To calculate the expected value of the profit from the bet in the game of craps, we can use the formula for expected value (EV):
\[ EV = (P(\text{win}) \times \text{profit if win}) + (P(\text{lose}) \times \text{profit if lose}) \]
From the problem:
- \( P(\text{win}) = \frac{1}{36} \)
- \( \text{profit if win} = 30 \) (you win $30, but your original $1 bet is returned, leading to a total return of $31, but profit is considered as $30)
- \( P(\text{lose}) = \frac{35}{36} \) (since there are 36 outcomes, and 35 of them are losses)
- \( \text{profit if lose} = -1 \) (you lose your bet of $1)
Now we can plug these values into the expected value formula:
\[ EV = \left(\frac{1}{36} \times 30\right) + \left(\frac{35}{36} \times -1\right) \]
Calculating each term:
-
For the win: \[ \frac{1}{36} \times 30 = \frac{30}{36} = \frac{5}{6} \approx 0.8333 \]
-
For the loss: \[ \frac{35}{36} \times -1 = -\frac{35}{36} \approx -0.9722 \]
Now we can sum these results:
\[ EV = 0.8333 - 0.9722 \approx -0.1389 \]
Rounding to two decimal places, the expected value of your profit is:
\[ \text{EV} \approx -0.14 \]
Thus, the expected value of your profit is \(-0.14\).