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A fair die is rolled twice. Let X denote the sum of the values. What is E[X|A] where A is the event that the first roll was a 6.

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We will see that the expected value for X given that the first number is a 6, is:

E[X|A] = 9.5

How to get the expected value?

We know that X represents the sum of both values. The first value is already 6, so we have that fixed:

X = 6 + p

Now, remember that all the numbers in the dice have the same probability of rolling up, then the expected value of p is:

E(p) = (1 + 2 + 3 + 4 + 5 + 6)*(1/6) = 3.5

Then the expected value of X, given the event A.

E[X|A] = 6 + p = 6 + 3.5 = 9.5

So the expected value for X is 9.5

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