Page 215 - Probability Demystified

204 CHAPTER 11 Game Theory

Thus, if the vendor selects Main St. 5 times out of 11, he will make
$18 2 :
11
Let s be the probability that the inspector shows up at Main St. and
1 s be the probability that the inspector shows up at Railroad Ave.
Then
50s þ 100ð1 sÞ¼ 75s 50ð1 sÞ

50s þ 100 100s ¼ 75s 50 þ 50s

150s þ 100 ¼ 125s 50

150s 125s þ 100 ¼ 125s 125s 50

275s þ 100 ¼ 50
275s þ 100 100 ¼ 50 100

275s ¼ 150

275s 150
¼
275s 275
6
s ¼
11

The inspector’s optimal strategy will be to show up at the Main St. 6
times out of 11.

5. Let s be the probability that Player B defends against the long shot
and 1 s be the probability that Player B defends against the lay-up.
Then
0:1s þ 0:4ð1 sÞ¼ 0:7s þ 0:2ð1 sÞ

0:1s þ 0:4 0:4s ¼ 0:7s þ 0:2 0:2s

0:3s þ 0:4 ¼ 0:5s þ 0:2

0:3s 0:5s þ 0:4 ¼ 0:5s 0:5s þ 0:2

0:8s þ 0:4 ¼ 0:2