Find the expected value, u, for the sum of two fair dice. The probability distribution for the sum of two dice is given in the table. u = 1

2=1/36, 3=2/36, 4=3/36, 5=4/36, 6=5/36, 7=6/36, 8=5/36, 9=4/36, 10=3/36, 11=2/36, 12=1/36

To find the expected value (u) for the sum of two fair dice, we need to calculate the weighted average of the sum of the dice based on their probabilities.

The expected value (u) can be calculated using the formula:

u = (sum of (x * P(x))), for all possible values of x

where x represents the sum of the dice and P(x) represents the probability of that sum.

In this case, we are given the probability distribution for the sum of two dice in the table. We can use this table to calculate the expected value (u).

Let's calculate step by step:

u = (2 * 1/36) + (3 * 2/36) + (4 * 3/36) + (5 * 4/36) + (6 * 5/36) + (7 * 6/36) + (8 * 5/36) + (9 * 4/36) + (10 * 3/36) + (11 * 2/36) + (12 * 1/36)

Now, let's calculate the product of each sum (x) and its corresponding probability (P(x)):

u = (2/36) + (6/36) + (12/36) + (20/36) + (30/36) + (42/36) + (40/36) + (36/36) + (30/36) + (22/36) + (12/36)

Now, let's simplify the equation:

u = (137/36)

Therefore, the expected value (u) for the sum of two fair dice is approximately 3.81.