Given two IID N(0,1) variables X and Y, what's $P(X+2Y \geq 0 | X \geq 0)$?

Question

My attempt is that $$ \begin{aligned} P(X+2Y \geq 0 \mid X \geq 0) &= P(Y \geq 0) + P(Y\leq0 \text{ and }|Y| < X/2 \mid X \geq 0)\\ &= 1/2 + P(-X/2 <Y < 0 \mid X \geq 0)\\ &= 1/2 + \int_0^\inf (\int_{-x/2}^0 f_y(y)dy) f_x(x)dx \end{aligned} $$

Which doesn't seem easy to evaluate. Is my thought process correct up to this point or am I completely off the track? How should I proceed from here?

Answer 1

The fact that $X$ and $Y$ are normally distributed is irrelevant; all that matters is that they are IID symmetric random variables with mean $0$.

Another way to put it is this: What is the fraction of the region in the right half-plane that also satisfies the inequality $X + 2Y \ge 0$?