In the game of craps, a pair of dice are rolled, and people bet on the outcome. For example, you can bet that the sum of the dice will be 3(also called "Ace Deuce"). The probability that you win is 1 over 18, and if you win, your profit is $15. 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 in this game of craps with the given conditions, we need to consider the probabilities and outcomes of winning and losing.

1. Winning Probability: The probability of winning by betting on a sum of 3 is given as \( \frac{1}{18} \).

2. Losing Probability: The probability of losing is \( 1 - \frac{1}{18} = \frac{17}{18} \).

3. Profit if you win: If you win, you make a profit of $15.

4. Loss if you lose: If you lose, you lose $1 (which is effectively a profit of -$1).

Now we can calculate the expected value \( E \):

\[ E = \text{(Probability of winning)} \times \text{(Profit if win)} + \text{(Probability of losing)} \times \text{(Profit if lose)} \]

Substituting the values we have:

\[ E = \left( \frac{1}{18} \times 15 \right) + \left( \frac{17}{18} \times (-1) \right) \]

Calculating each term:

1. Winning term:
   \[ \frac{1}{18} \times 15 = \frac{15}{18} \approx 0.8333 \]

2. Losing term:
   \[ \frac{17}{18} \times (-1) = -\frac{17}{18} \approx -0.9444 \]

Now we add the two results to find the expected value:

\[ E \approx 0.8333 - 0.9444 \approx -0.1111 \]

Rounding to two decimal places, the expected value of your profit is:

\[ \text{Expected Value} \approx -0.11 \]

Thus, the expected value of your profit is -0.11.