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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 $0.328$.

Dan
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    How strong is your numerical evidence for $\frac13$? I get $0.328\pm0.001$, so $\frac13$ seems unlikely. – joriki Feb 22 '23 at 12:14
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    @joriki Before your comment, I ran about several sets of $10,000$ trials each and I kept getting numbers mostly beginning with $0.33$. But after reading your comment, I ran $50$ trials of $10,000$ trials each, and I got an average of $0.3281$. Good call. – Dan Feb 22 '23 at 13:26
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    Unfortunately I don't see a way to calculate this. If it was $\frac13$, I'd try a bit harder, but since it isn't, there way well not be a closed form. – joriki Feb 22 '23 at 14:54
  • @joriki I feel the same way, about trying harder if it were $\frac13$. Here is a related question about the relationship between the expectation and the probability. – Dan Feb 23 '23 at 12:42

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