Strategy for Dice Roll Earnings

Solution & Explanation

To solve the problem of maximizing the expected winnings in the dice game, we need to consider the strategy of when to keep the first roll and when to opt for a reroll.

Step-by-Step Analysis:

1. Understand the Expected Value of a Single Die Roll:

   • A fair six-sided die has outcomes 1, 2, 3, 4, 5, and 6.

   • The expected value (EV) of a single roll is calculated as:

     $\text{EV} = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5$

   This value represents the average outcome if you were to roll the die many times.

2. Decide When to Reroll:

   • If the first roll results in a number less than the expected value (1, 2, or 3), it is statistically beneficial to reroll since the expected outcome of a new roll is 3.5, which is higher than 1, 2, or 3.
   • If the first roll results in a 4, 5, or 6, do not reroll, as these numbers are greater than the expected value of 3.5.

3. Calculate the Expected Value of the Strategy:

   • Scenario 1: First roll is 4, 5, or 6:

     • Probability of this scenario: $\frac{3}{6} = \frac{1}{2}$

     • Expected value for this scenario: average of 4, 5, and 6:

       $\text{EV}_{\text{keep}} = \frac{4 + 5 + 6}{3} = 5$

   • Scenario 2: First roll is 1, 2, or 3 and is rerolled:

     • Probability of this scenario: $\frac{3}{6} = \frac{1}{2}$
     • Expected value for rerolling: 3.5 (since it's a fresh roll of a fair die)

4. Combine Both Scenarios to Find Overall Expected Value:

   • Combine the expected values from both scenarios using their probabilities:

     $\text{Overall EV} = \left( \frac{1}{2} \times 5 \right) + \left( \frac{1}{2} \times 3.5 \right)$

     $\text{Overall EV} = 2.5 + 1.75 = 4.25$

Therefore, the expected amount of money you would win, using the optimal strategy of rerolling only when the first roll is 1, 2, or 3, is 4.25.

This solution effectively balances the probability of each scenario with its expected outcome, ensuring that the strategy maximizes the average winnings over many games.