Question

Two persons A and B are rolling die on the condition that the person who gets $3$ will win the game. If A starts the game, then find the probability of A and B respectively to win the game.

Answer 1

Probability of win$=\frac{1}{6}$

Probability of lose $=\frac{5}{6}$

if $A$ starts the game

Probability that $A$ wins $=\frac{1}{6}+\frac{5}{6}\times \frac{5}{6}\times \frac{1}{6}+\frac{5}{6}\times \frac{5}{6}\times \frac{5}{6}\times \frac{5}{6}\times \frac{1}{6}.........$ so on

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

Using infinite $GP$, we get

Probability that $A$ wins $\frac{1}{6}\times \frac{1}{1-{\left(\frac{5}{6}\right)}^2}=\frac{6}{11}$

Probability that $B$ wins $=1-\frac{6}{11}=\frac{5}{11}$