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## More on expected value

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# Law of large numbers

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## Video transcript

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.