Number of Random Fractional Reals whose Total Exceeds 1

Theorem

Let real numbers be selected at random following a continuous uniform distribution from the interval $\closedint 0 1$ until their total sum is greater than $1$.

The expectation of the number of selections is Euler's number $e$.

Proof

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Sources

• 1990: Nick Mackinnon: Another Surprising Appearance of e (The Mathematical Gazette Vol. 74: pp. 167 – 169)  www.jstor.org/stable/3619372

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ \ldots$