Suppose you roll a die and earn whatever face you get. What is the expected return? Now suppose you have a chance to roll a second die. If you roll, you forfeit your earnings from the first round. When should you roll the second time?

Question Analysis

This question is a probability and decision-making problem that requires calculating the expected value of rolling a die. The first part asks for the expected return from rolling a single die, while the second part introduces a decision-making scenario where you must determine whether to roll a second die, forfeiting the first roll's earnings. The goal is to maximize the expected return.

Answer

Expected Return from Rolling One Die:

1. A standard die has six faces, numbered 1 through 6.

2. The expected value (EV) is calculated by multiplying each outcome by its probability and summing the results.

3. Since each face has an equal probability of 1/6, the EV for one die roll is:

   [
   EV = \frac{1}{6}(1) + \frac{1}{6}(2) + \frac{1}{6}(3) + \frac{1}{6}(4) + \frac{1}{6}(5) + \frac{1}{6}(6)
   ]

   [
   EV = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5
   ]

   Therefore, the expected return from rolling one die is 3.5.

Decision to Roll a Second Die:

1. The second roll also has an expected value of 3.5, as it is the same fair die.
2. You should roll the second die if your first roll is less than the expected value of 3.5 because you have a chance to improve your outcome.
3. Specifically, if the first roll is 1, 2, or 3, the expected gain from rolling again is higher than the current result. If the first roll is 4, 5, or 6, you should keep the initial result since rolling again is less likely to yield a better outcome.

Conclusion:

• Keep the first roll if the result is 4, 5, or 6.
• Roll again if the result is 1, 2, or 3.