Question

In: Statistics and Probability

Fair game: The game with expected earning/loss equal to zero. A dice game costs $10 to...

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?

Expert Solution

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.

orchestra answered 3 years ago

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