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Law of Large Numbers/Law of Averages

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Tutorial

Source: Huff and Geis, How to Take a Chance.

Hi. This tutorial covers the law of large numbers, as well as the law of averages. So let's start with the law of large numbers. So it's defined as a law that states as the number of observations of a random event increases, the experimental probability of the event will approach the theoretical probability. So let's take a look at an example of this.

So we're going to start with a very simple chance experiment is we're just going to flip a coin. And we're going to let A be the event that the coin lands heads. So if we're thinking about the theoretical probability, if we flip a coin, we ended up with tails there. So there's a 50-50 chance. So 50-50 means that the probability ends up being 1/2, which is equivalent to 0.5. So the probability of flipping a coin and it landing heads is 0.5.

So now let's actually run this experiment and calculate the experimental probability. So what I'm going to do is I'm going to the experiment, first, do it five times and come up with the experimental probability for five flips of the coin. And then I'm going to do it for 20 flips and come up with the experimental probability. And hopefully, the experimental probability will be closer when we do 20 flips than if we just to do five flips.

So let's start with five flips. So I'm going to flip the coin five times. So heads. Flip it again, and that ended up being tails another tails. So that's three flips-- heads. And the last flip is tails. So the probability of getting heads-- so remember A is the event that we get heads-- ends up being 2 out of 5-- the number of favorable outcomes divided by the number of total outcomes. So as a decimal, that is 0.4. So close to 0.5 but still 0.1 off.

So now let's do 20 flips of the coin. So now that we've flipped it 20 times, we actually got it to turn out pretty perfect-- 10 and 10. So this probability ends up being 10 out of 20, which is 0.5. So what the law of large numbers isn't saying is that as you get higher and higher, you're going to get exactly 0.5. It turned out that we did get 0.5, which is very close to that theoretical probability of 0.5.

But generally, what will happen is as you increase the number of observations, this experimental probability should approach 0.5. So in this case, we did get it to work out pretty nicely and a good way of showing the law of large numbers there.

The next thing we have is what's called the law of averages. So the law of averages is a little different. Now it's a false law. So a lot of people call it the law of averages, but it's not actually a true probability. So it's a false law that states that the short run behavior of an event matches the long-term behavior of that same event. This is also known as the gambler's fallacy or the gambler's ruin.

So let's take a look at-- this is actually a historical example. So August 18, 1931 at the Casino in Monte Carlo, black came up a record 26 times in succession in roulette. So roulette is the game where you spin the wheel and the ball rotates around the game until it lands. And it can either land in red, black, or I think there's two that are green. But you bet on red or black. And if the ball lands on red, you win. Or if you bet on red and the ball lands on red, then you win.

So there was a near panicky rush to bet on red beginning about the time black had come up a phenomenal 15 times. In application of the law of averages doctrine, players doubled and tripled their stakes. This doctrine, leading them to believe that after black came up the 20th time that there was not a chance in a million of another repeat. In the end, the unusual run enriched the casino by some millions of francs.

So basically, since it was happening-- since black had shown up so many times, people thought it was due to turn up red. But remember that this is just short-term behavior. 26, 27 times-- that's still in the short run. It should have come up-- red and black should come up about the same amount in the long run but certainly not in the short run.

So because people were believing of this law of averages, they ended up losing, and the casino ended up making a lot of money off of these people. So that is kind of a good example of the law of averages. So this has been your tutorial on the law of large numbers and the law of averages. Thanks for watching.