12-Sided Die Probability Challenge

Solution & Explanation

The problem involves a non-standard 12-sided die with specific probabilities assigned to each face. The challenge is to determine the optimal number each player should choose to maximize their chance of winning, given the unique probability distribution.

Problem Breakdown:

1. Die Characteristics:

   • Face 12 has a probability of 40% (P(X=12) = 0.4).
   • The remaining 60% probability is evenly distributed across faces 1 through 11.
   • Probability for each of the other faces: $P(X=i) = \frac{0.6}{11} \approx 0.0545$ for $i \in \{1, 2, \ldots, 11\}$.

2. Objective:

   • Each player picks a number from 1 to 12.
   • The winner is the player whose chosen number is closest to the rolled die outcome.

3. Strategy:

   • The strategy involves picking a number that minimizes the expected distance between the chosen number and the rolled face.

Optimal Number Selection:

• Expectation Calculation:

  • Calculate the expected value (mean) of the die roll: $\mathbb{E}(X) = \sum_{i=1}^{11} i \left( \frac{0.6}{11} \right) + 12 \cdot 0.4 = \left( \frac{66 \cdot 0.6}{11} \right) + 4.8 = 8.4$
  • The expected value of 8.4 suggests that the outcome is more likely to be around 8 or 9, but this doesn't directly inform optimal choice.

• Median Approach:

  • The median minimizes the mean absolute error (distance) between the chosen number and the true value.
  • The cumulative probability up to 10 (excluding 12): $P(X \leq 10) = 10 \cdot \frac{0.6}{11} \approx 0.545$
  • Including 12: $P(X \geq 11) = 0.4 + 2 \cdot \frac{0.6}{11} \approx 0.454$
  • Since choosing 10 gives us a more balanced probability distribution on either side, it is the median and hence the optimal choice.

Conclusion:

• Player 1 Strategy:

  • Choose 10 as it provides the most balanced probability distribution, maximizing the chance of winning regardless of Player 2's choice.

• Player 2 Strategy:

  • If Player 1 chooses 10, Player 2 should choose 9 or 11, depending on the distribution of remaining probabilities.

Why 10 is Optimal:

• Choosing 10 results in a nearly equal probability of the roll being less than or greater than the chosen number, making it the most strategic choice when aiming to minimize the expected distance to the rolled outcome.

This analysis demonstrates how probability distributions and strategic thinking can guide optimal decision-making in games of chance.