You can actually beat 50% in the envelope game

Two envelopes hold amounts $a$ and $2a$ for some unknown positive $a$; you know nothing about $a$. You pick one at random and open it to see $X$. You may keep it or switch to the other.

Blindly switching (or blindly keeping) obviously gives the larger envelope with probability exactly $\tfrac12$. But consider this strategy: before playing, draw a random threshold $T$ from a distribution that is positive everywhere on $(0, \infty)$ (say an exponential). After opening, switch if $X < T$ and keep if $X \ge T$.

Show that this strategy gives you the larger envelope with probability strictly greater than $\tfrac12$.