E[X] and E[X^2] with Conditional Expectation

Question

$\newcommand{\E}{\operatorname{\mathbb E}}$ $\newcommand{\Var}{\operatorname{\mathbb Var}}$ If $\E[X] = {^1\!/\!_3}(\E[X\mid Y=1] + \E[X\mid Y=2] + \E[X\mid Y=3]) = 10$

Where $\E[X|Y=1] = 2,\; \E[X|Y=2] = 3+\E[X],\; \E[X|Y=3] = 5+\E[X]$

is $\E[X^2|Y=1] = 4,\; \E[X^2|Y=2] = 9 + 6\E[X] + 6\E[X^2],$ and so on?

This is to find the $\Var(X)$.

where $\Var(X) = \E[X^2] - (\E[X])^2$

Question: How do you find the Variance of this given that $\E[X] = 10$?

Answer 1

This looks like the well-known prisoner is trapped in a cell with 3 doors.

$$E(X^2)=E(2^2)1/3+E[(3+X)^2]1/3+E[(5+X)^2]1/3 $$ Expand the terms, take expectation term-by-term, use $E(X)=10$ and solve for $E(X^2).$