Quadratic with Rayleigh and exponential coefficients

Let $B$ have the Rayleigh density $f(b)=b\,e^{-b^2/2}$ for $b>0$, and let $C\sim\text{Exp}(2)$ (rate $2$) be independent of $B$. What is the probability that $x^2+Bx+C=0$ has no real roots?

$\tfrac12$. No real roots means $B^2<4C$, i.e. $C>B^2/4$. Given $B=b$ the exponential tail gives $e^{-2\cdot b^2/4}=e^{-b^2/2}$, and averaging over the Rayleigh density yields $\int_0^\infty b\,e^{-b^2}\,db=\tfrac12$.