Question: a. What are the odds for rolling a sum of 8 in a single roll of two fair dice? b. If you bet $1…

a. What are the odds for rolling a sum of 8 in a single roll of two fair dice? b. If you bet $1 that a sum of 8 will turn up, what should the house pay (plus returning your $1 dollar bet) if a sum of 8 turns up for the game to be fair? … a. The odds for rolling a sum of 8 are (Type a fraction. Simplify your answer.)

Solution

a. To find the odds for rolling a sum of 8 with two fair dice: 1. Calculate the total number of outcomes when rolling two dice. \[ 6 \times 6 = 36 \] 2. Find the combinations that result in a sum of 8: (2,6), (3,5), (4,4), (5,3), and (6,2). 3. Count these successful outcomes. There are 5 combinations. 4. Calculate the probability of rolling an 8. \[ \frac{5}{36} \] 5. The odds are given by the ratio of successful to unsuccessful outcomes (success : failure). \[ \frac{5}{36} \text{ to } \frac{31}{36} \] 6. Simplify the odds. \[ 5 \text{ to } 31 \] b. For the game to be fair, the expected payout should equal the bet. You bet $1, and the probability of winning is \(\frac{5}{36}\). 1. Let \( x \) be the amount the house pays if you win (including returning the $1 bet). 2. Set up the equation for the expected value being zero: \[ \left(\frac{5}{36} \times x\right) - \left(\frac{31}{36} \times 1\right) = 0 \] 3. Solve for \( x \): \[ \frac{5}{36} x = \frac{31}{36} \] \[ x = \frac{31}{5} \] 4. Simplify: \[ x = 6.2 \] Thus, the house should pay $5.20 plus the $1 bet for a total of $6.20 to make the game fair.