I need help you with a problem of conditional probability joint continuous

Question

$f_{X,Y}(x,y)=x+y, 0\le x\le 1, 0\le y\le 1$

$P(X>\frac{1}{5}|Y=y)$?

My solution

$f(x|y)=\frac{x+y}{x+\frac{1}{2}}$

$P(X>\frac{1}{5}|Y=y)=$

$\int _{\frac{1}{5}}^1\:\frac{x+y^2}{x+\frac{1}{2}}dx$

My procedure is correct?

Answer 1

It's almost correct. The last step should use the number $\frac15$:

$f(x|y) = \frac{f(x,y) }{f(y)}= \frac{f(x,y) }{\int_{0}^1 f(x,y)dx} = \frac{x+y}{y+\frac{1}{2}}$

$P(X>\frac{1}{5}|Y=y)=\int_{\frac15}^1\:\frac{x+y^2}{y+\frac{1}{2}}dx$.