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

such that A gains one dollar when a head

occurs and B gains one dollar if a

comes up. Assume that player A needs

while player B needs

more dollars to win the game (both

and

are non-negative integers)

Let

be the event that

heads occur before

tails.

occurs if and only if there are at least

heads in the fi…rst

trials. Explain why this reasoning is correct. Then use the binomial distribution to write an expression for

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