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:
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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 \( P(\text{Lose}) = 1 - P(\text{Win}) = 1 - \frac{1}{36} = \frac{35}{36} \).
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Determine the outcomes:
- If you win, you profit $30 (since you also get back your initial $1 bet).
- If you lose, you lose $1.
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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
\[ EV = \left(\frac{1}{36} \times 30\right) + \left(\frac{35}{36} \times -1\right) \]
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Calculate each part:
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For winning: \[ \frac{1}{36} \times 30 = \frac{30}{36} = \frac{5}{6} \approx 0.8333 \]
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For losing: \[ \frac{35}{36} \times (-1) = -\frac{35}{36} \approx -0.9722 \]
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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: \[ EV = \frac{30}{36} - \frac{35}{36} = \frac{30 - 35}{36} = \frac{-5}{36} \]
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Calculate the decimal value: \[ EV \approx -0.1389 \]
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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.