Page 208 - Probability Demystified

CHAPTER 11 Game Theory 197

5
Then the probability of making a successful shot when p ¼ is
8

5 5
0:1p þ 0:7ð1 pÞ¼ 0:1 þ 0:71
8 8

5 3
¼ 0:1 þ 0:7
8 8
13
¼
40
13
Hence Player A will be successful 13 of the time. The value of the game is .
40 40
Player B should defend against a long shot 1 4 of the time and against a
lay-up shot 3 of the time. The solution is left as a practice question. See
4
Question 5.

PRACTICE

1. A simplified version of football can be thought of as two types of
plays. The offense can either run or pass and the defense can defend
against a running play or a passing play. The payoff yards gained for
each play are shown in the payoff box. Find the optimal strategy for
each and determine the value of the game.

Defense
Offense Against Run Against Pass

Run 2 5

Pass 10 6

2. In a game of paintball, a player can either hide behind a rock or in
a tree. The other player can either select a pistol or a rifle. The
probabilities for success are given in the payoff box. Determine the
optimal strategy and the value of the game.