Sevens Are Everywhere

The number \(7\) appears everywhere. There are \(7\) days in a week, \(7\) wonders of the ancient world, and \(7\) continents on the planet earth.

If a positive integer less than or equal to \(n\) is chosen randomly, the probability that it contains a \(7\) in it's decimal expansion is \(p_n.\) As \(n\) grows large, what does \(p_n\) approach?

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