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

The question involves understanding the concept of expected value and decision-making under uncertainty. The first part asks for the expected return from rolling a single die, which involves calculating the average outcome. The second part introduces a decision point: whether to roll a second die, forfeiting the first roll's earnings, based on the potential expected outcome of the second roll. This requires comparing the expected values to make an optimal decision.

Answer

To address the first part of the question, we calculate the expected return from rolling a single six-sided die.

Calculating Expected Return (Single Roll):

• A fair six-sided die has faces numbered from 1 to 6.
• The expected return can be calculated by taking the average of all possible outcomes.

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

Thus, the expected return from one roll of the die is 3.5.

Decision to Roll a Second Die:

• If you decide to roll the second die, you forfeit the earnings from the first roll.
• The expected return from any single roll remains 3.5, as calculated above.

When to Roll the Second Die:

• You should roll the second die if the outcome of the first roll is less than the expected value of 3.5. This is because rolling again offers a potential outcome that, on average, matches or exceeds the expected value.
• Specifically, if you roll a 1, 2, or 3 on the first die, it is statistically beneficial to roll again, as these values are below the expected average of 3.5.

In summary, the decision to roll a second time should be based on the outcome of the first roll: roll again if you get a 1, 2, or 3, since the expected value of a new roll (3.5) is higher than these outcomes.