Digits to Infinity

Consider all positive integers. What percentage of these numbers contain the digit 3? Assume the numbers extend towards infinity.

Answer

Two things:

  • For an \(n\)-digit integer, the probability it has at least one “3” is

    \[ 1 - \bigl(\tfrac{9}{10}\bigr)^n, \]

    which tends to \(1\) as \(n\to\infty\).

  • Because there are infinitely many integers, those with large \(n\) vastly outnumber the small‐digit ones.

As a result, the overall fraction of integers containing a 3 converges to \(1\).

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