Benford's law, first digit equal to 1

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Benford’s law states that in many data collections the probability that the first significant digit is $d$ equals $\log_{10}\!\left(1+\dfrac1d\right)$. What is the probability that the first digit is $1$?

Solution

Just set $d=1$: $P=\log_{10}\!\left(1+\frac11\right)=\log_{10}2\approx \boxed{0.301}.$ Surprisingly the digit $1$ appears as the first digit about $30\%$ of the time, far more than the $\frac19\approx 11\%$ one would expect from a uniform distribution.