Originally Posted by

**akolman** Hello, I am lost with the following problem.

Consider two players, player A and player B, in a game that involves tossing a (not necessarily fair) coin with $\displaystyle p = P(H)$ such that A gains one dollar when a head $\displaystyle (H)$ occurs and B gains one dollar if a $\displaystyle T (tail)$ comes up. Assume that player A needs $\displaystyle n$ while player B needs $\displaystyle m$ more dollars to win the game (both $\displaystyle n$ and $\displaystyle m$ are non-negative integers)

Let $\displaystyle A$ be the event that $\displaystyle n$ heads occur before $\displaystyle m$ tails. $\displaystyle A$ occurs if and only if there are at least $\displaystyle n$ heads in the fi…rst $\displaystyle n +m- 1$ trials. Explain why this reasoning is correct. Then use the binomial distribution to write an expression for $\displaystyle P(A) = P_{n,m}.$

I know it is a very long and wordy problem, any hint will be appreciated, Thanks.