The argument I presented earlier to show that at least two humans must have the exact same number of hairs on their body is not directly related to the law of large numbers. The law of large numbers is a statistical concept that deals with the behavior of the average of a large number of independent random variables. It states that as the number of trials increases, the average of the observed outcomes will tend to converge to the expected value of the random variable.

However, the pigeonhole principle used in the argument is a combinatorial principle that states that if there are more pigeons than pigeonholes, then there must be at least one pigeonhole with more than one pigeon. In the case of humans and the number of hairs on their bodies, the principle is applied to show that there are more humans than there are possible distinct numbers of hairs on their bodies, so at least two humans must share the same number.

So while the pigeonhole principle and the law of large numbers both deal with counting and probability, they are not the same thing.
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