Calculate Expected Value in dice game.

Question

We play a game in 2 stages:

In stage one, we throw a dice until we get number 6, Let N represent the number of times played until we got 6 for the first time.

In stage two, we throw N dices (each one only once).

Question: Let $X$ represent the sum of results we got in stage 2, calculate $E(X|N=n)$:

What I know? I know that $N$ is $\operatorname{Geo}(1/6)$ and this $E(N)=1/(1/6)=6$ to continue I need to know the distribution of $X|N=n$, can I get help?

Answer 1

If we throw $n$ dice, then the expected value of their sum is $3.5n$. This follows directly from the fact that the average score on one die is $3.5$ (and expectation is linear).

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Let $A_i$ equal the outcome of the $i$th roll of the die. $E(A_i)$ can be calculated in the following way:$$\frac{1+2+3+4+5+6}{6}=3.5 \, .$$ Let $B$ equal the sum of $n$ rolls. \begin{align} E(B)&=E(A_1)+E(A_2)+\ldots+E(A_n) \\ &= \underbrace{3.5 + 3.5 + \ldots + 3.5}_{\text{$n$ times}} \\ &= 3.5n \, . \end{align}