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 combines elements of probability and decision-making under uncertainty. The first part of the question asks about the expected value of a single die roll, which is a fundamental concept in probability. The second part of the question introduces a strategic decision: whether to accept the outcome of the first roll or to take a risk by rolling a second die, forfeiting the first result. This requires analyzing the expected value of both scenarios to determine the optimal decision.

Answer

Expected Return from a Single Die Roll:

• A fair six-sided die has faces numbered from 1 to 6. The expected value (or expected return) of a single die roll is calculated by taking the average of all possible outcomes, weighted by their probabilities.

• The formula for the expected value (E) of a single die roll is:

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

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

When to Roll the Second Die:

• When deciding whether to roll the second die, we must consider the expected value of the second roll, which is also 3.5, as it is a fair six-sided die.

• The decision to roll the second die should be based on whether the result of the first roll is less than the expected value of the second roll (3.5). If the first roll is less than 3.5, rolling again gives a chance to achieve a higher return.

• Therefore, you should roll the second die if the result of the first roll is less than or equal to 3. This is because:

  • Roll Again if First Roll is ≤ 3: The expected value of a second roll (3.5) is greater than these results, offering a potential benefit.
  • Keep the First Roll if 4, 5, or 6: These values are already higher than the expected value of a second roll.

In summary, you maximize your expected return by rolling the second die if your first roll is 3 or less.