Question

\(26.3 A\) and \(B\) each have two unbiased four-faced dice, the four faces being numbered \(1,2,3,4\). Without looking, \(B\) tries to guess the sum \(x\) of the numbers on the bottom faces of \(A\) 's two dice after they have been thrown onto a table. If the guess is correct \(B\) receives \(x^{2}\) euros, but if not he loses \(x\) euros. Determine \(B\) 's expected gain per throw of \(A\) 's dice when he adopts each of the following strategies: (a) he selects \(x\) at random in the range \(2 \leq x \leq 8\); (b) he throws his own two dice and guesses \(x\) to be whatever they indicate; (c) he takes your advice and always chooses the same value for \(x\). Which number would you advise? 26.4 Use the method of induction to prove equation (26.16), the probability addition law for the union of \(n\) general events.

Short answer

The expected gain for B is maximized with y = 5

Step-by-step solution

Understanding the problem

Determine the expected gain for B per throw of A's dice under three different strategies: random guessing, guessing based on his own dice, and choosing the best fixed value.

Calculate outcomes for A's dice

The two dice A throws can result in sums ranging from 2 to 8. Calculate the probability of each sum: \[P(2) = \frac{1}{16}, P(3) = \frac{2}{16}, P(4) = \frac{3}{16}, P(5) = \frac{4}{16}, P(6) = \frac{3}{16}, P(7) = \frac{2}{16}, P(8) = \frac{1}{16}\]

Strategy (a) - Random selection

The gain for any sum x is x² euros if guessed correctly, and -x euros if guessed incorrectly. The expected gain for each potential sum is: \[EG_{random} = \frac{1}{7} \sum_{x=2}^{8} \left( P(x) \cdot x^2 + (1 - P(x)) \cdot -x \right) \]

Calculate expected gains for random selection

Compute the expected gain for random guessing using the formula from Step 3: \[EG_{random} = \frac{1}{7} \left( \frac{1}{16} (4 - 2) + \frac{2}{16} (9 - 3) + \frac{3}{16} (16 - 4) + \frac{4}{16} (25 - 5) + \frac{3}{16} (36 - 6) + \frac{2}{16} (49 - 7) + \frac{1}{16} (64 - 8) \right)\]

Strategy (b) - Using B's dice

In this case B guesses x based on his own dice. Expected gain depends on the same probabilities with the difference being the value B guesses matches A with probabilities above each time.

Calculate expected gains for B's dice

If B guesses the sum on his own dice, the probabilities of his own results matching each sum are the same as those for A. Therefore, apply the same expected gain calculation as for Strategy (a).

Strategy (c) - Best fixed value

Choose the fixed value x that maximizes B's expected gain. For fixed guess y, his expected gain is given by: \[EG_{fixed}(y) = \sum_{x=2}^{8} P(x) \cdot \left( \delta_{xy} \cdot y^2 + (1 - \delta_{xy}) \cdot -y \right) = P(y) \cdot y^2 + (1 - P(y)) \cdot -y\] Solve for the value that yields the highest expectation.

Key concepts

Probabilities of Dice Sums

When rolling two four-faced dice, the possible sums range from 2 to 8. Each sum has a different probability based on how many combinations can result in that sum. The probabilities can be calculated as follows:
- Sum of 2: Only one combination (1+1), so its probability is \[ P(2) = \frac{1}{16} \]
- Sum of 3: Two combinations (1+2, 2+1), so its probability is \[ P(3) = \frac{2}{16} \] ... and so forth until
- Sum of 8: Only one combination (4+4), giving a probability of \[ P(8) = \frac{1}{16} \] Each sum probability is important for calculating the expected gains since it shows how likely each outcome is when A rolls the dice.

Expected Gain Calculation

The expected gain is a crucial concept in probability and refers to the average amount one can expect to win or lose in a given situation over many trials. To calculate the expected gain for each strategy, you'll need to take into account both the probability of each dice sum and the amounts won or lost for correct or incorrect guesses.
1. **Random Selection Strategy**: If B guesses the sum randomly, the expected gain for each possible sum x can be calculated by considering both the positive gain when he guesses correctly and the negative loss when he guesses incorrectly. The formula used here is: \[ EG_{random} = \frac{1}{7} \sum_{x=2}^{8} \left( P(x) \cdot x^2 + (1 - P(x)) \cdot -x \right) \]
2. **Using B's Dice Strategy**: Here, B guesses the sum based on his own dice. The probabilities of his dice results match those of A's dice. Thus, we compute the expected gain similarly to the random selection but knowing that the same probabilities apply.
3. **Fixed Value Strategy**: To find the best fixed value for B to always guess, we need to determine the fixed guess y that gives the maximum expected gain using: \[ EG_{fixed}(y) = \sum_{x=2}^{8} P(x) \cdot \left( \delta_{xy} \cdot y^2 + (1 - \delta_{xy}) \cdot -y \right) = P(y) \cdot y^2 + (1 - P(y)) \cdot -y \] where \( \delta_{xy} \) is 1 if x equals y, and 0 otherwise. This equation helps determine which fixed sum provides the highest average gain over time.

Probability Addition Law

The probability addition law helps to calculate the probability of the union of multiple events. For instance, if we want to find the probability that at least one of several events occurs, we use the probability addition law.
The union of two events A and B can be calculated as: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
This generalizes to n events, making sure we don't double-count the overlap in events. For three events, the formula is: \[ P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C) \] Understanding these concepts is essential for accurately predicting outcomes and making informed decisions in probability-based scenarios.