Let $X,Y$ be independent random variables such that $X\sim\chi_{n-1}^{2}, Y\sim\chi_{1}^{2}$ are chi-squared distributed (where $n\geq2$ is a natural number). I am trying to evaluate $\mathbb{P}[X\leq Y]$ as a function of $n$, at least asymptotically when $n\to\infty$. Obviously this probability decays to 0, but I'd like to be able to say something more quantative.

I was able to reduce this into a purely analytic question: Denoting $Z=X-Y$, I'm seeking for $F_Z(0)$. Expressing $F_Z$ using $f_X,F_Y$ I found that $$\mathbb{P}[X\leq Y]=\frac{1}{2^{\frac{n-1}{2}}\Gamma(\frac{n-1}{2})}\int_{0}^{\infty}\left(erfc(\sqrt{x})x^{\frac{n-3}{2}}e^{-\frac{x}{2}}\right)dx $$ where $erfc$ is the complementary error function.

To my suprise, wolfram spits out nice looking algebraic numbers for odd $n$, which gives me hope that this integral can be expressed as a (simpler) function of $n$. Any help, either evaluating the integral or calculating the asymptotic probability in some other way will be much appreciated.