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 .

Question

(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.