Question

A pair of fair dice is rolled together till a sum a either 5 or 7 is obtained. If p denoted the probability that 7 comes before 5, find 15p

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 a sum of 7 occurs.
We have $P\left(A\right)=\frac{6}{36}=\frac{1}{6},P\left(B\right)=\frac{4}{36}=\frac{1}{9},P\left(C\right)=\frac{26}{36}=\frac{13}{18}$.
Thus,
$p=P(A$ or $(C\cap A)$ or $(C\cap C\cap A)$ or $....)$
$=P\left(A\right)+P\left(C\right)P\left(A\right)+P\left(C{)}^{2}P\right(A)+....$
$=\frac{P\left(A\right)}{1-P\left(C\right)}=\frac{1/6}{1-13/18}=\frac{3}{5}$.