# Question

In Problem 70, suppose that the coin is tossed n times. Let X denote the number of heads that occur. Show that

P{X = i} = 1/n + 1 i = 0, 1, . . . , n

Make use of the fact that

when a and b are positive integers.

Problem 70

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;

(b) Both flips result in heads.

P{X = i} = 1/n + 1 i = 0, 1, . . . , n

Make use of the fact that

when a and b are positive integers.

Problem 70

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;

(b) Both flips result in heads.

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