Conditional Probability: $P(X|Y\leq y)=$ $\sum\limits_{k=1}^{y} P(X|Y=k)$?

Question

Suppose $X \sim Binomial (Y,\delta)$. For the random variable $X$ can I compute the following conditional probability as follows? $P(X=x|Y\leq y)=$ $\sum\limits_{k=1}^{y} P(X=x|Y=k)$, where Y is a discrete random variable having the value of positive integers.

Answer 1

Yes, you can, provided that the summands are disjoint, i.e. that $$ \left( {X \cap Y = k} \right) \cap \left( {X \cap Y = j} \right) = \emptyset \quad \left| {\;k \ne j} \right. $$