Binomial and Geometric · Superchallenge problem

Two evenly matched players, A and B, play a series of games for a prize of $\$128$; each game is won by either player with probability $\dfrac12$, independently, and the first player to win $7$ games takes the prize. The series is stopped when A has $4$ wins and B has $3$. They split the prize in proportion to each player's probability of winning had the series continued. How many dollars does A receive?