Question

A pair of fair dice is rolled together till a sum of either $5$ or $7$ is obtained. If p denotes the probability that $7$ comes before $5$, find $5p.$

Answer 1

Let $A$ denote the event that a sum of $7$ occurs,
$B$ the event that a sum of $5$ occurs
And $C$ the event that neither a sum of $5$ nor of $7$ occurs.
We have
$P\left(A\right)=\frac{6}{36}=\frac{1}{6},P\left(B\right)=\frac{4}{36}=\frac{1}{9},P\left(C\right)=\frac{26}{36}=\frac{13}{18}p=P\left(A\right)+P\left(C\right).P\left(A\right)+P{\left(C\right)}^2.P\left(A\right)+...=\frac{P\left(A\right)}{1-P\left(C\right)}=\frac{1/6}{1-13/18}=\frac{3}{5}\Rightarrow 5p=3$