Final answer:
To find the variance of the product of two dice rolls, calculate the probabilities for each outcome, then find the expected value of the product and the individual dice rolls. Finally, use the variance formula to calculate the variance.
Step-by-step explanation:
To find the variance of the product of two dice rolls, we first need to understand the probabilities associated with each outcome. When two fair six-sided dice are rolled, there are a total of 36 possible outcomes. The product of the dice rolls can range from 1 to 36. We can calculate the probability of each outcome by counting the number of ways to obtain that product and dividing it by the total number of outcomes. From there, we can calculate the expected value and the variance.
To calculate the variance, we need to subtract the expected value of the product from the square of the expected value of the individual dice rolls. The equation for variance is Var(X) = E(X^2) - [E(X)]^2.
Let's go through an example:
- Calculate the probabilities for each product outcome: 1, 2, 3, ..., 36.
- Calculate the expected value of the product by multiplying each outcome by its probability and summing them up.
- Calculate the expected value of each individual dice roll by summing up the values and dividing by the number of faces on the dice.
- Calculate E(X^2) by squaring the expected value of each product outcome and summing them up.
- Calculate the variance using the formula Var(X) = E(X^2) - [E(X)]^2.