Flying Toupees!

An infinite number of men, all named Ronald Frump and all wearing toupees, line up in a row (a very long row!)

The first man's toupee flies off his head.

If the first man's toupee flies off his head, there is a \(\frac{1}{2}\) probability that the second man's toupee will fly off his head.

If the second man's toupee flies off his head, there is a \(\frac{1}{3}\) probability that the third man's toupee will fly off his head.

And so on...

In fact, in general rule is that...

If the \( (n-1)^\text{th}\) man's toupee flies off his head, then there is a \(\frac{1}{n}\) probability that the \(n^\text{th}\) man's toupee will fly off his head.

So, now for the question...

What is the expected number of toupees that will fly off heads?

Give your answer to 5 decimal places.

Assumption: The only chance a man has of his toupee flying off his head is if the man before him had his toupee fly off as per the above probabilities. Otherwise, the toupee will stay firmly in place.


Image credit: http://www.solikeyouknow.com/
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