SECTION 3.2 Conditional Probability and the Multiplication Rule 151 Using the Multiplication Rule to Find Probabilities In a recent year, there were 19,326 U.S. MD medical school seniors who applied to first-year post-graduate residency programs and submitted their residency program choices. Of these seniors, 18,108 were matched with residency positions, with about 75.6% getting one of their top three choices. Medical students rank the residency programs in their order of preference, and program directors in the United States rank the students. The term “match” refers to the process whereby a student’s preference list and a program director’s preference list overlap, resulting in the placement of the student in a residency position. (Source: National Resident Matching Program) 1. Find the probability that a randomly selected senior was matched with a residency position and it was one of the senior’s top three choices. 2. Find the probability that a randomly selected senior who was matched with a residency position did not get matched with one of the senior’s top three choices. 3. Would it be unusual for a randomly selected senior to be matched with a residency position and that it was one of the senior’s top three choices? SOLUTION Let A = 5matched with residency position6 and B = 5matched with one of top three choices6. So, P(A) = 18,108 19,326 and P(B|A) = 0.756. 1. The events are dependent. P1A and B2 = P1A2 # P1B0 A2 = a 18,108 19,326b10.7562 ≈ 0.708 So, the probability that a randomly selected senior was matched with one of the senior’s top three choices is about 0.708. 2. To find this probability, use the complement. P1B′0 A2 = 1 - P1B0 A2 = 1 - 0.756 = 0.244 So, the probability that a randomly selected senior was matched with a residency position that was not one of the senior’s top three choices is 0.244. 3. It is not unusual because the probability of a senior being matched with a residency position that was one of the senior’s top three choices is about 0.708, which is greater than 0.05. In fact, with a probability of 0.708, this event is likely to happen. TRY IT YOURSELF 5 In a jury selection pool, 65% of the people are female. Of these 65%, one out of four works in a health field. 1. Find the probability that a randomly selected person from the jury pool is female and works in a health field. Is this event unusual? 2. Find the probability that a randomly selected person from the jury pool is female and does not work in a health field. Is this event unusual? Answer: Page A38 EXAMPLE 5
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