Suppose that three contestants on a quiz show are each given

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?

Solution

The probbaility that exactly two of the contestants answered the question correctly is:

p*p=p^2

p is uniformly distributed over the interval (0, 1].