Rolling a die - Conditional Probability

Question

A die is thrown repeatedly.

Let $X$ ~ First 5 is thrown and $Y$ ~ First 6 is thrown

Calculate $\mathbb{E}(X|Y=3)$

You may use the identity: $\sum_{n=k}^\infty nz^{n-k} = \frac{1}{(1-z)^2}+\frac{k-1}{1-z}$

I know from the definition of expectation, we have:

$\mathbb{E}(X|Y=3) = (1*\frac{1}{5})+(2*\frac{4}{5} * \frac{1}{5}) + (3* \frac{4}{5}* \frac{4}{5} * 0) + (5* \frac{4}{5} * \frac{4}{5} * 1 * \frac{5}{6} * \frac{1}{6}) + (6* \frac{4}{5} * \frac{4}{5} * 1 * \frac{5}{6} * \frac {5}{6} * \frac{1}{6}) + ...$, where every following term, has an extra '$*\frac{5}{6}$' term and constant increases by 1.

However I am unsure of how to apply this to the identity given to find the value of the infinite sum?

Answer 1

While it is not necessary, if the question expects you to use the given identity then here is how you would go about it -

$ \small \displaystyle E(X|Y = 3) = 1 \cdot \frac{1}{5} + 2 \cdot \frac{4 \cdot 1}{5^2} + 4 \cdot \frac{4 \cdot 4}{5^2} \cdot \frac{1}{6} + 5 \cdot \frac{4 \cdot 4}{5^2} \cdot \frac{5 \cdot 1}{6^2} + ...$

$ \small \displaystyle = \frac{13}{25} + \frac{4 \cdot 4}{5^2} \cdot \frac{1}{6} \left (4 \left(\frac{5}{6}\right)^0 + 5 \cdot \left(\frac{5}{6}\right)^1 + ...\right)$

Now note that comparing the below against the given identity,

$ \small \displaystyle 4 \left(\frac{5}{6}\right)^0 + 5 \cdot \left(\frac{5}{6}\right)^1 + ...$

$ \displaystyle \small z = \frac{5}{6}, k = 4$ and $\sum \limits_{n=k}^\infty nz^{n-k} = 54$

and we get, $ \displaystyle \small E(X|Y = 3) = \frac{157}{25}$

Answer 2

$\mathbb{P}(X=1|Y=3)=\frac{1}{5},\mathbb{P}(X=2|Y=3)=\frac{4}{25},\mathbb{P}(X>3|Y=3)=1−(\frac{1}{5}+\frac{4}{25})=\frac{16}{25}.$

Then we multiply these by the expected results, i.e. $1,2$ and $9$, giving $\mathbb{E}(X|Y=3)=(1∗\frac{1}{5})+(2∗\frac{4}{25})+(9∗\frac{16}{25})=6.28$