If you roll a die, what is the probable reward? When should you contemplate rolling a second die and forfeiting your initial incentive?
Question Analysis
This question combines elements of probability and decision-making, requiring an understanding of expected value and risk assessment. The first part of the question asks about the expected reward from rolling a single die, which involves calculating the average outcome (expected value) of a single roll. The second part introduces a decision-making scenario: under what circumstances should you consider rolling a second die, thereby giving up the reward from the first roll? This involves comparing the potential outcomes and expected rewards from rolling again versus keeping the initial result.
Answer
Expected Reward from One Die Roll:
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A standard die has six faces, numbered 1 through 6.
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Each face has an equal probability of 1/6.
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The expected value (EV) of a single roll is calculated as the average of all possible outcomes:
[
EV = \frac{1}{6} \times (1 + 2 + 3 + 4 + 5 + 6) = \frac{21}{6} = 3.5
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Therefore, the probable reward (expected value) when rolling one die is 3.5.
Contemplating Rolling a Second Die:
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You should consider rolling a second die if the potential reward of doing so outweighs the value of your current roll. This involves comparing the current roll value to the expected value of rolling again.
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If your initial roll is less than 3.5, the general strategy would suggest contemplating a second roll, as the expected value of rolling again (3.5) is higher than your current result.
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However, if your initial roll is 4, 5, or 6, the expected value of rolling again is lower (3.5), and it might be wiser to keep your initial roll unless other factors (e.g., specific game rules or strategies) influence your decision.
Conclusion:
- Roll Again: If initial roll < 3.5
- Keep Initial Roll: If initial roll ≥ 4
This approach helps maximize your expected reward over multiple trials. Always consider the context and any additional rules that might affect decision-making.