In rolling a pair of dice, suppose you win $5 if the sum is 10 but lose $2 if the roll is "doubles." If the price to play is $1, would it be a fair game? Justify your answer.

The probability of throwing a "10" with a pair of dice is 3/36 = 1/12. The probability of throwing "doubles" is 6/36 = 1/6.

Probable gain for $1 bet =
(1/12)*5 - (1/6)*2 = $ 1/12

The game favors the bettor

To determine whether the game is fair, we need to calculate the expected value of playing the game. The expected value represents the average amount of money you can expect to win or lose over the long run.

To find the expected value, we need to consider the probability of each outcome and multiply it by the corresponding monetary outcome.

Let's break down the possibilities and their probabilities:

1. Getting a sum of 10: There are three ways to achieve this: (4, 6), (5, 5), and (6, 4), as there are four possible outcomes on each die. The probability of each outcome is 1/36 since we have a total of 36 possible outcomes for two dice. Winning $5 happens in all three cases, so the total expected value for this outcome is (1/36) * $5 = $5/36.

2. Rolling doubles: There are six possible combinations to roll doubles, such as (1, 1), (2, 2), etc. The probability of rolling doubles is 6/36 since there are six favorable outcomes out of 36. Losing $2 occurs in all six cases, so the total expected value for this outcome is (6/36) * (-$2) = -$12/36.

3. Getting any other sum: To calculate the remaining probability, we subtract the probabilities of rolling 10 or doubles from 1. So the probability of getting any other sum is (1 - (1/36) - (6/36)) = 29/36. Since there is no mention of any other outcome, we can assume that it results in no gain or loss. Hence, the expected value for this outcome is (29/36) * $0 = $0.

To calculate the overall expected value, we add up the expected values of all the outcomes:

Expected value = ($5/36) + (-$12/36) + ($0)
= ($5 - $12 + $0)/36
= -$7/36

The negative result indicates that, on average, you would lose money by playing this game. Therefore, it is not a fair game to play.

By considering the probabilities and outcomes of each roll, we were able to calculate the expected value and determine the fairness of the game.