Deciding on a Second Dice Roll

Solution & Explanation

To determine when it is advantageous to take a second roll when playing a dice game, we can utilize the concept of expected value. The expected value provides a measure of the average outcome of a random event, which in this case is a dice roll.

Understanding the Expected Value

A standard die has six faces, numbered from 1 to 6. Each face has an equal probability of appearing when the die is rolled. Therefore, the expected value $E[x]$ of a single roll of a fair six-sided die can be calculated as:

$E[x] = \frac{1}{6} \times (1 + 2 + 3 + 4 + 5 + 6) = \frac{21}{6} = 3.5$

This means that, on average, a single roll of a die will yield a result of 3.5 points.

Decision Criteria for the Second Roll

When deciding whether to roll the second die, the key is to compare the result of the first roll to the expected value of 3.5. Here’s the reasoning:

1. If the first roll results in a score less than 3.5 (i.e., 1, 2, or 3):

   • The expected outcome of rolling again is higher than the current score, as the average score from a new roll is 3.5.
   • Therefore, it is advantageous to roll the second die, as you have a chance to increase your score.

2. If the first roll results in a score greater than or equal to 3.5 (i.e., 4, 5, or 6):

   • The expected outcome of rolling again is lower than or equal to the current score.
   • Therefore, it is disadvantageous to roll the second die, as you are more likely to decrease your score.

Probabilistic Justification

To further justify this decision, consider the probabilities of rolling a higher number than the first roll:

• First roll is 1: Probability of rolling higher on the second roll is $\frac{5}{6} \approx 83\%$
• First roll is 2: Probability of rolling higher is $\frac{4}{6} \approx 67\%$
• First roll is 3: Probability of rolling higher is $\frac{3}{6} = 50\%$
• First roll is 4: Probability of rolling higher is $\frac{2}{6} \approx 33\%$
• First roll is 5: Probability of rolling higher is $\frac{1}{6} \approx 17\%$
• First roll is 6: Probability of rolling higher is $0\%$

Given these probabilities, it is advantageous to roll again only if the first roll is 1, 2, or 3, where the chance of improving your score is significant.

Conclusion

In conclusion, the decision to roll the second die should be based on the result of the first roll compared to the expected value of 3.5. Roll again if the first result is less than 4; otherwise, keep the score from the first roll.