4.1. Suppose that three contestants on a quiz show are each given the same question, and that...

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4.1. Suppose that three contestants on a quiz show are each given the same question, and that each answers it correctly, independently of the others, with probability p. But the difficulty of the question is itself a random variable, so let us suppose, for the sake of illustration, that p is uniformly distributed over the interval (0, 1]. What is the probability that exactly two of the contestants answer the question correctly?

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