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If a person gains or loses an amount equ...

If a person gains or loses an amount equal to the number appearing when an unbiased die is rolled once, according to wether the number is even or odd, how much money can he expect per game in the long run ?

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To solve the problem of how much money a person can expect to gain or lose per game in the long run when rolling an unbiased die, we can follow these steps: ### Step 1: Identify the outcomes of rolling a die When rolling a die, the possible outcomes are the numbers 1, 2, 3, 4, 5, and 6. ### Step 2: Classify outcomes into gains and losses - **Gains**: The person gains the amount equal to the number rolled if the number is even. The even numbers are 2, 4, and 6. - **Losses**: The person loses the amount equal to the number rolled if the number is odd. The odd numbers are 1, 3, and 5. ### Step 3: Calculate the expected gain - For even numbers (gains): - Gain when rolling a 2: +2 - Gain when rolling a 4: +4 - Gain when rolling a 6: +6 The probability of rolling each even number is \( \frac{1}{6} \). Calculating the expected gain: \[ E(\text{gain}) = \left( \frac{1}{6} \times 2 \right) + \left( \frac{1}{6} \times 4 \right) + \left( \frac{1}{6} \times 6 \right) \] \[ E(\text{gain}) = \frac{2}{6} + \frac{4}{6} + \frac{6}{6} = \frac{12}{6} = 2 \] ### Step 4: Calculate the expected loss - For odd numbers (losses): - Loss when rolling a 1: -1 - Loss when rolling a 3: -3 - Loss when rolling a 5: -5 The probability of rolling each odd number is \( \frac{1}{6} \). Calculating the expected loss: \[ E(\text{loss}) = \left( \frac{1}{6} \times -1 \right) + \left( \frac{1}{6} \times -3 \right) + \left( \frac{1}{6} \times -5 \right) \] \[ E(\text{loss}) = -\frac{1}{6} - \frac{3}{6} - \frac{5}{6} = -\frac{9}{6} = -1.5 \] ### Step 5: Calculate the overall expected value Now, we combine the expected gain and expected loss to find the overall expected value: \[ E(\text{total}) = E(\text{gain}) + E(\text{loss}) = 2 - 1.5 = 0.5 \] ### Conclusion The expected amount of money the person can expect to gain per game in the long run is **0.5**. ---
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