What is the probability of $B^2\ge 4AC$ if A, B and C are independent exponential random variables with parameter 1?

Question

I tried the following integral, is it correct?

$$\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\frac{b^2}{4a}}e^{-b}e^{-a}e^{-c}\,dc\,da\,db$$

Answer 1

An easier approach (but please double-check my calculations): fix $A,\,C$ so$$P(B\ge2\sqrt{AC})=\exp(-2\sqrt{AC}).$$With $a=s\cos^4\theta,\,b=s\sin^4\theta$, averaging gives$$\begin{align}\int_{[0,\,\infty)^2}\exp(-(\sqrt{a}+\sqrt{c})^2)dadc&=4\int_0^{\pi/2}\cos^3\theta\sin^3\theta d\theta\int_0^\infty s\exp(-s)ds\\&=4\frac{1}{12}1\\&=\frac13\end{align}$$by Beta and Gamma functions. Python agrees with me.