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.
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Winning Probability: The probability of winning by betting on a sum of 3 is given as \( \frac{1}{18} \).
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Losing Probability: The probability of losing is \( 1 - \frac{1}{18} = \frac{17}{18} \).
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Profit if you win: If you win, you make a profit of $15.
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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:
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Winning term:
\[ \frac{1}{18} \times 15 = \frac{15}{18} \approx 0.8333 \] -
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.