Question

A and B throw alternatively a pair of dice, A wins if he throws a sum of 6 before B throws a sum of 7 and B wins if he throws sum of 7 before A throws a sum of 6. Find their respective chance of winning, If A begins.

Answer 1

$P\left(Sum6\right)=\frac{5}{36}$

$\left(1,5\right),\left(5,1\right),\left(2,4\right),\left(4,2\right),\left(3,3\right)$

$P\left(Sum7\right)=\frac{6}{36}=\frac{1}{6}$

$\left(1,6\right),\left(6,1\right),\left(2,5\right),\left(5,2\right),\left(3,4\right),\left(4,3\right)$

$\therefore P\left(Awins\right)=\frac{5}{36}+\frac{31}{36}\times \frac{5}{6}\times \frac{5}{36}+\frac{31}{36}\times \frac{5}{6}\times \frac{31}{36}\times \frac{5}{6}\times \frac{5}{36}$

$=\frac{\frac{5}{36}}{1-\frac{31}{36}\times \frac{5}{6}}=\frac{30}{61}$