In: Statistics and Probability
Fair game: The game with expected earning/loss equal to zero.
A dice game costs $10 to play. You get to roll two dice and if you roll a 2 or a 12 you win $100, if you roll a 7 you win $20 and there is no prize otherwise. Is this a fair game?
In a fair game, expected earning is zero. The dice game costs $10 to play. Also, rolling a 2 or a 12 will give the player $100, and rolling a 7 would give $20. No prize for any other outcome. The probability space for this game will be . Each outcome is equally likely having probability .
Now, below table organizes the probability of outcomes of sums of the dice.
Sum | Probability |
2 | |
3 | |
4 | |
5 | |
6 | |
7 | |
8 | |
9 | |
10 | |
11 | |
12 |
It is to be noted that, the total earning have to be zero for this to be a fair game. The total expectation would include the debit of $10 to play the game. Hence, we have the following total expected earning as , where Xi is the earning from getting the sum equal to the i. Hence, we have or or or or or or or . Hence, not only the expected earning is not zero, it is negative, indicating that playing would lead to a loss in earning of the player. Hence, the game is not fair.