Flipping Coins

There's a machine that keeps flipping two unbaised coins at once (i.e. 4 possible outcome per operation). Two guys, A and B. A wins if a (H,H) combination comes up. B wins if two consecutive (H,T) combinations come up (by (H,T) I mean one head and one tail from either of the coins). What's the probability of A winning?



Solution:

This one is tricky is you condition by when the game ends, because the states get mangled up. Easier is to do the following.

Let P(A) be the probability that A wins. Then, we have:

P(A) = 0.25 + 0.5 * (0.25 * 1 + 0.5 * 0 + 0.25 * P(A) ) + 0.25 * P(A)

explanation:

1st term on right: 0.25: Prob of (H,H)
2nd term on right: if either (H,T) or (T,H), which has prob 0.5, A wins if (H,H) (0.25 prob), loses if either (H,T) or (T,H) again (0.5 prob), or the game essentially resets if (T,T), (0.25 prob).
3rd term on right: if (T,T), game essentially restarts, since (T,T) doesn't help B.

Solving for P(A) we get 60%.