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First occurrence of a decimal digit

Under uniform length on

[0, 1]

, the position

K

of the first occurrence of a fixed decimal digit has

Pr (K = k) = \frac{1}{10} (\frac{9}{10})^{k - 1} .

The set of expansions in which the digit never occurs has length zero, and

E K = \sum_{k \geq 1} \frac{k}{10} (\frac{9}{10})^{k - 1} = 10.

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