is on many levels, one of the most intuitive laws in

是在很多層面上，最直觀的規律之一。

mathematics and in probability theory.

數學和概率論方面。

But because it's so applicable to so many things, it's often a

但因為它適用於很多事情，所以它常常是一個。

misused law or sometimes, slightly misunderstood.

被誤用的法律或有時，稍有誤解。

So just to be a little bit formal in our mathematics, let

所以，為了讓我們的數學更正式一點，讓

me just define it for you first and then we'll talk a little

我就先給你下個定義，然後我們再聊一聊。

bit about the intuition.

位的直觀感受。

So let's say I have a random variable, X.

所以，讓我們'說我有一個隨機變量，X。

And we know its expected value or its population mean.

而我們知道它的期望值或它的人口平均值。

The law of large numbers just says that if we take a sample

大數定律只是說，如果我們取一個樣本。

of n observations of our random variable, and if we were

我們的隨機變量的n個觀測值，如果我們是

to average all of those observations-- and let me

平均所有這些觀察 - -讓我

define another variable.

定義另一個變量。

Let's call that x sub n with a line on top of it.

讓我們把那個x子n叫做上面有一條線。

This is the mean of n observations of our

這是我們的n次觀測的平均值。

random variable.

隨機變量。

So it's literally this is my first observation.

所以它'的字面意思是這是我的第一個觀察。

So you can kind of say I run the experiment once and I get

所以，你可以說，我運行的實驗一次，我得到。

this observation and I run it again, I get that observation.

這個觀察結果，我再運行一次，就會得到這個觀察結果。

And I keep running it n times and then I divide by my

我繼續運行n次，然後我除以我的。

number of observations.

意見的數量；

So this is my sample mean.

所以這是我的樣本平均值。

This is the mean of all the observations I've made.

這是我所有觀察結果的平均值'。

The law of large numbers just tells us that my sample mean

大數定律只是告訴我們，我的樣本平均值

will approach my expected value of the random variable.

將接近我對隨機變量的期望值。

Or I could also write it as my sample mean will approach my

或者我也可以寫成我的樣本平均值將接近我的

population mean for n approaching infinity.

n接近無窮大時的人口平均值。

And I'll be a little informal with what does approach or

而我'會有點非正式的做法，什麼是方法或。

what does convergence mean?

融合是什麼意思？

But I think you have the general intuitive sense that if

但我想你有一般的直觀感受，如果說

I take a large enough sample here that I'm going to end up

我在這裡採取了足夠大的樣本，我'要結束了。

getting the expected value of the population as a whole.

得到整體人口的預期值。

And I think to a lot of us that's kind of intuitive.

我想對我們很多人來說，這是一種直覺。

That if I do enough trials that over large samples, the trials

如果我做了足夠多的試驗，在大的樣本中，試驗...

would kind of give me the numbers that I would expect

會種給我的數字，我希望

given the expected value and the probability and all that.

給定的預期值和概率和所有這些。

But I think it's often a little bit misunderstood in terms

但我認為它經常有點誤解的方面'。

of why that happens.

的原因。

And before I go into that let me give you

在我說這個之前，讓我給你說說

a particular example.

一個特別的例子。

The law of large numbers will just tell us that-- let's say I

大數定律會告訴我們--------比如說我

have a random variable-- X is equal to the number of heads

有一個隨機變量--X等於人頭數。

after 100 tosses of a fair coin-- tosses or flips

百發百中

of a fair coin.

幣的公平。

First of all, we know what the expected value of

首先，我們可以知道什麼是預期值

this random variable is.

這個隨機變量是。

It's the number of tosses, the number of trials times

這是'的折騰次數，試驗次數。

the probabilities of success of any trial.

任何審判的成功概率；

So that's equal to 50.

所以，這'等於50。

So the law of large numbers just says if I were to take a

所以，大數定律只是說，如果我拿一個。

sample or if I were to average the sample of a bunch of these

的樣本，或者說如果我把這些樣本的平均數。

trials, so you know, I get-- my first time I run this trial I

試驗，所以你知道，我... ... 我第一次參加這個試驗時，我...

flip 100 coins or have 100 coins in a shoe box and I shake

拋出100個硬幣或有100個硬幣在鞋盒裡，我搖動。

the shoe box and I count the number of heads, and I get 55.

鞋盒裡，我數了數人頭的數量，得到55個。

So that Would be X1.

所以這將是X1。

Then I shake the box again and I get 65.

然後我再搖一搖盒子，我得到65。

Then I shake the box again and I get 45.

