27 years ago I wrote,

There are some 10^{10} people in the world. Roughly speaking, I am among the top 10^{5} most successful. Accounting for this improbably high position in the hierarchy is a troubling problem for me. A tantalizing idea suggests itself: can one argue somehow that in a random population of *n* people, the expected position to find an introspective person of my sort is on the order of √*n* from the top?

Well, here’s an argument that gets exactly that answer. The person in position #1 will be pretty interested in his or her good fortune. Suppose we imagine that person #2 is half as interested, person #3 is 1/3 as interested, and so on: essentially, each person’s interest in their luck is proportional to their luck. Then the total amount of human interest in positional luck, integrated over the whole of humanity, is about log *n*.

Now pick a human being at random, weighted by their interest in their luck. The middle position is ½log *n*, which is the logarithm of √*n*. So there you have it. If I am person number 10^{5}, then half the population’s interest in their luck is to be found in people higher up than I am, and half in people lower down, making me quite entirely typical. Throwing in a few trillion dogs, squirrels, and mosquitoes, with their lower levels of introspection, won’t change the result.

[12 April 2011]

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