Finding conditional expectation on the basis of conditional probabilities

Question

Knowing that

1. $P(Y=1\mid X=5)=1/3$,
2. $P(Y=5\mid X=5)=2/3$.

Calculate $E(Y\mid X=5)$ and $E(XY^2\mid X=5)$. How to solve this question? I have no idea whatsoever.

Answer 1

Since the conditional probabilities sum up to $1$ then that is all for $Y\mid X=5$. So $$E[Y\mid X=5]=1\cdot P(Y=1\mid X=5)+5\cdot P(Y=5\mid X=5)=1\cdot\frac13+5\cdot\frac23=\frac{11}3$$ and \begin{align}E[XY^2\mid X=5]&=E[5Y^2\mid X=5]=5E[Y^2\mid X=5]\\[0.2cm]&=5\left((1)^2\cdot P(Y=1\mid X=5)+(5)^2\cdot P(Y=5\mid X=5)\right)\\[0.2cm]&=5\left(1\cdot\frac13+25\cdot\frac23\right)=\frac{255}3\end{align}

Answer 2

Hint:

Let $Z$ be a random variable with $P(Z=1)=\frac13$ and $P(Z=5)=\frac23$.

Then $Y$ and $XY^2$ both under condition $X=5$ "behave" like $Z$ and $5Z^2$.