- To calculate the expected value, multiply each price by its corresponding probability.
- Sum the products: \(24 \times \frac{1}{6} = 4\), \(30 \times \frac{1}{2} = 15\), and \(36 \times \frac{1}{3} = 12\).
- Add these contributions: \(4 + 15 + 12 = 31\).
- The expected value of a T-shirt is \(\boxed{\$31}\).
Explanation
- Analyze the data and understand the expected value formula
To find the expected value of a T-shirt, we use the definition of the expected value of a discrete random variable. The expected value, \(E[X]\), is calculated by taking the sum of all possible outcomes multiplied by their respective probabilities:
\[E[X] = \sum x_i \cdot P(x_i)\]
In this problem, the possible prices are \(x_1 = 24\), \(x_2 = 30\), and \(x_3 = 36\). The associated probabilities are \(P(x_1) = \frac{1}{6}\), \(P(x_2) = \frac{1}{2}\), and \(P(x_3) = \frac{1}{3}\). First, let's verify that these probabilities sum to \(1\):
\[\frac{1}{6} + \frac{1}{2} + \frac{1}{3} = \frac{1}{6} + \frac{3}{6} + \frac{2}{6} = \frac{6}{6} = 1\]
Since the sum is \(1\), we are ready to calculate the expected value.
- Calculate the expected value
Now, we substitute the given values into the expected value formula:
\[E[X] = (24 \times \frac{1}{6}) + (30 \times \frac{1}{2}) + (36 \times \frac{1}{3})\]
Breaking this down:
- The contribution from the \(\$24\) T-shirt is \(24 \times \frac{1}{6} = 4\).
- The contribution from the \(\$30\) T-shirt is \(30 \times \frac{1}{2} = 15\).
- The contribution from the \(\$36\) T-shirt is \(36 \times \frac{1}{3} = 12\).
Adding these results together gives:
\[E[X] = 4 + 15 + 12 = 31\]
Therefore, the expected value of a T-shirt is \(\$31\).
Examples
If you are running a retail store, calculating the expected value of an item's price helps you estimate the average revenue per sale. For instance, if you sell three different styles of T-shirts with prices \(P_1, P_2, P_3\) and know the proportion of sales for each style (the probabilities \(Pr_1, Pr_2, Pr_3\)), the expected value \(E[X] = \sum P_i \cdot Pr_i\) provides a reliable forecast for your total daily revenue by multiplying this average price by the total number of shirts expected to be sold.