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Chapter 6: Problem 54

Three friends \((\mathrm{A}, \mathrm{B}\), and \(\mathrm{C})\) will participate in a round-robin tournament in which each one plays both of the others. Suppose that \(P(\) A beats \(B)=.7, P(\) A beats \(C)=.8\), \(P(\mathrm{~B}\) beats \(\mathrm{C})=.6\), and that the outcomes of the three matches are independent of one another. a. What is the probability that \(\mathrm{A}\) wins both her matches and that B beats C? b. What is the probability that A wins both her matches? c. What is the probability that A loses both her matches? d. What is the probability that each person wins one match? (Hint: There are two different ways for this to happen.)

Short Answer

Expert verified
a) The probability that A wins both matches and B beats C is \(0.7 * 0.8 * 0.6 = 0.336\). b) The probability that A wins both her matches is \(0.7 * 0.8 = 0.56\). c) The probability that A loses both her matches is \(0.3 * 0.2 = 0.06\). d) The probability that each person wins one match is \((0.7 * 0.6 * 0.2) + (0.3 * 0.4 * 0.8) = 0.268\).

Step by step solution

01

Part (a) Solution

The task is to find the probability that A wins both matches and B beats C. Since these events are independent, the combined probability is simply the product of individual probabilities. So take the probability of A winning against B, which is 0.7, multiply it by the probability of A winning against C, which is 0.8, and finally multiply it by the probability of B winning against C, which is 0.6.
02

Part (b) Solution

The task is to find the probability that A wins both her matches. Since the matches are independent, simply multiply the probability of A winning against B, which is 0.7, by the probability of A winning against C, which is 0.8.
03

Part (c) Solution

The task is to find the probability that A loses both her matches. The event of A losing a match is the complement of A winning. So the probability of A losing against B is \(1 - P(B)\), which is 0.3, and against C it's \(1 - P(C)\), which is 0.2. As these are also independent, simply multiply these two probabilities.
04

Part (d) Solution

The task is to find the probability that each person wins one match. There are two cases for this: either A beats B, B beats C, and C beats A, or C beats B, B beats A, and A beats C. Compute the probabilities for each case by multiplying the probabilities of those individual matches, and then add them for the final probability, as these two cases are mutually exclusive and therefore the probabilities can be added.

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Most popular questions from this chapter

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