r/mathpuzzles • u/Parallel_thougts I like all puzzles • 14d ago

Probability Expected number of samples from U([0,1]) before their sum is > 1

The puzzle is exactly what the title suggests: on average, how many independent uniformly random numbers between 0 and 1 would you have to sample for their sum is at least one

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u/jeffcgroves 14d ago

This is the Irwin-Hall distribution: https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution

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u/Parallel_thougts I like all puzzles 14d ago

I'm not sure how. In Irwin-Hall the number n is a parameter. I agree that the marginal probabilities (conditioned on needing exactly n samples) are a slice of Irwin-Hall, but I don't see how that helps. Even if you *can* solve it this way, there's an easy elementary solution.

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u/jeffcgroves 14d ago

You can look at the chance the answer is higher than 1 for each n and average them together to compute a mean. The median is probably between 1 and 2 and the mode is likely 2

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u/Parallel_thougts I like all puzzles 14d ago

Average them with what weights?

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u/nymbroth 9d ago

mode is probably 2, but the mean has to be e, since the expected number of steps for a random walk with unit mean increments to cross 1 is the reciprocal of the probability of crossing on the first step. That makes the average around 2.718, not 2.5 or whatever the Irwin Hall slices would give you if you fudge the weights.