Conditional variance

In the last example, we saw that the conditional distribution of X, which was a uniform over a smaller range (and in some sense, less uncertain), had a smaller variance, i.e., Var(XA)Var(X). Here is an example where this is not true. Let Y be uniform on {0,1,2} and let B be the event that Y belongs to {0,2}.

a) What is the variance of Y?

Var(Y)=

b) What is the conditional variance Var(YB)?

Var(YB)=