to think about independent
events than to deal with conditional probabilities.
 It seems that most people’s natural intuition for
probabilities breaks down when it comes to
conditional probabilities.
Don’t fall into this trap: whenever you see
probabilities multiplied together, stop and ask
whether you think they are really independent.
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Slide 15 - 14
Drawing Without Replacement
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Sampling without replacement means that once one
individual is drawn it doesn’t go back into the pool.
 We often sample without replacement, which doesn’t
matter too much when we are dealing with a large
population.
 However, when drawing from a small population, we
need to take note and adjust probabilities accordingly.
Drawing without replacement is just another instance of
working with conditional probabilities.
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Slide 15 - 15
Practice with probabilities
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Consider drawing cards from a standard deck of cards,
with jokers removed.
One card is drawn. What is the probability it is an ace or a
red card?
Two cards are drawn without replacement. What is the
probability they are both aces?
If cards are drawn without replacement, what is the
probability I get 5 hearts in a row?
I draw a card and tell you it is red. What is the probability it
is a heart?
I draw a card and tell you it is a heart. What is the
probability it is red?
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Slide 15 - 16
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Are “red card” and “spade” independent? Mutually
exclusive?
Are “red card” and “ace” independent? Mutually
exclusive?
Are “face card” and “king” independent? Mutually
exclusive?
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Slide 15 - 17
Checking for Independence
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Students in grades 4, 5, and 6 were surveyed as to what
their primary goal was.
Grades
Popular
Sports
total
Boy
117
50
60
227
Girl
130
91
30
251
Total
247
141
90
478
If we select a student at random, what is
P(girl)
P(girl ∩ popular)
P(grades|boy)
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Slide 15 - 18
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P(sports|girl)
P(sports|boy)
P(sports)
If events are independent, the conditional
probability should be the same as the overall
probability.
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Slide 15 - 19
Death penalty
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28
Favor
Oppose
Republican
0.26
0.04
Democrat
0.12
0.24
Other
0.24
0.10
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#38
23% of adults smoke cigarettes.
57% of smokers and 13% of non-smokers
develop a certain lung condition by age 60.
How do these statistics indicate that lung
condition and smoking are not independent?
What is the probability that a randomly selected
60 year old has this lung condition?
#40 What is the probability that someone with the
condition was a smoker?
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Slide 15 - 21
Building a contingency table
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Drivers suspected of drunk driving are given tests
to determine blood alcohol content. 78% of
suspects get a breath test, 36% a blood test, and
22% get both.
Breath test
No breath test
total
Blood test
No blood test
total
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Are any of these numbers marginal probabilities?
Are any joint probabilities?
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Slide 15 - 22
Building a contingency table
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Are giving a blood test and a breath test mutually
exclusive?
Are giving the two tests independent?
Breath test
No breath test
total
Blood test
No blood test
total
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Slide 15 - 23
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In our text book, 48% of pages have a data
display, 27% have an equation, 7% have both.
Make a contingency table for the variables display
and equation.
What is the probability for a randomly selected
page with an equation to also have a data
display?
Are these disjoint events?
Are these independent events?
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Slide 15 - 24
Tree Diagrams
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A tree diagram helps us think through conditional
probabilities by showing sequences of events as
paths that look like branches of a tree.
Making a tree diagram for situations with
conditional probabilities is consistent with our
“make a picture” mantra.
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Slide 15 - 25
Tree Diagrams (cont.)
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Figure 15.5 is a nice
example of a tree
diagram and shows how
we multiply the
probabilities of the
branches together.
All the final outcomes
are disjoint and must
add up to one.
We can add the final
probabilities to find
probabilities of
compound events.
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Slide 15 - 26
Making a tree diagram
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A recent study found that in 77% of all accidents
the driver was wearing a seatbelt. 92% of those
drivers escaped serious injury, while only 63% of
those not wearing seatbelts escaped injury.
Construct a tree diagram.
What is the probability that a driver who was
seriously injured wasn’t wearing a seatbelt?
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Slide 15 - 27
What Can Go Wrong?
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Don’t use a simple probability rule where a
general rule is appropriate:
 Don’t assume that two events are independent
or disjoint without checking that they are.
Don’t find probabilities for samples drawn without
replacement as if they had been drawn with
replacement.
Don’t reverse conditioning naively.
Don’t confuse “disjoint” with “independent.”
Copyright © 2010 Pearson Education, Inc.
Slide 15 - 28
What have we learned?
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The probability rules from Chapter 14 only work
in special cases—when events are disjoint or
independent.
We now know the General Addition Rule and
General Multiplication Rule.
We also know about conditional probabilities and
that reversing the conditioning can give surprising
results.
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Slide 15 - 29
What have we learned? (cont.)
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Venn diagrams, tables, and tree diagrams help
organize our thinking about probabilities.
We now know more about independence—a
sound understanding of independence will be
important throughout the rest of this course.
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