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 involves understanding the concept of expected value in probability and decision-making based on maximizing expected returns. The problem is divided into two parts: calculating the expected return from rolling a single die, and determining the optimal strategy for deciding whether to roll a second die, considering the potential loss of the first round's earnings.

Answer

Calculating the Expected Return from One Die Roll:

1. A fair six-sided die has outcomes ranging from 1 to 6.
2. The expected value (E) is calculated by multiplying each outcome by its probability and summing the results:
   [
   E = \sum ( \text{outcome} \times \text{probability} ) = 1 \times \frac{1}{6} + 2 \times \frac{1}{6} + \ldots + 6 \times \frac{1}{6}
   ]
3. Simplifying the calculation:
   [
   E = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5
   ]
4. Thus, the expected return from rolling a die is 3.5.

Deciding When to Roll the Second Die:

• If you roll a second die, you forfeit your earnings from the first round. Therefore, you should only consider rolling the second die if the expected value of doing so is greater than your current earnings.
• The expected value from rolling the die again is also 3.5 (as calculated from a single roll).
• Therefore, you should roll the second die if your first roll results in a value less than 3.5. Since you cannot roll a fraction, the practical decision rule is:
  • Roll the second die if your first roll results in a 1, 2, or 3.
  • Do not roll the second die if your first roll results in a 4, 5, or 6.

Conclusion:
The optimal strategy is to roll the second die if the first roll yields 1, 2, or 3, as this maximizes your expected return by potentially achieving a value higher than 3.5.