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4 Tutorials that teach Law of Large Numbers/Law of Averages

Law of Large Numbers/Law of Averages

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Author: Sophia Tutorial
Description:

This lesson will explain the law of large numbers and the law of averages.

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Tutorial

This tutorial is going to deal with an important statistical concept called:

1. Law of Large Numbers
2. Law of Averages

1. LAW OF LARGE NUMBERS

Suppose you flip a coin and determine the long-term relative frequency of heads. You're going to determine heads to be a win. If you flip heads, you're will go up to the next dot from this segment. And if you flip tails, you will go down to the lose dot from this segment.

So suppose the first coin you flip is heads. You'll go up to the first dot, as high as you can go. So if you flip heads again, you're only going to just go to the second dot.

Suppose the third time, you flip tails. That means you will follow that down. And then you flip heads again, you will follow it back up. And so you can continue looking at what happens in the long term by simulating the rest.

You will start to notice is these swings at the beginning are pretty wild swings. They're very large shifts, whereas over to the right, these are not very big of shifts.

And what you may notice is that the long term relative frequency of heads seems to be settling in right around 50%.

The law of large numbers says that the more times you run a chance experiment, the relative frequency of an event approaches the true probability of that event. And you're going to get closer and closer to the right answer the larger the number of trials becomes. So you can keep going, and you would probably get closer and closer and closer to 50% as you kept going to the right.

Now, this isn't to say that odd things don't happen. Look closely. You did have runs of three and four heads during our experiments. So it's not to say that things don't happen that are unusual. Four heads in a row is fairly unusual, but it happens. You also had a couple of different runs of three tails in a row.

So unusual things can happen in the short term, but predictably, what will happen in the long term is that this blue line will start to settle in right around 50%.

• Law of Large Numbers
• A mathematical rule that states that as the number of trials of a chance experiment increase, the experimental probability of an event becomes closer to the true probability of that event.

2. LAW OF AVERAGES

On the radio, you might hear a sports announcer say one or both of these two things:

This first saying is saying that he's due, which means he's going to get a hit this time because he hasn't gotten a hit so far. The second one is saying that he's going to get a hit this time because he has gotten a hit. None of these make any sense. Both of these logically are fallacies.

They apply the law of large numbers, which means that maybe this player gets a hit one out of every three times he's at bat in the long term. But what they're trying to do is apply the law of large numbers to these five at-bats that this player has had.

This is sometimes called the law of averages. And it's not actually a mathematical term. It's a psychological game people play with themselves to convince themselves that favorable outcomes are just over the horizon

You see the "law of averages" a lot in the casino. Remember, it's not actually a mathematical term, but rather a way for individual's to convince themselves of a favorable outcome.

The law of averages - the false law of averages - is also called the gambler's fallacy or the gambler's ruin. People can convince themselves that favorable outcomes are about to happen even when they're not necessarily going to happen, because they're applying the law of large numbers to small numbers of trials.

• "Law of Averages"/Gambler's Fallacy/Gambler's Ruin
• A misapplication of the Law of Large Numbers, where people try to apply long-run probabilities to short-run events. The false "Law of Averages" is not a mathematical phenomenon, but rather a psychological trick people play on themselves to convince themselves that favorable outcomes are about to occur, using past behavior to influence their reasoning.

The Law of Large Numbers is important. It states that over the long term, the relative frequency of an event is the probability of that event. The key is distinguishing long term from short term.

And the law of large numbers sometimes will be inappropriately applied to short term events. Sometimes that's called the law of averages and the gambler's fallacy. And it's important not to fall into that trap of applying the predictable nature of large numbers to the unpredictable nature of short term events.

Good luck!

Source: This work is adapted from Sophia author jonathan osters.

Terms to Know
Law of Averages/Gambler's Fallacy/Gambler's Ruin

A misapplication of the Law of Large Numbers, where people try to apply long-run probabilities to short-run events. The false "Law of Averages" is not a mathematical phenomenon, but rather a psychological trick people play on themselves to convince themselves that favorable outcomes are about to occur, using past behavior to influence their reasoning.

Law of Large Numbers

A mathematical rule that states that as the number of trials of a chance experiment increase, the experimental probability of an event becomes closer to the true probability of that event.