Let's learn a little bit about the law of large numbers, which
is on many levels, one of the most intuitive laws in
mathematics and in probability theory.
But because it's so applicable to so many things, it's often a
misused law or sometimes, slightly misunderstood.
So just to be a little bit formal in our mathematics, let
me just define it for you first and then we'll talk a little
bit about the intuition.
So let's say I have a random variable, X.
And we know its expected value or its population mean.
The law of large numbers just says that if we take a sample
of n observations of our random variable, and if we were
to average all of those observations-- and let me
define another variable.
Let's call that x sub n with a line on top of it.
This is the mean of n observations of our
random variable.
So it's literally this is my first observation.
So you can kind of say I run the experiment once and I get
this observation and I run it again, I get that observation.
And I keep running it n times and then I divide by my
number of observations.
So this is my sample mean.
This is the mean of all the observations I've made.
The law of large numbers just tells us that my sample mean
will approach my expected value of the random variable.
Or I could also write it as my sample mean will approach my
population mean for n approaching infinity.
And I'll be a little informal with what does approach or
what does convergence mean?
But I think you have the general intuitive sense that if
I take a large enough sample here that I'm going to end up
getting the expected value of the population as a whole.
And I think to a lot of us that's kind of intuitive.
That if I do enough trials that over large samples, the trials
would kind of give me the numbers that I would expect
given the expected value and the probability and all that.
But I think it's often a little bit misunderstood in terms
of why that happens.
And before I go into that let me give you
a particular example.
The law of large numbers will just tell us that-- let's say I
have a random variable-- X is equal to the number of heads
after 100 tosses of a fair coin-- tosses or flips
of a fair coin.
First of all, we know what the expected value of
this random variable is.
It's the number of tosses, the number of trials times
the probabilities of success of any trial.
So that's equal to 50.
So the law of large numbers just says if I were to take a
sample or if I were to average the sample of a bunch of these
trials, so you know, I get-- my first time I run this trial I
flip 100 coins or have 100 coins in a shoe box and I shake
the shoe box and I count the number of heads, and I get 55.
So that Would be X1.
Then I shake the box again and I get 65.
Then I shake the box again and I get 45.
And I do this n times and then I divide it by the number
of times I did it.
The law of large numbers just tells us that this the
average-- the average of all of my observations, is going
to converge to 50 as n approaches infinity.
Or for n approaching 50.
I'm sorry, n approaching infinity.
And I want to talk a little bit about why this happens
or intuitively why this is.
A lot of people kind of feel that oh, this means that if
after 100 trials that if I'm above the average that somehow
the laws of probability are going to give me more heads
or fewer heads to kind of make up the difference.
That's not quite what's going to happen.
That's often called the gambler's fallacy.
Let me differentiate.
And I'll use this example.
So let's say-- let me make a graph.
And I'll switch colors.
This is n, my x-axis is n.
This is the number of trials I take.
And my y-axis, let me make that the sample mean.
And we know what the expected value is, we know the expected
value of this random variable is 50.
Let me draw that here.
This is 50.
So just going to the example I did.
So when n is equal to-- let me just [INAUDIBLE]
here.
So my first trial I got 55 and so that was my average.
I only had one data point.
Then after two trials, let's see, then I have 65.
And so my average is going to be 65 plus 55 divided by 2.
which is 60.
So then my average went up a little bit.
Then I had a 45, which will bring my average
down a little bit.
I won't plot a 45 here.
Now I have to average all of these out.
What's 45 plus 65?
Let me actually just get the number just
so you get the point.
So it's 55 plus 65.
It's 120 plus 45 is 165.
Divided by 3.
3 goes into 165 5-- 5 times 3 is 15.
It's 53.
No, no, no.
55.
So the average goes down back down to 55.
And we could keep doing these trials.
So you might say that the law of large numbers tell this,
OK, after we've done 3 trials and our average is there.
So a lot of people think that somehow the gods of probability
are going to make it more likely that we get fewer
heads in the future.
That somehow the next couple of trials are going to have to
be down here in order to bring our average down.
And that's not necessarily the case.
Going forward the probabilities are always the same.
The probabilities are always 50% that I'm
going to get heads.
It's not like if I had a bunch of heads to start off with or
more than I would have expected to start off with, that all of
a sudden things would be made up and I would get more tails.
That would the gambler's fallacy.
That if you have a long streak of heads or you have a
disproportionate number of heads, that at some point
you're going to have-- you have a higher likelihood of having a
disproportionate number of tails.
And that's not quite true.
What the law of large numbers tells us is that it doesn't
care-- let's say after some finite number of trials your
average actually-- it's a low probability of this happening,
but let's say your average is actually up here.
Is actually at 70.
You're like, wow, we really diverged a good bit from
the expected value.
But what the law of large numbers says, well, I don't
care how many trials this is.
We have an infinite number of trials left.
And the expected value for that infinite number of trials,
especially in this type of situation is going to be this.
So when you average a finite number that averages out to
some high number, and then an infinite number that's going to
converge to this, you're going to over time, converge back
to the expected value.
And that was a very informal way of describing it, but
that's what the law or large numbers tells you.
And it's an important thing.
It's not telling you that if you get a bunch of heads that
somehow the probability of getting tails is going
to increase to kind of make up for the heads.
What it's telling you is, is that no matter what happened
over a finite number of trials, no matter what the average is
over a finite number of trials, you have an infinite
number of trials left.
And if you do enough of them it's going to converge back
to your expected value.
And this is an important thing to think about.
But this isn't used in practice every day with the lottery and
with casinos because they know that if you do large enough
samples-- and we could even calculate-- if you do large
enough samples, what's the probability that things
deviate significantly?
But casinos and the lottery every day operate on this
principle that if you take enough people-- sure, in the
short-term or with a few samples, a couple people
might beat the house.
But over the long-term the house is always going to win
because of the parameters of the games that they're
making you play.
Anyway, this is an important thing in probability and I
think it's fairly intuitive.
Although, sometimes when you see it formally explained like
this with the random variables and that it's a little
bit confusing.
All it's saying is that as you take more and more samples, the
average of that sample is going to approximate the
true average.
Or I should be a little bit more particular.
The mean of your sample is going to converge to the true
mean of the population or to the expected value of
the random variable.
Anyway, see you in the next video.