Let $X$ have standard deviation $\sigma$ and $Y$ standard deviation $\tau$. I can write $$P(X>0\mid X+Y>0) = \frac{P(X>0, X+Y>0)}{P(X+Y>0)}$$ As all variables are zero-mean and symmetric, $P(X+Y>0)=1/2$, so that $$P(X>0\mid X+Y>0) = 2 P(X>0, X+Y>0) = 2 P(X>0, Y>-X)$$ Using the independence of $X$ and $Y$ and the notation $\Phi$ and $\phi$ for the standard normals, I get $$P(X>0\mid X+Y>0) = 2 \int_0^\infty \phi\left(\frac{x}{\sigma}\right) \left(1-\Phi\left(-\frac{x}{\tau}\right)\right) dx.$$ But from here on I do not know how to continue to get a simple expression. Any suggestions?
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It might help to draw the picture in 2 dimensions. Draw the line $x+y=0$ and look at symmetry to tell you what "fraction" of the region $x+y >0$ has $x>0$. – Paul Apr 03 '20 at 17:58
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3@Ian, the upper half of the 4th quadrant still has $x>0$. The answer is actually $3/4$. – Paul Apr 03 '20 at 18:28