Expected Value Question

Question

A player pays $4 to play the following game: He rolls a die and receives $6 if he tosses and a 5 and $2 for anything else. Find the player’s expected net winnings for the game. Is the game fair? If not, what would the cost of the game be so that it is fair?

Answer 1

On 5, the net result is +2 and -2 otherwise

Prob(5) = 1/6

1/6 * 2 + 5/6 * -2 = 2/6 + -10/6 = -8/6 = -4/3 = $-1.33

let x be the cost to play the game

1/6 (x-6) + 5/6 * (x-2) = 0

(1/6)x - 1 + (5/6)x - 10/6 = 0

(1/6)x - 6/5 + (5/6)x - 10/6 = 0

x - 16/6 = 0

x = 16/6 = 8/3 = $2.66

check:

(1/6)( 6 - 2.66) + (5/6)(-0.66) = (1/6)(6 - 2 and 2/3) - (5/6)(-2/3)

= (1/6)( 3 and 1/3) - 10/18

= (1/6)(10/3) - 10/18

= 10/18 - 10/18

= 0