You want to invent a gambling game in which a person rolls t

You want to invent a gambling game in which a person rolls two dice and is paid some money if the sum is 7, but otherwise he loses his money. How much should you pay him for winning a $1 bet if you want this to be a fair game, that is, to have expected value 0?

Solution

There are 6*6 = 36 possibilities in throwing 2 dice.

There are 6 ways to produce a 7:

1,6
2,5
3,4
4,3
5,2
6,1

Thus, P(7 sum) = 6/36 = 1/6 = P(win).

Thus, P(loss) = 1 - 1/6 = 5/6.

Let x = winning prize. [Hence, you actually win x - 1 dollars.]

Thus, the expected value is

E(x) = (1/6)(x - 1) + (5/6)(-1) = 0

1/6x - 1/6 - 5/6 = 0

1/6x - 1 = 0

Hence,

x = 6 [YOU SHOULD GIVE HIM $6 IF HE WINS.]