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Improper integration by truncation

Under uniform length on $[0,1]$, the position $K$ of the first occurrence of a fixed decimal digit has
$$
\Pr(K=k)=\frac1{10}\left(\frac9{10}\right)^{k-1}.
$$
The set of expansions in which the digit never occurs has length zero, and
$$
\mathbb E K=\sum_{k\geq1}\frac{k}{10}\left(\frac9{10}\right)^{k-1}=10.
$$


First occurrence of a decimal digit

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 12D / b / iii / Solution

First occurrence of a decimal digit

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First occurrence of a decimal digit

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