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Section 4-4

Section 4-4. Multiplication Rule: Basics. NOTATION. P ( A and B ) = P (event A occurs in a first trial and event B occurs in a second trial). EXAMPLES. Suppose that you first toss a coin and then roll a die. What is the probability of obtaining a “Head” and then a “2”?

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Section 4-4

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  1. Section 4-4 Multiplication Rule: Basics

  2. NOTATION P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial)

  3. EXAMPLES • Suppose that you first toss a coin and then roll a die. What is the probability of obtaining a “Head” and then a “2”? • A bag contains 2 red and 6 blue marbles. Two marbles are randomly selected from the bag, one after the other, without replacement. What is the probability of obtaining a red marble first and then a blue marble?

  4. CONDITIONAL PROBABILITY • If event B takes place after it is assumed that event A has taken place, we notate this by B|A. This is read “B, given A.” • P(B|A) represents the probability of event B occurring after it is assumed that event A has already occurred.

  5. INDEPENDENT AND DEPENDENT EVENTS • Two events A and B are independent if the occurrence of one event does not affect the probability of the occurrence of the other. • Several events are independent if the occurrence of any does not affect the occurrence of the others. • If A and B are not independent, they are said to be dependent.

  6. FORMAL MULTIPLICATION RULE P(A and B) = P(A) · P(B|A) NOTE: If events A and B are independent, then P(B|A) = P(B) and the multiplication rule simplifies to P(A and B) = P(A) · P(B)

  7. APPLYING THE MULTIPLICATION RULE

  8. INTUITIVE MULTIPLICATION RULE When finding the probability that event A occurs in one trial and 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.

  9. EXAMPLES • What is the probability of drawing an “ace” from a standard deck of cards and then rolling a “7” on a pair of dice? • In the 105th Congress, the Senate consisted of 9 women and 91 men, If a lobbyist for the tobacco industry randomly selected two different Senators, what is the probability that they were both men? • Repeat Example 2 except that three Senators are randomly selected.

  10. EXAMPLE In a survey of 10,000 African-Americans, it was determined that 27 had sickle cell anemia. • Suppose we randomly select one of the 10,000 African-Americans surveyed. What is the probability that he or she will have sickle cell anemia? • If two individuals from the group are randomly selected, what is the probability that both have sickle cell anemia? • Compute the probability of randomly selecting two individuals from the group who have sickle cell anemia, assuming independence.

  11. TREATING DEPENDENT EVENTS AS INDEPENDENT:THE 5% GUILDELINE FOR CUMPBERSOME CALCULATIONS 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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