Random amounts added until the total tops one hundred

Question

You draw independent amounts, each uniform on [0, 100], and add them up, stopping when the running total first exceeds 100. On average, how many amounts do you draw?

Accepted Answer

Scaling does not change a waiting-count. Divide everything by 100: each amount Ui uniform on [0,100] becomes Ui/100, uniform on [0,1], and the target 100 becomes 1. The count of draws until the sum first exceeds the target is identical, because dividing every quantity by 100 does not change when the total crosses the line. So this is exactly "uniform [0,1] draws until the sum exceeds 1," whose answer is E[N] = \sum{n=0}^{\infty} \frac{1}{n!} = e \approx 2.718. The scale of the numbers is a red herring; only the ratio of target to range matters, and here it is 1. Takeaway. Waiting counts are s…

Random amounts added until the total tops one hundred

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