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3-7. An urn contains 15 balls numbered 1 through 15. A ball is drawn from the urn and kept aside. Another ball is then drawn from the same urn. What is the probability that the first ball bears an even number and the second ball bears an odd number? 3-8. An urn contains 17 balls marked LOSE and 3 balls marked WIN. You and an opponent take turns selecting a single ball at random from the urn without replacement. July 16, 2014 M01_HOGG2358_09_GE_C01 page 36 36 Chapter 1 Probability The person who selects the third WIN ball wins the game.

The proofs of parts (b) and (c) are left as exercises. Before extending the definition of independent events to more than two events, we present the following example. 4-4 An urn contains four balls numbered 1, 2, 3, and 4. One ball is to be drawn at random from the urn. Let the events A, B, and C be defined by A = {1, 2}, B = {1, 3}, and C = {1, 4}. Then P(A) = P(B) = P(C) = 1/2. Furthermore, P(A ∩ B) = 1 = P(A)P(B), 4 P(A ∩ C) = 1 = P(A)P(C), 4 1 = P(B)P(C), 4 which implies that A, B, and C are independent in pairs (called pairwise independence).

90; however, this test indicates that an impurity is there when it is not about 5% of the time. The chemist produces compounds with the impurity about 20% of the time; that is, 80% do not have the impurity. A compound is selected at random from the chemist’s output. The test indicates that an impurity is present. What is the conditional probability that the compound actually has an impurity? 5-8. A store sells four brands of tablets. The least expensive brand, B1 , accounts for 40% of the sales.