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3 The shaded region is B − AB. Thus A ∪ B = A ∪ (B − AB). 15 Suppose that in a community of 400 adults, 300 bike or swim or do both, 160 swim, and 120 swim and bike. What is the probability that an adult, selected at random from this community, bikes? Solution: Let A be the event that the person swims and B be the event that he or she bikes; then P (A ∪ B) = 300/400, P (A) = 160/400, and P (AB) = 120/400. 65. 16 A number is chosen at random from the set of integers {1, 2, . . , 1000}. , either 3 or 5 or both)?

Is Q a probability on S? Is R a probability on S? Why or why not? In its general case, the following exercise has important applications in coding theory, telecommunications, and computer science. 32. (The Hat Problem) A game begins with a team of three players entering a room one at a time. For each player, a fair coin is tossed. If the outcome is heads, a red hat is placed on the player’s head, and if it is tails, a blue hat is placed on the player’s head. The players are allowed to communicate before the game begins to decide on a strategy.

Was Rockford right to agree with the reporter? Explain why or why not? 8. A company has only one position with three highly qualified applicants: John, Barbara, and Marty. However, because the company has only a few women employees, Barbara’s chance to be hired is 20% higher than John’s and 20% higher than Marty’s. Find the probability that Barbara will be hired. 24 9. 10. Chapter 1 Axioms of Probability In a psychiatric hospital, the number of patients with schizophrenia is three times the number with psychoneurotic reactions, twice the number with alcohol addictions, and 10 times the number with involutional psychotic reaction.

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