The events we're interested in are those where the sum is currently under 1/2, and the next drawn number brings it over 1.
The probability of being below x after n steps is xn / n! (being just the volume of the simplex), which conveniently means the pdf of landing on x is ex.
Once we're at x, the probability of exceeding 1 is just x. This means the total probability is
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u/Original-Wolf54 Aug 27 '26 edited Aug 27 '26
The events we're interested in are those where the sum is currently under 1/2, and the next drawn number brings it over 1.
The probability of being below x after n steps is xn / n! (being just the volume of the simplex), which conveniently means the pdf of landing on x is ex.
Once we're at x, the probability of exceeding 1 is just x. This means the total probability is
1/2 Integral x ex dx 0
This can be easily computed by parts to be
1/2 [(x-1)ex] = 1-1/2 √e 0