Given two independent random variables $X$ and $Y$. Does the statement
$P( X + Y = k | Y = k) = P(X = 0)$
hold true always?
My logic is
$P( X + Y = k | Y = k) \\= \frac{P(X + Y = k \cap Y = k)}{P(Y=k)} \\= \frac{P(X+k= k \cap Y=k)}{P(Y=k)} \\= \frac{P(X=0 \cap Y =k)}{P(Y=k)} \\= \frac{P(X=0)P( Y =k)}{P(Y=k)} \\= P(X=0) $
but I'm not sure if the second and third line are correct.