Question

For each situation below, you are betting on the outcome of the roll of two dice. In each case, determine the expected value of your bet, and whether or not the bet is favourable. Make sure to properly justify your answer.

(a) If you roll an eleven (i.e. the sum of the faces is 11), you win $9. If exactly one of the dice is a two, you lose $2. Otherwise, no money is exchanged.
(b) If the PRODUCT of the numbers on the dice is odd, you win $3. Otherwise (i.e. if the product is even), you lose your $1.

Answer 1

(a) To determine the expected value of the bet, we need to calculate the probability of each outcome and multiply it by the corresponding payout or loss.

The possible outcomes for rolling two dice are the sums ranging from 2 to 12. Let's analyze the three scenarios described:

1. If you roll an eleven, you win $9. The probability of rolling an eleven is 2/36 since there are two ways to obtain an 11 (5-6 and 6-5) out of 36 possible combinations. So, the payout for this scenario is (2/36) * $9 = $0.50.

2. If exactly one of the dice is a two, you lose $2. The probability of this happening is (10/36) since there are 10 combinations where only one of the dice is a two (2-1, 2-3, 2-4, 2-5, 2-6, 1-2, 3-2, 4-2, 5-2, 6-2). The loss for this scenario is (10/36) * $2 = $0.56.

3. In all other cases, no money is exchanged. The remaining outcomes have a probability of (24/36) since there are 24 combinations where none of the above conditions occur. Therefore, the payout for this scenario is (24/36) * $0 = $0.

To calculate the expected value, we sum up the payouts for each scenario:

Expected value = $0.50 - $0.56 + $0 = -$0.06

Since the expected value is negative (-$0.06), the bet is not favorable. On average, you can expect to lose $0.06 per bet.

(b) In this case, the payout and loss depend on whether the product of the numbers on the dice is odd or even.

1. If the product is odd, you win $3. To determine the probability of this, we need to consider the combinations where the product is odd. There are 18 such combinations (1-1, 1-3, 1-5, 1-7, 1-9, 2-1, 2-3, 2-5, 2-7, 2-9, 3-1, 3-3, 3-5, 3-7, 4-1, 4-3, 5-1, 5-3). Therefore, the payout for this scenario is (18/36) * $3 = $1.50.

2. If the product is even, you lose $1. The probability of this happening is (18/36) since there are 18 combinations where the product is even. The loss for this scenario is (18/36) * $1 = $0.50.

To calculate the expected value, we sum up the payouts for each scenario:

Expected value = $1.50 - $0.50 = $1.00

Since the expected value is positive ($1.00), the bet is favorable. On average, you can expect to win $1.00 per bet.

Therefore, the bet in scenario (b) is favorable, while the bet in scenario (a) is not favorable.