The numbereis as ubiquitous as pi, turning up everywhere, and especially in probability theory. If you randomly select real numbers between 0 and 1, and continue until their sum exceeds 1, the expected number of choices ise.

-Martin Gardner

note in Calculus Made Easy, p. 153

Of course, this begs for a Python simulation:

A derivation would be nice, but that'll have to wait. I wonder about the variance.