然後我又搖了搖盒子，我得到了45。

And I do this n times and then I divide it by the number

我這樣做n次，然後我把它除以數字。

of times I did it.

的時候，我做了。

The law of large numbers just tells us that this the

大數定律只是告訴我們，這是個

average-- the average of all of my observations, is going

平均 - 平均我所有的觀察，是怎麼回事

to converge to 50 as n approaches infinity.

當n接近無窮大時，收斂到50。

Or for n approaching 50.

或者對於n接近50。

I'm sorry, n approaching infinity.

我'對不起，n接近無窮大。

And I want to talk a little bit about why this happens

我想說說為什麼會發生這種情況。

or intuitively why this is.

或直觀地瞭解到這是為什麼。

A lot of people kind of feel that oh, this means that if

很多人都有點覺得，哦，這意味著，如果。

after 100 trials that if I'm above the average that somehow

經過100次試驗，如果我高於平均水平，不知何故。

the laws of probability are going to give me more heads

概率法則會給我更多的人頭

or fewer heads to kind of make up the difference.

或更少的人頭，算是彌補了這一差距。

That's not quite what's going to happen.

那'不完全是'要發生的事。

That's often called the gambler's fallacy.

這'就是常說的賭徒'的謬論。

Let me differentiate.

讓我來區分一下。

And I'll use this example.

而我'就用這個例子。

So let's say-- let me make a graph.

所以，讓我們'say -- 讓我做一個圖。

And I'll switch colors.

而我'會換顏色。

This is n, my x-axis is n.

這是n，我的x軸是n。

This is the number of trials I take.

這是我參加的試驗次數。

And my y-axis, let me make that the sample mean.

而我的Y軸，讓我把它變成樣本平均值。

And we know what the expected value is, we know the expected

我們知道什麼是預期值，我們知道預期的。

value of this random variable is 50.

這個隨機變量的值是50。

Let me draw that here.

讓我把它畫在這裡。

This is 50.

這是50。

So just going to the example I did.

所以就以我做的例子來說明。

So when n is equal to-- let me just [INAUDIBLE]

所以，當n等於 -- 讓我[聽不清]。

here.

在這裡。

So my first trial I got 55 and so that was my average.

所以我的第一次試驗我得了55分，所以這是我的平均成績。

I only had one data point.

我只有一個數據點。

Then after two trials, let's see, then I have 65.

然後經過兩次試驗，讓'看看，那麼我有65。

And so my average is going to be 65 plus 55 divided by 2.

所以我的平均數將是65加55除以2。

which is 60.

這是60。

So then my average went up a little bit.

所以後來我的平均水平就上升了一點。

Then I had a 45, which will bring my average

然後，我有一個45，這將使我的平均。

down a little bit.

下了一點。

I won't plot a 45 here.

我不會在這裡策劃一個45號。

Now I have to average all of these out.

現在我得把這些東西平均起來。

What's 45 plus 65?

什麼是45加65？

Let me actually just get the number just

讓我實際上只是得到的數字只是

so you get the point.

所以你得到的點。

So it's 55 plus 65.

所以是55加65。

It's 120 plus 45 is 165.

它'的120加45是165。

Divided by 3.

除以3。

3 goes into 165 5-- 5 times 3 is 15.

3進入165 5 -- 5次3是15。

It's 53.

這是53。

No, no, no.

不，不，不。

55.

55.

So the average goes down back down to 55.

所以平均數又回落到55。

And we could keep doing these trials.

我們可以繼續做這些試驗。

So you might say that the law of large numbers tell this,

所以你可以說，大數法則告訴。

OK, after we've done 3 trials and our average is there.

好了，在我們'做了3次試驗，我們的平均水平在那裡。

So a lot of people think that somehow the gods of probability

所以，很多人認為，不知為何，概率之神。

are going to make it more likely that we get fewer

將使我們更有可能得到更少的。

heads in the future.

頭在未來。

That somehow the next couple of trials are going to have to

接下來的幾場審判都要以某種方式進行

be down here in order to bring our average down.

為了讓我們的平均水平下降，在這裡下。

And that's not necessarily the case.

而事實卻未必如此。

Going forward the probabilities are always the same.

往後的概率總是一樣的。

The probabilities are always 50% that I'm

概率永遠是50%，我'米。

going to get heads.

會得到頭。

It's not like if I had a bunch of heads to start off with or

它不像如果我有一堆頭開始或

more than I would have expected to start off with, that all of

比我一開始預想的更多，所有的

a sudden things would be made up and I would get more tails.

