A sponsor provides two prizes for a raffle. The first prize winner gets to
choose a probability p [1/e, 1 − 1/e]. A sequence of independent coin flips with probability p for a head are then made. The winner receives £10 for each flip up to and including the first head. The same coin is tossed in another independent
sequence and the second prize winner receives £5 for each flip up to and including
the first head in this sequence.
(i) Write down the probability mass functions of the amounts of the two payments
U for the first prize and V for the second prize.
For U, I was thinking along the lines of letting be Bernoulli variables and letting N be a different random variable but couldn't quite get this to work. Could someone start me off?