A and B throw one dice for a stake of Rs.11, which is to be won by the player who first throws a six.
The game ends when stake is won by A or B. If A has the first throw, what are their respective expectations?

The probability of throwing a 6 = $$\ \frac{\ \ 1}{\ 6}$$

The probability of not throwing a 6 = $$\ \frac{\ \ 5}{\ 6}$$

The probability that A wins = $$\ \frac{\ \ 1}{\ 6}+\ \frac{\ 5}{\ 6}\times\ \ \frac{\ 5}{6}\times\ \ \frac{\ 1}{6}+\ \frac{\ 5}{6}\ \times\ \ \frac{\ 5}{6}\times\ \ \frac{\ 5}{6}\times\ \ \frac{\ 5}{6}\times\ \frac{\ 1}{6}\ +\ ...............$$

= $$\ \frac{\ 1}{6}\left(1+\left(\ \frac{\ 5}{6}\right)^2+\left(\ \frac{\ 5}{6}\right)^4.........\right)$$

= $$\ \ \frac{\ 1}{6}\left(\ \dfrac{\ 1}{\ 1-\ \frac{\ 25}{36}}\right)$$ = $$\ \frac{\ 6}{11}$$

The expected return of A  = 11*$$\ \frac{\ 6}{11}$$ = 6

Hence the expected return of B = 11-6=5