Problem 7 - Algebra and Trigonometry

Question

Find the expected value (or expectation) of the games described.? $$ \begin{array}{l}{\text { A die is rolled. Tom wins } \$ 2 \text { if the die shows an even num- }} \\ {\text { ber and he pays } \$ 2 \text { otherwise. }}\end{array} $$

Step-by-Step Solution

Answer

Expected value of Tom's game is $0.

Step 1: Identify Possible Outcomes

A die has 6 faces with numbers 1 through 6. Tom wins $2 if the die shows an even number (2, 4, or 6) and loses $2 if it shows an odd number (1, 3, or 5).

Step 2: Determine the Probability of Each Outcome

There are 3 even numbers (2, 4, 6) and 3 odd numbers (1, 3, 5), so the probability of rolling an even number is \( \frac{3}{6} = \frac{1}{2} \) and rolling an odd number is also \( \frac{3}{6} = \frac{1}{2} \).

Step 3: Calculate the Value for Each Outcome

If an even number is rolled, Tom wins \( +2 \). If an odd number is rolled, Tom loses \( -2 \). So the outcomes are +2 and -2.

Step 4: Use the Formula for Expected Value

The expected value \( E(X) \) is calculated using the formula: \[ E(X) = x_1 \cdot P(x_1) + x_2 \cdot P(x_2) \] where \( x_1 \) is \(+2\) with a probability of \( \frac{1}{2} \) and \( x_2 \) is \(-2\) with a probability of \( \frac{1}{2} \).

Step 5: Calculate Expected Value

Substitute the values into the formula: \[ E(X) = (2)(\frac{1}{2}) + (-2)(\frac{1}{2}) \] \[ E(X) = 1 - 1 \] \[ E(X) = 0 \]

Key Concepts

Probability

Probability is the foundation for determining how likely an event is to occur. In the example of rolling a die, probability helps us determine the chance of getting either an even or an odd number. When dealing with a fair six-sided die, each number—1 through 6—has the same likelihood of appearing. Therefore, the probability of each number is \( \frac{1}{6} \).

To determine the probability of an even or an odd roll, we need to consider all possible outcomes:

• Even numbers: 2, 4, 6
• Odd numbers: 1, 3, 5

Both categories contain three numbers out of the six possible outcomes, so the probability of rolling an even number is the sum of their individual probabilities: \( \frac{3}{6} \) or \( \frac{1}{2} \). The same is true for odd numbers, also \( \frac{1}{2} \).

Understanding probability helps in making informed predictions about the likely results of random phenomena, which is crucial for calculating expected values.

Outcome Analysis

Outcome analysis involves examining all potential results of a particular event to assess the possible gains or losses. In our example, Tom can either win or lose money based on the number rolled on the die. When he rolls:

• An even number (2, 4, or 6), he wins $2.
• An odd number (1, 3, or 5), he loses $2.

Each possible result is termed an "outcome," and it's important to assign a monetary value to each. This way, we not only understand what happens during each scenario but also quantify the impact.

By analyzing possible outcomes, we can plan for different results and weigh their probabilities, giving us a clearer picture of potential risks and rewards. This sets the stage for calculating expected value, helping us to make more strategic decisions.

Expected Value Calculation

Expected value, also known as expectation or mean in statistics, gives us an average measure of the expected outcome when an event is replicated many times. It leverages probabilities and values of outcomes to offer a clearer financial picture. To calculate this, we multiply each outcome's value by its probability and sum up the results.

Using our die-rolling example, the formula is:\[ E(X) = x_1 \cdot P(x_1) + x_2 \cdot P(x_2) \]Where:

• \( x_1 = +2 \), probability \( P(x_1) = \frac{1}{2} \)
• \( x_2 = -2 \), probability \( P(x_2) = \frac{1}{2} \)

The calculation is thus:\[ E(X) = (2)(\frac{1}{2}) + (-2)(\frac{1}{2}) \]\[ E(X) = 1 - 1 \]\[ E(X) = 0 \]This calculation tells us that, on average, Tom neither wins nor loses money over time. The expected value is a powerful tool for anticipating long-term results, adapting strategies, and managing potential risks.