You and your friend are playing the following game: two fair six sided dice are rolled; if the total showing is divisible by 3 , you pay your friend \( \$ 30 \). How much should he pay you when the total is not divisible by 3 if you want to make the game fair? A fair game is one in which your expected winnings are \( \$ 0 \). Your Answer:

Let X be the amount your friend pays you when the dice total is not divisible by 3. Step 1: Count the outcomes. When rolling two six-sided dice, there are 36 equally likely outcomes. Outcomes where the total is divisible by 3: • Sum of 3: 2 outcomes (1,2) and (2,1) • Sum of 6: 5 outcomes (1,5), (2,4), (3,3), (4,2), (5,1) • Sum of 9: 4 outcomes (3,6), (4,5), (5,4), (6,3) • Sum of 12: 1 outcome (6,6) Total outcomes = 2 + 5 + 4 + 1 = 12. Probability the total is divisible by 3 = 12/36 = 1/3. Probability the total is not divisible by 3 = 1 − 1/3 = 2/3. Step 2: Determine the expected value. You pay your friend $30 when the total is divisible by 3 and receive X dollars from your friend when it’s not. The expected value (E) for you:   E = (Probability not divisible by 3) × (Amount received) + (Probability divisible by 3) × (− Amount paid)   E = (2/3)X + (1/3)(−30) For a fair game, the expected value should equal 0:   (2/3)X − 10 = 0 Step 3: Solve for X.   (2/3)X = 10   X = 10 × (3/2)   X = 15 Thus, your friend should pay you $15 when the total is not divisible by 3 for the game to be fair.