A random number p such that 0<p<1 is selected at random from a uniform distribution U[0,1]. A biased coin is then constructed such that the probability of heads on a single flip is p (thus 1-p for a tails). This coin is flipped twice and the result is HH. If the coin is flipped a third time, what is the probability of a third head? More precisely, what is the fair value of a lottery ticket that pays one dollar if the third flip is a head, and zero otherwise? What would you pay for it/ sell it for?