7.11 Exercise 5

In the casino game keno, a machine chooses at random 20 numbers between 1 and 80 (inclusive) without replacement. Players try to predict which numbers will be chosen. Players don’t try to guess all 20, though; generally, they’ll try to predict between 1 and 10 of the chosen numbers. The amount won depends on the number of guesses they made and the number of guesses that were correct.

1. At one casino, a player can try to guess just 1 number. If that number is among the 20 selected, the player wins $2; otherwise, the player loses $1. What is the expected value?
2. At the same casino, if a player makes 2 guesses and they’re both correct, the player wins $14; otherwise, the player loses $1. What is the expected value?
3. Players can also make 3 guesses. If 2 of the 3 guesses are correct, the player wins $1. If all 3 guesses are correct, the player wins $42. Otherwise, the player loses $1. What is the expected value?
4. Which of these games is the best for the player? Which is the best for the casino?