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Warm Up1) If the probability that a company will win a contract is .3, what is the

probability that it will not win the contract?

2) Suppose the probability that a construction company will be awarded a certain contract is .25, the probability that it will be awarded a second contract is .21, and the probability that it will get both contracts is .13. What is the probability that the company will win at least one of the two contracts?

3) If we randomly select one of the numbers 0-9 for the last digit of a phone number, find the probability that among the 10 possible outcomes, five are odd (1, 3, 5, 7, 9) or three are above 6 (7, 8, 9).

4) If the probabilities that Jane, Tom, and Mary will be chosen chairperson of the board are .5, .3, and .2 respectively, find the probability that the chairperson will be either Jane or Mary.

5) If one card is drawn from an ordinary deck of 52 playing cards, what is the probability that it will be either a club or a face card (king, queen, or jack)?

6) If the probabilities are, respectively, 0.92, 0.33, and 0.29 that a person vacationing in Washington, D.C. will visit the Capitol building, the Smithsonian Institution, or both, what is the probability that a person vacationing there will visit at least one of these buildings?

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Section 4-4 Multiplication Rule:

Basics

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Key Concept

If the outcome of the first event A somehow affects the probability of the second event B, it is important to adjust the probability of B to reflect the occurrence of event A.

The rule for finding P(A and B) is called the multiplication rule.

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Notation

P(A and B) =

P(event A occurs in a first trial and

event B occurs in a second trial)

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Tree DiagramsA tree diagram is a picture of the possible outcomes of a procedure, shown as line segments emanating from one starting point. These diagrams are helpful if the number of possibilities is not too large.

This figure summarizes the possible outcomes for a true/false followed by a multiple choice question.

Note that there are 10 possible combinations.

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Key Point – Conditional Probability

The probability for the second event B should take into account the fact that the first event A has already occurred.

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Notation for Conditional Probability

P(B A) represents the probability of event B occurring after it is assumed that event A has already occurred (read B A as “B given A.”)

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Definitions

Independent Events

Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. (Several events are similarly independent if the occurrence of any does not affect the probabilities of occurrence of the others.) If A and B are not independent, they are said to be dependent.

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Formal Multiplication Rule

P(A and B) = P(A) • P(B A)

(dependent events)

P(A and B) = P(A) • P(B)

(independent events)

Note that if A and B are independent events, P(B A) is really the same as P(B).

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Intuitive Multiplication Rule

When finding the probability that event A occurs in one trial and event B occurs in the next trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B takes into account the previous occurrence of event A.

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Examples-Independent Eventsa) Find the probability of tossing a coin 3 times and

getting all heads.b) The probability that a student will receive a state

grant is 1/3, while the probability that she will be awarded a federal grant is ½. If whether or not she receives one grant is not influenced by whether or not she receives the other, what is the probability of her receiving both grants?

c) Suppose a reputed psychic in an extrasensory perception experiment has called heads or tails correctly on ten successive tosses of a coin. What is the probability that guessing would have yielded this perfect score?

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Examples-dependent eventsa) The event that your car will start and the event

that you will get to class on time

b) A jury consists of nine persons who are native-born and three persons who are foreign-born. If two of the jurors are randomly picked for an interview, what is the probability that they will both be foreign born?

c) For b, what is the probability that if three of the twelve jurors are randomly picked for an interview, they will all be foreign-born?

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Examples-Independent Events

d) The probability that a car will skid on a bridge on a rainy day is .75. Today the weather station announced that there is a 20% chance of rain. What is the probability that it will rain today and that a car will skid on the bridge?

e) The probability that Ted will enroll in an English class in 1/3. If he does enroll in an English class, the probability that he would enroll in a math class is 1/5. What is the probability that he enrolls in English and Math?

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Applying the Multiplication Rule

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Small Samples from Large Populations

If a sample size is no more than 5% of the size of the population, treat the selections as being independent (even if the selections are made without replacement, so they are technically dependent).

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Summary of Fundamentals

In the addition rule, the word “or” in P(A or B) suggests addition. Add P(A) and P(B), being careful to add in such a way that every outcome is counted only once.

In the multiplication rule, the word “and” in P(A and B) suggests multiplication. Multiply P(A) and P(B),

but be sure that the probability of event B takes into account the previous occurrence of event A.