Random numbers added until the total tops one half

Question

You draw independent numbers, each uniform on [0, 1], adding them up, and stop as soon as the running total first exceeds 	frac12. On average, how many numbers do you draw?

Accepted Answer

Let N be the number of draws. You are still going after n draws exactly when the first n still sum to at most 	frac12: P(N n) = P\!\left(U1 + \cdots + Un \le 	frac12ight). For a threshold t \le 1, the region where n uniforms sum to at most t is a scaled simplex with volume \frac{t^n}{n!}. With t = 	frac12: P(N n) = \frac{(1/2)^n}{n!}. Summing the tail, E[N] = \sum{n \ge 0} P(N n): E[N] = \sum{n=0}^{\infty} \frac{(1/2)^n}{n!} = e^{1/2} = \sqrt{e} \approx 1.6487. In general, the expected number of uniform [0,1] draws to exceed a threshold t \in [0,1] is e^{t}; the original problem is the c…

Random numbers added until the total tops one half

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