Let $X$ and $Y$ be continuous random variables with joint density function
$$f(x,y) = \begin{cases}24xy& \text{for } x>0,\; y>0,\; 0<x+y<1\\ 0 &\text{otherwise} \end{cases}$$
What is the conditional probability $$P\left(X < \frac 12 \;\left|\; Y = \frac 14\right)\right.$$
Any help would be much appreciated. Thank you.
Let $\left[x<\frac12\right]$ denote the function that takes value $1$ if $x<\frac12$ and takes value $0$ otherwise.
Then: $$\Pr\left(X<\frac12\mid Y=\frac14\right)=\frac{\int\left[x<\frac12\right]f(x,\frac14)dx}{\int f(x,\frac14)dx}\tag1$$
The function $g$ prescribed by:$$u\mapsto\frac{f(u,\frac14)}{\int f(x,\frac14)dx}\tag2$$ can be interpreted as the PDF of $X$ under condition $Y=\frac14$.
Observe that $(1)$ can also be written as: $$\int\left[x<\frac12\right]g(x)dx\tag3$$