Question

Would you pay \(\$ 10\) per throw of two dice if you were to reccive a number of dollars equal to the product of the numbers on the dice? Himt: What is your expectation? If it is more than \(\$ 10\), then the game would be favorable for you.

Short answer

Yes, since the expected value (\( 12.25 \)) is greater than the cost per throw (\( 10 \)).

Step-by-step solution

- Calculate Total Outcomes

First, determine the total number of outcomes when two dice are thrown. Each die has 6 faces, so the total number of outcomes is given by: \( 6 \times 6 = 36 \) possible outcomes.

- List Outcomes with Their Products

List all possible results and their respective products. For example, if die 1 shows 1 and die 2 shows 1, the product is 1. Continue this pattern: (1,1):1, (1,2):2, (1,3):3, ..., (6,5):30, (6,6):36.

- Calculate Expected Value

Calculate the expected value (E) of the product of the numbers. The expected value is the sum of all possible products divided by the number of outcomes.\( E = \frac{1 + 2 + 3 + ... + 36}{36} \).

- Calculate Each Product Frequency

Count how many times each product appears in the list of outcomes. This will help with the weighted average calculation.

- Calculate Weighted Sum

Multiply each product with its frequency and sum all values.For example: \( 1 \times 1 + 2 \times 2 + 3 \times 2 + ... + 36 \times 1 \).

- Divide by Total Outcomes

Divide the weighted sum by the total outcomes (36) to get the expected value.\( E = \frac{441}{36} = 12.25 \).

- Compare Expected Value to Cost

Compare the calculated expected value to the cost per throw. Since \( 12.25 > 10 \), the game is favorable.

Key concepts

Probability Distribution

Understanding probability distribution is vital when dealing with any probabilistic model, like our dice-throwing game. Probability distribution represents how the probabilities are assigned to all possible outcomes of a random variable.
For our two dice, each outcome (e.g., rolling a 1,2, or a 6,6) has a probability. Since each die has 6 faces, there are a total of 36 outcomes. These outcomes are distributed equally, making each outcome have a probability of \(\frac{1}{36}\). By organizing these probabilities into a distribution, we can explore more complex statistics like expected value.

Dice Outcomes

Let's delve into the outcomes when two dice are rolled. Each die has 6 faces, yielding a total of 36 possible combinations. The outcomes range from (1,1) to (6,6).
When we say the outcome is (1, 1), it means the first die shows 1 and the second die shows 1, resulting in a product of 1. Similarly, if the outcome is (6,5), the product is 30.
This step is crucial because to calculate the expected value, we need all the products from the possible outcomes. This can be visualized as:

• (1,1): 1
• (1,2): 2
• (1,3): 3
• ...
• (6,5): 30
• (6,6): 36

This list helps us lay the groundwork for the next steps, such as finding each product's frequency.

Mathematical Expectation

Mathematical expectation, or expected value, is an average value you can expect from a probabilistic experiment, such as rolling dice multiple times.
For our dice game, we calculate the expected value by summing all possible products of the dice outcomes and dividing by the number of outcomes (36).
The weighted sum formula, used here, takes into account the frequency of each product. First, list the products and count their frequencies. For example: 1 appears 1 time, 2 appears 2 times, and so forth. Multiply each product by its frequency, sum them up, and then divide by the total number of outcomes: \[E = \frac{\text{Sum of (product * frequency)}}{36} = \frac{441}{36} = 12.25 \]
The result helps us make decisions about whether the game is financially favorable.

Game Theory

Game theory involves strategizing in situations where success depends not only on your actions but also on the actions of others. In our dice game, game theory principles help us decide whether to play or not.
To apply game theory, calculate the expected value and compare it to the cost per throw. Given our expected value of 12.25 dollars per throw exceeds the cost of 10 dollars per throw, it indicates a favorable game situation.
The decision rule here is simple: Play the game if the expected value is higher than the cost. This basic idea is an example of how game theory can be practically applied to real-life situations to maximize gains and minimize losses.