Question

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

Short answer

Calculate probabilities, find the expected value, and compare to $10.

Step-by-step solution

Determine Possible Outcomes

Identify the possible outcomes when throwing two dice. Each die has 6 faces, so there are a total of 6 × 6 = 36 possible outcomes.

Calculate the Product Values

List the possible products for each pair of outcomes. For example, (1,1) gives 1 and (6,6) gives 36. Make a table of products from (1,1) to (6,6).

Determine Frequency of Each Product

Count the frequency of each product. Some products like 1 (only from (1,1)) appear once, while others like 2 (from (1,2) and (2,1)) appear more frequently.

Calculate Expected Value

Use the formula for expected value: \[E(X) = \sum (value \times probability)\] Calculate the probability for each product (frequency / 36) and sum the products of these probabilities and their values.

Compare Expected Value to Cost

Compare the expected value to the cost per throw (\(10). If the expected value is more than \)10, the game is favorable; otherwise, it is not.

Key concepts

Probability

Probability is a fundamental concept used in various fields, including mathematics, statistics, and game theory. It's basically the measure of how likely an event is to occur. For instance, when you throw a die, the probability of landing on any specific number (like 3) is \(\frac{1}{6}\) because one die has six faces.
In the context of this exercise, we are concerned with two dice. When two dice are thrown, you get a variety of outcomes. The probability of each specific combination, such as (1,1) or (3,4), is determined by multiplying the probabilities for each die, which are both \(\frac{1}{6}\). Therefore, each unique outcome has a probability of \(\frac{1}{36}\). This is because the total number of possible outcomes when throwing two dice is 6 × 6 = 36.
Knowing these probabilities is essential for calculating expected values, where we determine the average outcome considering all possible events and their probabilities.

Dice Outcomes

Every roll of a pair of dice results in a combination of two faces, one from each die. The possible outcomes range from (1,1) to (6,6), making a total of 36 distinct pairs. Each outcome, however, leads to a specific product of the two numbers. For example:

• The pair (1,1) yields a product of 1.
• The pair (2,1) yields a product of 2.
• The pair (6,6) yields a product of 36.

Some products appear more frequently than others because different pairs of dice can produce the same product. For instance, both (2,3) and (3,2) produce the product 6. Understanding the frequency of each product outcome is crucial for calculating the expected value. Simply put, it tells us how often a particular result comes up, which we use to determine the average (expected) outcome in probabilistic terms.

Expected Value Calculation

The expected value (EV) is an important concept used to determine if a game or investment is favorable. It represents the average outcome if the same event were repeated many times. To find the EV when throwing two dice, follow these steps:

• List all possible products from each pair of dice outcomes.
• Calculate the frequency of each product.
• Determine the probability of each product by dividing its frequency by 36 (total outcomes).

Mathematically, the expected value formula is \[E(X) = \sum (value \times probability)\].
This can be interpreted as the sum of all possible values each multiplied by its probability. By doing this, you incorporate both the likelihood of different outcomes and their respective values, allowing us to calculate an average payoff. If this value is higher than the cost of playing the game (\textdollar 10 in this case), then the game is considered favorable.

Game Theory

Game theory is the study of mathematical models of strategic interaction among rational decision-makers. In simpler terms, it helps us evaluate situations where individuals make decisions that impact each other. The dice game in the exercise is a perfect example of a game theory application.
When deciding whether or not to play the game, we use game theory to calculate the expected value and weigh it against the cost. If the EV exceeds the cost of playing, rational players would choose to play since it offers a positive gain over many repetitions. Conversely, if the EV is lower than the cost, players should avoid the game to prevent potential losses.
Understanding game theory helps in making decisions based on logical analysis rather than intuition. It emphasizes the importance of considering all possible outcomes, their probabilities, and their impacts, leading to more informed and strategic decisions.