4.4 Exercise 47

Two persons play a game consecutively until one of them is successful or there are ten unsuccesful plays. Let E_i be the event of a success on the ith play of the game. Suppose $\left\{E_{i} : 1 \leq i\right\}$ is an independent class with $P \left(E_{i}\right) = p_{1}$ for i odd and $P \left(E_{i}\right) = p_{2}$ for i even. Let A be the event the first player wins, B be the event the second player wins, and C be the event that neither wins.

1. Express $A , B$, and C in terms of the E_i.
2. Determine $P \left(A\right)$, $P \left(B\right)$, and $P \left(C\right)$ in terms of p₁, p₂, $q_{1} = 1 - p_{1}$, and $q_{2} = 1 - p_{2}$. Obtain numerical values for the case $p_{1} = 1 / 4$ and $p_{2} = 1 / 3$.
3. Use appropriate facts about the geometric series to show that $P \left(A\right) = P \left(B\right)$ iff $p_{1} = p_{2} / \left(1 + p_{2}\right)$.
4. Suppose $p_{2} = 0 . 5$. Use the result of part (c) to find the value of p₁ to make $P \left(A\right) = P \left(B\right)$ and then determine $P \left(A\right)$, $P \left(B\right)$, and $P \left(C\right)$.