Two people want to play a game in which the expected amount of money that each of them wins is equal to zero. After having chosen a number x, the game is played as follows: Player 1 rolls a fair die, independently, three times.
- If none of the three rolls results in 6, then Player 1 pays one dollar to Player 2.
- If exactly one of the rolls results in 6, then Player 2 pays one dollar to Player 1.
- If exactly two rolls result in 6, then Player 2 pays two dollars to Player 1.
- If all three rolls result in 6, then Player 2 pays x dollars to Player 1.
Determine the value of x.
So I determined the probability of each roll outcome but I am not sure how to figure out what the value of x is with this information.
P(no sixes) = 125/216
P(1 six) = 75/216
P(2 sixes) = 15/216
P(3 sixes) = 1/216