20.5 The law of large numbers
We’ll continue our journey with a quite remarkable and famous result: the law of large numbers. You have probably already heard several faulty arguments invoking the law of large numbers. For instance, gamblers are often convinced that their bad luck will end soon because of said law. This is one of the most frequently misused mathematical terms, and we are here to clear that up.
We’ll do this in two passes. First, we are going to see an intuitive interpretation, then add the technical but important mathematical details. I’ll try to be gentle.
20.5.1 Tossing coins…
First, let’s toss some coins again. If we toss coins repeatedly, what is the relative frequency of heads in the long run?
We should have a pretty good guess already: the average number of heads should converge to P(heads) = p as well. Why? Because we saw this when studying the frequentist interpretation of probability in Section 18.2.7.
Our simulation...
