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Math  340                Exam 1                                  Fall 2000

Problem 1. (20 points)

Suppose that A and B are two events with P(A)=0.48 and P(A\cup B)=0.75.

a. For what value of P(B) would A and B be mutually exclusive?

b. For what value of P(B) would A and B be independent?

Problem 2. (20 points)

Consider two urns. The first urn contains 5 black balls and 2 white balls. The second urn contains
3 black balls and 8 white balls. An urn is chosen at random, and a ball is chosen at random from
that urn. What is the probability that the ball drawn is white?

Problem 3.(30 points)

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The fraction of the people in a population who have a certain disease is 0.5%. A diagnostic test is
available to test for the disease. But for a healthy person the chance of being falsely diagnosed as
having the disease is 0.25%, while for someone with the disease the chance of being falsely
diagnosed as healthy is 22%. Suppose the test is performed on a person selected at random from
the population.

a. What is the probability that the test shows a positive result (namely, the person is
diagnosed as diseased, perhaps correctly, perhaps not)?

b. Suppose the test shows a positive result. What is the probability that the person tested
has the disease?

Problem 4. (30 points)

a. A probabilistic experiment consists of drawing 100 cards with replacement from a full
deck of cards and counting the number of aces drawn.  Approximate the probability that we
observe at least 16 aces in 100 draws.

b. Now suppose that you repeat the experiment in part (a) every day for a year (assume a
non-leap year).  Approximate the probability that you observe 16 or more aces (on a day) at
least three times during the year.



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