Circle area versus ellipse area with exponential radii

An ellipse centred at the origin has horizontal semi-axis $A$ and vertical semi-axis $B$, where $A$ and $B$ are independent $\text{Exp}(1)$ random variables; its area is $\pi AB$. A circle is drawn whose diameter equals $A$. What is the probability that the circle has the larger area?

$1/5$. The circle's area is $\pi A^2/4$, which beats $\pi AB$ exactly when $A>4B$. For independent rate-$1$ exponentials, $\mathbb{P}[A>4B]=\int_0^\infty e^{-b}e^{-4b}\,db=\tfrac15$.