Calculating Expected Value for Two Dice

Solution & Explanation

The problem asks us to calculate the expected value of the sum of two dice: a 6-sided die and a 10-sided die. Let's break down the solution step-by-step.

Step 1: Understanding Expected Value

The expected value (EV) of a random variable is a measure of the center of its probability distribution. For a discrete random variable, it is calculated as:

$E(X) = \sum_{i=1}^{n} P(x_i) \cdot x_i$

where $x_i$ are the possible outcomes and $P(x_i)$ is the probability of each outcome.

Step 2: Calculating Expected Value for Each Die

6-Sided Die

• The possible outcomes are $1, 2, 3, 4, 5, 6$.
• Each outcome has a probability of $\frac{1}{6}$.

$E(X) = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6) = \frac{21}{6} = 3.5$

10-Sided Die

• The possible outcomes are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$.
• Each outcome has a probability of $\frac{1}{10}$.

$E(Y) = \frac{1}{10}(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) = \frac{55}{10} = 5.5$

Step 3: Calculating the Expected Value of Their Sum

The expected value of the sum of two independent random variables is the sum of their expected values. Thus, for the combined roll of the 6-sided die $X$ and the 10-sided die $Y$:

$E(X + Y) = E(X) + E(Y) = 3.5 + 5.5 = 9$

Conclusion

The expected value of the sum of the numbers rolled on a 6-sided die and a 10-sided die is $9$. This means that if you were to roll these two dice many times and average the sums, you would expect the average to be 9.