Coins with random bias

Suppose you are given a coin with random bias. The bias of the coin is uniformly distributed over [0, 1] and assume that the value of the bias does not change during the tosses.
(a) What is the probability that all three tosses are heads?
(b) What is the probability that the second toss is Head when the first toss turns out as Head?

Solution:
Let Xi, i=1, 2,3, denote the outcome of the i-th coin toss, then:

So the probability of three heads is 1/4.


So the probability of second toss being H when the first one turns out as H is 2/3.