4.4 Exercise 48

Three persons play a game consecutively until one achieves his objective. Let E_i be the event of a success on the ith trial, and suppose $\left\{E_{i} : 1 \leq i\right\}$ is an independent class, with $P \left(E_{i}\right) = p_{1}$ for $i = 1 , 4 , 7 , \hdots$, $P \left(E_{i}\right) = p_{2}$ for $i = 2 , 5 , 8 , \hdots$, and $P \left(E_{i}\right) = p_{3}$ for $i = 3 , 6 , 9 , \hdots$. Let $A , B , C$ be the respective events the first, second, and third player wins.

1. Express $A , B$, and C in terms of the E_i.
2. Determine the probabilities in terms of $p_{1} , p_{2} , p_{3}$, then obtain numerical values in the case $p_{1} = 1 / 4$, $p_{2} = 1 / 3$, and $p_{3} = 1 / 2$.