Page 213 - Probability Demystified

202 CHAPTER 11 Game Theory

Let p be the probability that Player A plays the ace and 1 p be the
probability that Player A plays the three. Then

3p 5ð1 pÞ¼ 5p þ 7ð1 pÞ
3p 5 þ 5p ¼ 5p þ 7 7p
8p 5 ¼ 12p þ 7
8p þ 12p 5 ¼ 12p þ 12p þ 7
20p 5 þ 5 ¼ 7 þ 5
20p ¼ 12
12 3
p ¼ ¼
20 5
3
The value of the game when p ¼ is
5

3 3
3p 5ð1 pÞ¼ 3 51
5 5

3 2
¼ 3 5
5 5
1
¼ or $0:20
5
Player A will lose on average $0.20 per game. Thus, the game is not
fair.
Let s be the probability that Player B plays the two and 1 s be the
probability that Player B plays the four; then
3s 5ð1 sÞ¼ 5s þ 7ð1 sÞ
3s 5 þ 5s ¼ 5s þ 7 7s
8s 5 ¼ 12s þ 7
8s þ 12s 5 ¼ 12s þ 12s þ 7
20s 5 ¼ 7
20s 5 þ 5 ¼ 7 þ 5
20s ¼ 12
20s 12
¼
20 20
12 3
s ¼ ¼
20 5
Player B should play the two, 3 times out of 5.