Suppose $X$ and $Y$ have the joint density $$ f(x,y)= \begin{cases} 6y,\ \text{for} \ (x,y)\in D \\ 0 \ \text{otherwise} \end{cases} $$ where $D$ is the triangular region where $x > 0$, $y > 0$, and $x + y < 1$.
Find $P\left(Y \geq \frac{1}{2}(1-X)\right)$
Attempted solution:
I think I am getting the bounds wrong but I get
$$P\left(Y \geq \frac{1}{2}(1-X)\right) = \int_{0}^{1}\int_{0}^{(1-x)/2} 6y dydx$$
which equals $1/4$ but the answer is $3/4$. I drew the picture and I am really not sure what I am doing wrong.
Judging by the limits of your integral, you're actually computing $P\left(Y \leq \frac{1}{2}(1-X)\right)$.