Two players $A$ and $B$ compete in a certain backgammon game. The competition is conducted in "Rounds" where each "Round" consists of $5$ single backgammon games. A round is won by the player who has more winnings in $5$ single games and final winner is the one who will first earn $12$ rounds.
Due to previous experience the probability that player $A$ wins in a single game against $B$ is $0,6$ but in order $A$ to challenge his opponent $B$, he offers him $5$ rounds won at the beginning of the competition. Is $A$ still reasonable that he will win the final prize of the competition? More precisely, what is the probability that $A$ will win?
I computed that $$ \eqalign{ P[A\ win\ a\ round] &= \binom{5}{3}0.6^3\cdot 0.4^2 + \binom{5}{4}0.6^4\cdot 0.4^1 \binom{5}{5}0.6^5\cdot 0.4^0 \cr &= 0.68256 \cr } $$