Final answer:
The mean or expectation of a random variable X representing the sum of the numbers on two dice can be found by calculating the average of all possible outcomes. By calculating the probability for each possible sum and multiplying it by the sum itself, we can find the weighted average. Therefore, the mean or expectation of X is approximately 0.742.
Step-by-step explanation:
The mean or expectation of a random variable X representing the sum of the numbers on two dice can be found by calculating the average of all possible outcomes. Each die has six faces numbered from 1 to 6, so there are 6 possible outcomes for each die.
The sum of two dice will range from 2 (1+1) to 12 (6+6), with a total of 11 possible outcomes. To find the mean, we calculate the weighted average of the outcomes, where the weight is the probability of each outcome. Since there is only one way to get a sum of 2 (1+1), the probability of getting a sum of 2 is 1/36. Similarly, there is only one way to get a sum of 12 (6+6), so the probability of getting a sum of 12 is also 1/36.
For all other sums, there are multiple ways to achieve them. For example, there are two ways to get a sum of 3 (1+2 and 2+1), so the probability of getting a sum of 3 is 2/36. By calculating the probability for each possible sum and multiplying it by the sum itself, we can find the weighted average:
(1/36) * 2 + (2/36) * 3 + (3/36) * 4 + (4/36) * 5 + (5/36) * 6 + (6/36) * 7 + (5/36) * 8 + (4/36) * 9 + (3/36) * 10 + (2/36) * 11 + (1/36) * 12
Simplifying this expression, we get:
2/36 + 6/36 + 12/36 + 20/36 + 30/36 + 42/36 + 40/36 + 36/36 + 30/36 + 22/36 + 12/36
Adding all the terms, we get:
294/36
Dividing this by the total number of possible sums, 11, we get the mean or expectation of X:
294/36 / 11 = 294/396 = 49/66 ≈ 0.742
Therefore, the mean or expectation of X is approximately 0.742.
Therefore answer is C. 7.