Section 4-5
Multiplication Rule:
Complements and
Conditional Probability
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Key Concept
In this section we look at the probability
of getting at least one of some specified
event; and the concept of conditional
probability which is the probability of an
event given the additional information
that some other event has already
occurred.
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Complements: The Probability
of “At Least One”
 “At least one” is equivalent to “one or
more.”
 The complement of getting at least one
item of a particular type is that you get
no items of that type.
To find the probability of at least one of
something, calculate the probability of none,
then subtract that result from 1. That is,
P(at least one) = 1 – P(none).
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Example
a) Find the probability that at least one of
five employees in San Francisco has a
listed phone number. In San Francisco,
39.5% of numbers are unlisted.
b) Find the probability of getting at least one
girl if a couple plans to have 3 kids.
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Definition
A conditional probability of an event is a
probability obtained with the additional
information that some other event has already
occurred. P(B A) denotes the conditional
probability of event B occurring, given that
event A has already occurred, and it can be
found by dividing the probability of events A
and B both occurring by the probability of
event A:
P(B A) =
P(A and B)
P(A)
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Intuitive Approach to
Conditional Probability
The conditional probability of B given A can be
found by assuming that event A has occurred
and, working under that assumption,
calculating the probability that event B will
occur.
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Examples
a) Find the probability of getting 3 aces
when 3 cards are selected without
replacement.
b) If 90% of the households in a certain
region have answering machines and 50%
have both answering machines and call
waiting, what is the probability that a
household chosen at random and found to
have an answering machine also has call
waiting?
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The table below gives the results of a survey of the drinking
and smoking habits of 1200 college students. Rows and
columns have been summed.
Smokes
Doesn’t
smoke
Total
Drink
beer
315
585
Doesn’t
drink
165
135
Total
900
300
1200
480
720
1. What is the probability that someone in the group
smokes?
2. What is the probability a student smokes given that she is
a beer drinker?
3. What is the probability that someone drinks?
4. What is the probability that students drink given that they
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smoke?
To prove independence…
Two events are independent if:
P(B A) = P(B)
Two events are dependent if:
P(B A) is not equal to P(B)
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Examples
a) The probabilities that a student will get
passing grades in Math, English, or in
both are (let M=Math, E=English):
P(M) = .70, P(E) = .80, P(both) = .56. Check
if events M and E are independent.
b) The probabilities that it will rain or snow
on Christmas (C), New Years’ (N) or both
are:
P(C)=.60, P(N)=.60, P(both) = .42. Check if
events C and N are independent.
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•
Suppose that a consumer research organization has studied the
service under warranty provided by the 200 tire dealers in a
large city, and that their findings are summarized in the
following table:
Good service under warranty
Name-brand
Off-brand
TOTAL
•
•
Poor service under warranty
64
106
78
94
TOTAL
80
120
Fill in the rest of the table.
If one of these tire dealers is randomly selected, find the probabilities of:
i) Choosing a name-brand dealer
ii) Choosing a dealer who provides a good service under warranty
iii) Choosing a name-brand dealer who provides good services under
warranty
iv) Limit the choices to name-brand dealer and find the probability of
choosing a dealer who will provide good service under warranty.
v) What do you notice about part ii) in contrast to part iv)?
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