A sponsor provides two prizes for a raffle. The first prize winner gets to

choose a probability p $\displaystyle \in$ [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 $\displaystyle X_1,X_2,...X_N$ be Bernoulli variables and letting N be a different random variable but couldn't quite get this to work. Could someone start me off?