Problem of the Week, #118

A colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

PROBLEM 118

Suppose that you draw numbers x_1,x_2,\dots from [0,1] randomly until x_1+x_2+\dots +x_n first exceeds 1. What is the probability that this happens on the fourth draw? That is, what is the probability that x_1+x_2+x_3 < 1 and x_1+x_2+x_3+x_4 > 1?

PROBLEM 118A

In the above problem, what is the expected number of draws until the sum first exceeds 1?

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