Random numbers added until the total tops one

Question

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

Accepted Answer

Let N be the number of draws. The trick is to count via the tail probabilities. You are still going after n draws exactly when the first n numbers have not yet reached 1: P(N n) = P(U1 + U2 + \cdots + Un \le 1). That event is the region of the unit cube where the coordinates sum to at most 1, a standard simplex, whose volume is famously P(U1 + \cdots + Un \le 1) = \frac{1}{n!}. Now use the tail-sum formula for a nonnegative integer variable, E[N] = \sum{n \ge 0} P(N n): E[N] = \sum{n=0}^{\infty} \frac{1}{n!} = 1 + 1 + \frac12 + \frac16 + \cdots = e \approx 2.718. Takeaway. The expected count …

Random numbers added until the total tops one

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