Question

$A$ and $B$ throw a pair of dice alternately. $A$ wins the game if he gets a total of $7$ and $B$ wins the game if he gets a total of $10$. If $A$ starts the game, then find the probability that $B$ wins.

Answer 1

$A$ win $7:1,6$

$6,1$

$p\left(A\right)=\frac{6}{36}=\frac{1}{6}$ $3,4$

$4,3$

$2,5$

$5,2$

$B$ win $10$

$4,6$ $P\left(B\right)=\frac{3}{36}$

$6,4$

$5,5$ $=\frac{1}{12}$

$B$ wins $\Rightarrow$ P(Alosc) P(B win)

$+P\left(Alosc\right)P\left(Blose\right)P\left(Alosc\right)P\left(Bwin\right)$

$+...$

$=\frac{5}{6}\times \frac{1}{12}+\frac{5}{6}\times \frac{11}{12}\times \frac{5}{6}\times \frac{1}{12}+...$

$=\frac{\frac{5}{72}}{1-\frac{55}{72}}=\frac{5}{17}$