I have having difficulties with this question
I have attached my question.
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thanks
$\displaystyle \mathbb{P}(X\leq 1/2 -Y |Y)=[1/2-Y]^+$ where [x]^+ is the maximum between x and 0 (because if Y is greater than 1/2, we cannot have a negative probability, and the prob of X being less than a negative number is 0).
Now
$\displaystyle
\mathbb{E}[1/2-Y]^+=\int_0^1 \max\{1/2-y,0\}\mathbb{P}(Y\in dy)=\int_0^{1/2} (1/2-y) dy
$
you can complete the rest.