Final answer:
When two dice are rolled, the sum can range from 2 to 12 with varying probabilities. Each outcome has a corresponding probability, and the mean of this distribution is 7.
Step-by-step explanation:
When two dice are rolled, each die has six sides, for a total of 6 x 6 = 36 possible outcomes. These outcomes can sum to any number between 2 and 12. We can create a probability distribution by listing out all the sums and their probabilities.
- Sum of 2: 1/36
- Sum of 3: 2/36
- Sum of 4: 3/36
- Sum of 5: 4/36
- Sum of 6: 5/36
- Sum of 7: 6/36
- Sum of 8: 5/36
- Sum of 9: 4/36
- Sum of 10: 3/36
- Sum of 11: 2/36
- Sum of 12: 1/36
The mean of the random variable is calculated by multiplying each sum by its corresponding probability and then adding all the products together. The mean (expected value) is often denoted as μ.
μ = (∑ X*P(X)) = (2*(1/36) + 3*(2/36) + ... + 12*(1/36)) = 7
The mean of the sum when rolling two dice is 7, which is the center of the distribution. This distribution shows that as you roll the dice more times, the resulting sum frequencies will tend to form a normal distribution around the mean, demonstrating the central limit theorem (CLT).