一下子事情就會被編造出來，我就會得到更多的尾巴。

That would the gambler's fallacy.

這將賭徒'的謬論。

That if you have a long streak of heads or you have a

如果你有一個長長的人頭或你有一個。

disproportionate number of heads, that at some point

過多的人頭，以至於在某些時候

you're going to have-- you have a higher likelihood of having a

你會有 - 你有一個較高的可能性 有一個。

disproportionate number of tails.

不成比例的尾數。

And that's not quite true.

而這並不完全正確。

What the law of large numbers tells us is that it doesn't

大數定律告訴我們的是，它不'。

care-- let's say after some finite number of trials your

照顧--比方說，經過一些有限的試驗，你的。

average actually-- it's a low probability of this happening,

其實平均--------這種情況發生的概率很低。

but let's say your average is actually up here.

但讓我們'說你的平均水平實際上是在這裡。

Is actually at 70.

其實是在70。

You're like, wow, we really diverged a good bit from

你'喜歡，哇，我們真的分歧了一個很好的位從

the expected value.

的預期值。

But what the law of large numbers says, well, I don't

但大數法則怎麼說，我不'；。

care how many trials this is.

關心這是多少個試驗。

We have an infinite number of trials left.

我們還有無限次的試驗。

And the expected value for that infinite number of trials,

而這無限次試驗的預期值。

especially in this type of situation is going to be this.

尤其是在這種情況下是會這。

So when you average a finite number that averages out to

所以，當你把一個有限的數平均起來，平均到

some high number, and then an infinite number that's going to

一些高數，然後一個無限的數字，'的要去

converge to this, you're going to over time, converge back

趨向於此，你會隨著時間的推移，趨向於此。

to the expected value.

到預期值。

And that was a very informal way of describing it, but

這是一種非常非正式的描述方式，但是... ...

that's what the law or large numbers tells you.

這'就是法律或大數告訴你的。

And it's an important thing.

而這'是一件重要的事情。

It's not telling you that if you get a bunch of heads that

它'不是告訴你，如果你得到了一群頭，。

somehow the probability of getting tails is going

某種程度上，得到尾巴的概率是要的

to increase to kind of make up for the heads.

來增加，以種彌補頭。

What it's telling you is, is that no matter what happened

它告訴你的是，不管發生了什麼事情

over a finite number of trials, no matter what the average is

在有限的試驗次數中，無論平均數是多少，都是如此。

over a finite number of trials, you have an infinite

在有限的試驗次數中，你有一個無限的

number of trials left.

剩下的審判次數。

And if you do enough of them it's going to converge back

如果你做了足夠多的人，它就會匯合回來。

to your expected value.

到你的預期值。

And this is an important thing to think about.

而這是一件很重要的事情，值得思考。

But this isn't used in practice every day with the lottery and

但這並不是用在實踐中天天中彩票和。

with casinos because they know that if you do large enough

因為他們知道，如果你做的足夠大的

samples-- and we could even calculate-- if you do large

樣本 -- 我們甚至可以計算 -- 如果你做大的。

enough samples, what's the probability that things

足夠的樣本，什麼'的概率的東西

deviate significantly?

偏差很大？

But casinos and the lottery every day operate on this

但賭場和彩票每天都在此基礎上進行操作。

principle that if you take enough people-- sure, in the

原則，如果你採取足夠的人 - 當然，在... ...

short-term or with a few samples, a couple people

短期或有幾個樣品，幾個人。

might beat the house.

可能會打敗房子。

But over the long-term the house is always going to win

但從長遠來看，房子總是要贏的。

because of the parameters of the games that they're

因為他們'遊戲的參數。

making you play.

讓你玩。

Anyway, this is an important thing in probability and I

總之，這是概率中很重要的一件事，我

think it's fairly intuitive.

認為它'相當直觀。

Although, sometimes when you see it formally explained like

雖然，有時當你看到它的正式解釋，如

this with the random variables and that it's a little

這與隨機變量，它是一個小的

bit confusing.

有點混亂。

All it's saying is that as you take more and more samples, the

所有它說的是，當你採取越來越多的樣品，。

average of that sample is going to approximate the

該樣本的平均數將近似於...

true average.

真正的平均值。

Or I should be a little bit more particular.

或者我應該更特別一點。

The mean of your sample is going to converge to the true

你的樣本的均值將趨近於真實的。

mean of the population or to the expected value of

人口的平均值或預期值。

the random variable.

的隨機變量。

Anyway, see you in the next video.

總之，下一個視頻裡見。