You have a fair penny. It takes a dollar to play and to win, you must flip 4 heads in a row for 10 dollars. Should you play?

Expected Payoff: ((1/2)^4)*10 = 0.625 Cost = 1 ==> No

I will consider 2 cases. First case: we pay a buck to flip the coin 4 times and if we don't get 4 heads, we again pay a dollar to flip the coin 4 times and the results of the first round do not count (i.e getting THHH for the first round and HTTT for the second won't do).So here we pay 1 dollar for 4 flips. In this case we would expect HHHH to happen once in 16 rounds (since the probability of having HHHH is 1/16). Then expected value of this game = -16 (the cost of playing 16 rounds) + 10 = -6. Hence, we shouldn't play. Second case: we pay a dollar to flip the coin 4 times and if we do not get 4 heads we pay another buck for ONE more flip until we get 4 heads in a row. Let us work out the expected number of flips to get 4 heads in a row. Let Ei = expected number of flips to get i heads in a row. Then E4 = 0.5 * E3 + 0.5 * E4 + 1 (since if we get a head then the problem is reduced to finding the expected number of flips to get 3 heads in a row (ie E3), if we get a tail, then we start from scratch (ie E4)). Similarly, E3 = 0.5 * E2 + 0.5 * E4 + 1; E2 = 0.5 * E1 + 0.5 * E4 + 1; E1 = 0.5 * E0 + 0.5 * E4 + 1; E0 = 0 since the expected number of flips required to get 0 heads is simply zero. Solving these equations, we get that E4 = 30, ie we expect to flip the coin 30 times until we get 4 heads in a row. Remember that the first 4 flips cost us a dollar and EACH subsequent flip is an extra dollar. So the cost of this game would be 1 + 26 = 27. So the epected value of this game = - 27 + 10 = - 17. Therefore, we shouldn't play this game.