The Law of Large Numbers
- How many times do you need to shuffle a deck of cards to make them truly random? How much uranium does it take to build a nuclear bomb? (explosion booming) How can you predict the next word in a sentence? And how does Google know which page you're actually searching for? Well, the reason we know the answer to all of these questions is because of a strange math feud in Russia that took place over 100 years ago. In 1905, socialist groups all across Russia rose up against the Tsar, the ruler of the empire.
They demanded a complete political reform, or failing that, that he stepped down from power entirely. - This divided the nation into two. So on one side you got the Tsarists, right? They wanted to defend the status quo and keep the Tsar in power. But then on the other side, you had the socialists who wanted this complete political reform. And this division was so bad that it crept into every part of society to the point where even mathematicians started picking sides.
- [Derek] On the side of the Tsar was Pavel Nekrasov, unofficially called the Tsar of Probability. Nekrasov was a deeply religious and powerful man, and he used his status to argue that math could be used to explain free will and the will of God. - His intellectual nemesis on the socialist side was Andrey Markov, also known as Andrey The Furious. Markov was an atheist and he had no patience for people who were being unrigorous, something he considered Nekrasov to be, because in his eyes, math had nothing to do with free will or religion.
So he publicly criticized Nekrasov's work, listing it among "the abuses of mathematics." Their feud centered on the main idea people had used to do probability for the last 200 years. And we can illustrate this with a simple coin flip. When I flip the coin 10 times, I get six times heads and four times tails, which is obviously not the 50/50 you'd expect. But if I keep flipping the coin, then at first the ratio jumps all over the place.
But after a large number of flips, we see that it slowly settles down and approaches 50/50. And in this case, after 100 flips, we end up on 51 heads and 49 tails, which is almost exactly what you would expect. This behavior that the average outcome gets closer and closer to the expected value as you run more and more independent trials is known as the law of large numbers. It was first proven by Jacob Bernoulli in 1713, and it was the key concept at the heart of probability theory right up until Markov and Nekrasov.
But Bernoulli only proved that it worked for independent events like a fair coin flip, or when you ask people to guess how much they think an item is worth, where one event doesn't influence the others. But now imagine that instead of asking each person to submit their guess individually, you ask people to shout out their answer in public. Well, in this case, the first person might think it's an extraordinarily valuable item, and say it's worth around $2,000, but now all the other people in the room are influenced by this value, and so, their guesses have become dependent.
And now the average doesn't converge to the true value, but instead it clusters around a higher amount. - And so, for 200 years, probability had relied on this key assumption, that you need independence to observe the law of large numbers. And this was the idea that sparked Nekrasov and Markov's feud. See, Nekrasov agreed with Bernoulli that you need independence to get the law of large numbers. But he took it one step further.
He said, if you see the law of large numbers, you can infer that the underlying events must be independent. - Take this table of Belgian marriages from 1841 to 1845. Now you see that every year the average is about 29,000. And so, it seems like the values converge and therefore that they follow the law of large numbers. And when Nekrasov looked at other social statistics like crime rates and birth rates, he noticed a similar pattern.
But now think about where all this data is coming from. It's coming from decisions to get married, decisions to commit crimes, and decisions to have babies, at least for the most part. So Nekrasov reasoned that because these statistics followed the law of large numbers, the decisions causing them must be independent. In other words, he argued that they must be acts of free will. So to him, free will wasn't just something philosophical, it was something you could measure. It was scientific.