What is the probability that $\prod\limits_{k=1}^\infty \left(1+\prod\limits_{i=1}^k u_i\right)>e$, where the $u$'s are i.i.d. $\text{Uniform}(0,1)$-variables ?
The product, $\prod\limits_{k=1}^\infty \left(1+\prod\limits_{i=1}^k u_i\right)$, has an expectation of $e$. I am wondering, what is the probability that it is greater than its expectation.
Excel simulations suggest that the answer is (simply) $\frac13$.
EDIT: After reading @joriki's comment, I ran more Excel simulations, and now it seems the answer is more like $0.328$.