Conditional Expectation - Aplications E(g(X)|Y)

Question

Y$\sim Be(1/5)$ and let E denote an arbitrary event, and define the indicator random variable X by

$X=\begin{cases} 1 & \text{if E occurs}\\0 & \text{if E does not occur}& \end{cases}$

If $P(X=1|Y=0)=1/3$, $P(X=1|Y=1)=1/4$ and $g(x)=x^2$. Caculate $P(\mathbb{E}(g(X)|Y)>1/2).$

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I caculate,

$\mathbb{E}(X^2|Y=0)=0^2P(X=0|Y=0)+1^2P(X=0|Y=0)=1/3$ $\mathbb{E}(X^2|Y=1)=0^2P(X=0|Y=1)+1^2P(X=1|Y=1)=1/4$

But, I have no idea what can I do next =S. So any help or guidance is appreciated :)

Answer 1

Indeed: $\mathsf{E}(X^2\mid Y=0)\;$ $=\;0^2\mathsf P(X=0\mid Y=0)+1^2\mathsf P(X=\color{red}{1}\mid Y=0)\;\\=\;{1/3}$

And too: $\mathsf{E}(X^2\mid Y=1)\;$ $=\;0^2\mathsf P(X=0\mid Y=1)+1^2\mathsf P(X=1\mid Y=1)\\=\;1/4$

So therefore, as BGM commented: $\mathsf E(X^2\mid Y) = \tfrac 13 \;\mathbf 1_{[Y=0]}+\tfrac 14\;\mathbf 1_{[Y=1]}$

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Thus its clear that: $\mathsf P(\mathsf E(X^2\mid Y)>\tfrac12)=0$