Question

A pair of fair dice is rolled together till a sum of $7$ or $11$ is obtained. Let $p$ denote the probability that $7$ comes before $11$. Find the value of $4p$

Answer 1

Let $A,B,C$ denote the events a sum of $7$, a sum of $11$, neither $7$ nor $11$ occurs.
Then $P\left(A\right)=1/6,P\left(B\right)=1/18$ and $P\left(C\right)=7/9$.
$p=P\left(A\right)+P(C\cap A)+P(C\cap C\cap A)+...$
$=\frac{P\left(A\right)}{1-P\left(C\right)}=\frac{1/6}{1-7/9}=\frac{3}{4}$.