In Problem 1, suppose that the coin is tossed n times. Let X denote the number of heads th...

In Problem 1, suppose that the coin is tossed n times. Let X denote the number of heads that occur. Show that

Problem 1

Consider an urn containing a large number of coins, and suppose that each of the coins has some probability p of turning up heads when it is flipped. However, this value of p varies from coin to coin. Suppose that the composition of the urn is such that if a coin is selected at random from it, then the p-value of the coin can be regarded as being the value of a random variable that is uniformly distributed over [0, 1]. If a coin is selected at random from the urn and flipped twice, compute the probability that

(a) the first flip results in a head;

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(b) both flips result in heads.