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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”

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).

“At least one” is equivalent to “one or more.”

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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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DefinitionA 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.

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 smoke?

Drink beer

Doesn’t drink

Total

Smokes 315 165 480

Doesn’t smoke

585 135 720

Total 900 300 1200

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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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Examplesa) 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 Poor service under warranty TOTAL

Name-brand Off-brandTOTAL

• 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 dealerii) 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)? 

64 80

78 120

106 94