Question: a. What are the odds for rolling a sum of 3 in a single roll of two fair dice? b. If you bet $1…

a. What are the odds for rolling a sum of 3 in a single roll of two fair dice?

b. If you bet $1 that a sum of 3 will turn up, what should the house pay (plus returning your $1 dollar bet) if a sum of 3 turns up for the game to be fair?

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a. The odds for rolling a sum of 3 are (Type a fraction. Simplify your answer.)

Solution

To solve part (a) of the problem, we need to find the odds of rolling a sum of 3 with two fair dice: 1. Enumerate possible outcomes for two dice rolls: each die has 6 faces, so there are \(6 \times 6 = 36\) possible outcomes. 2. Identify outcomes that result in a sum of 3. These are: - (1, 2) - (2, 1) 3. Count these successful outcomes: there are 2 successful outcomes. 4. Calculate the probability of rolling a sum of 3: \[ \text{Probability} = \frac{\text{Number of successful outcomes}}{\text{Total number of outcomes}} = \frac{2}{36} = \frac{1}{18} \] 5. Convert the probability to odds: - Odds are calculated as the ratio of successful outcomes to unsuccessful outcomes. - Unsuccessful outcomes are \(36 - 2 = 34\). \[ \text{Odds} = \frac{\text{Successful outcomes}}{\text{Unsuccessful outcomes}} = \frac{2}{34} = \frac{1}{17} \] The odds for rolling a sum of 3 are \(\frac{1}{17}\). For part (b) of the problem: 1. Determine the fair payout for the bet. 2. If the probability of winning is \(\frac{1}{18}\), then the payout should equalize the expected value to zero for fairness. 3. Set up the equation: - Let \(x\) be the payout. The equation for a fair game is: \[ \left(\frac{1}{18} \right)(x + 1) = \left(\frac{17}{18}\right)(1) \] 4. Solve for \(x\): \[ x + 1 = 17 \] \[ x = 16 \] The house should pay $16 (plus returning the $1 bet) for the game to be fair.