Two players take turns shooting at a target, with each shot by player i hitting the target with probability pi,
and miss with probability qi = 1 -pi, i = 1,2. We assume that the players’ shots are (mutually) independent.
Shooting ends when two consecutive shots hit the target (i.e. either when player 1 hits the target right after
player hits it, or the other way round).
If player 1 starts, we denote N1 the number of shots taken. if player 2 starts, we denote N2 the
number of shots taken.
Let mu i = E[Ni] denote the mean number of shots taken when player i shoots first, i = 1,2. Let Hi denote the
event {the i-th shot is a hit}.
(a) Express E[N1|(H1 intersection with H2)], E[N1|(H1 intersection with H2^c] and E[N1|H1^c ] in terms of mu 1,mu 2, p1, p2,q1,q2.
No need to necessarily use all the parameters. give a concise
explanation of your reasoning.
(b) Show that mu 1(1 -p1q2) = 1+ p1 + mu 2q1.
You might want to prove first that mu 1 = 2p1 p2 + (2+ mu 1)p1q2 + (1+ mu 2)q1 (using the results of
the previous question).
(c) Assume that p1 = 1/2 and p2 = 1/4 . Find mu 1 and mu 2.
First, derive formula for mu 2 similar as the one of the previous question (no justification needed
for this step); then solve a linear system of equations.