Question

In the game of craps, a pair of dice are rolled, and people bet on the outcome. For example, you can bet $1 that the dice will total 12. The probability that you win is 1 over 36, and if you win, your profit is $30. If you lose, you lose $1 .

(a) What is the expected value of your profit? Round the answer to two decimal places.

Answer 1

To calculate the expected value of your profit when betting on the outcome of the dice in the game of craps, we can follow these steps:

1. **Determine the probabilities:**
- The probability of winning (getting a total of 12) is \( P(\text{Win}) = \frac{1}{36} \).
- The probability of losing (not getting a total of 12) is \( P(\text{Lose}) = 1 - P(\text{Win}) = 1 - \frac{1}{36} = \frac{35}{36} \).

2. **Determine the outcomes:**
- If you win, you profit $30 (since you also get back your initial $1 bet).
- If you lose, you lose $1.

3. **Calculate the expected value (EV):**
\[
EV = (P(\text{Win}) \times \text{Profit if win}) + (P(\text{Lose}) \times \text{Profit if lose})
\]
Plugging in the values:
- Profit if win = $30
- Profit if lose = -$1

\[
EV = \left(\frac{1}{36} \times 30\right) + \left(\frac{35}{36} \times -1\right)
\]

4. **Calculate each part:**
- For winning:
\[
\frac{1}{36} \times 30 = \frac{30}{36} = \frac{5}{6} \approx 0.8333
\]

- For losing:
\[
\frac{35}{36} \times (-1) = -\frac{35}{36} \approx -0.9722
\]

5. **Combine the values:**
\[
EV = \frac{5}{6} - \frac{35}{36}
\]

To combine these fractions, convert \(\frac{5}{6}\) to have a common denominator of 36:
\[
\frac{5}{6} = \frac{30}{36}
\]

Now, combine:
\[
EV = \frac{30}{36} - \frac{35}{36} = \frac{30 - 35}{36} = \frac{-5}{36}
\]

6. **Calculate the decimal value:**
\[
EV \approx -0.1389
\]

7. **Round to two decimal places:**
\[
EV \approx -0.14
\]

Therefore, the expected value of your profit when betting on a total of 12 in craps is approximately **-0.14**.