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# Intro to theoretical probability

Video transcript

What I want to do in
this video is give you at least a basic
overview of probability. Probability, a word that
you've probably heard a lot of, and you are probably a
little bit familiar with it. But hopefully,
this will give you a little deeper understanding. Let's say that I have
a fair coin over here. And so when I talk
about a fair coin, I mean that it has
an equal chance of landing on one
side or another. So you can maybe view it
as the sides are equal, their weight is the
same on either side. If I flip it in
the air, it's not more likely to land on
one side or the other. It's equally likely. And so you have one
side of this coin. So this would be
the heads I guess. Try to draw George Washington. I'll assume it's a
quarter of some kind. And the other side, of
course, is the tails. So that is heads. The other side right
over there is tails. And so if I were
to ask you, what is the probability-- I'm
going to flip a coin. And I want to know what is the
probability of getting heads. And I could write
that like this-- the probability
of getting heads. And you probably, just
based on that question, have a sense of what
probability is asking. It's asking for some
type of way of getting your hands around an event
that's fundamentally random. We don't know whether
it's heads or tails, but we can start to
describe the chances of it being heads or tails. And we'll talk about different
ways of describing that. So one way to think
about it, and this is the way that
probability tends to be introduced in textbooks,
is you say, well, look, how many different, equally
likely possibilities are there? So how many equally
likely possibilities. So number of equally--
let me write equally-- of equally likely possibilities. And of the number of
equally possibilities, I care about the number that
contain my event right here. So the number of possibilities
that meet my constraint, that meet my conditions. So in the case of the
probability of figuring out heads, what is the number of
equally likely possibilities? Well, there's only
two possibilities. We're assuming that the coin
can't land on its corner and just stand straight up. We're assuming
that it lands flat. So there's two
possibilities here, two equally likely
possibilities. You could either get heads,
or you could get tails. And what's the number
of possibilities that meet my conditions? Well, there's only one,
the condition of heads. So it'll be 1/2. So one way to think about it
is the probability of getting heads is equal to 1/2. If I wanted to write
that as a percentage, we know that 1/2 is
the same thing as 50%. Now, another way to think about
or conceptualize probability that will give you
this exact same answer is to say, well, if I were to
run the experiment of flipping a coin-- so this flip, you
view this as an experiment. I know this isn't the kind of
experiment that you're used to. You know, you normally think an
experiment is doing something in chemistry or physics
or all the rest. But an experiment
is every time you do, you run this random event. So one way to think
about probability is if I were to do this
experiment, an experiment many, many, many times-- if
I were to do it 1,000 times or a million times or a billion
times or a trillion times-- and the more the better--
what percentage of those would give me
what I care about? What percentage of those
would give me heads? And so another way to think
about this 50% probability of getting heads is if I
were to run this experiment tons of times, if I were
to run this forever, an infinite number of times,
what percentage of those would be heads? You would get this 50%. And you can run that simulation. You can flip a coin. And it's actually
a fun thing to do. I encourage you to do it. If you take 100 or 200
quarters or pennies, stick them in a big
box, shake the box so you're kind of simultaneously
flipping all of the coins, and then count how many of
those are going to be heads. And you're going to see that the
larger the number that you are doing, the more
likely you're going to get something
really close to 50%. And there's always some
chance-- even if you flipped a coin a million times, there's
some super-duper small chance that you would get all tails. But the more you
do, the more likely that things are going to
trend towards 50% of them are going to be heads. Now, let's just apply
these same ideas. And while we're starting with
probability, at least kind of the basic, this is
probably an easier thing to conceptualize. But a lot of times, this is
actually a helpful one, too, this idea that if you run the
experiment many, many, many, many times, what
percentage of those trials are going to give you
what you're asking for. In this case, it was heads. Now, let's do another
very typical example when you first
learn probability. And this is the idea
of rolling a die. So here's my die
right over here. And of course, you have, you
know, the different sides of the die. So that's the 1. That's the 2. And that's the 3. And what I want to do--
and we know, of course, that there are-- and I'm
assuming this is a fair die. And so there are six equally
likely possibilities. When you roll this, you could
get a 1, a 2, a 3, a 4, a 5, or a 6. And they're all equally likely. So if I were to ask you,
what is the probability given that I'm rolling a fair
die-- so the experiment is rolling this fair die, what is
the probability of getting a 1? Well, what are the number of
equally likely possibilities? Well, I have six equally
likely possibilities. And how many of those
meet my conditions? Well, only one of them meets
my condition, that right there. So there is a 1/6
probability of rolling a 1. What is the probability
of rolling a 1 or a 6? Well, once again, there are six
equally likely possibilities for what I can get. There are now two possibilities
that meet my conditions. I could roll a 1 or
I could roll a 6. So now there are
two possibilities that meet my constraints,
my conditions. There is a 1/3 probability
of rolling a 1 or a 6. Now, what is the
probability-- and this might seem a little silly
to even ask this question, but I'll ask it just
to make it clear. What is the probability
of rolling a 2 and a 3? And I'm just talking
about one roll of the die. Well, in any roll of the die,
I can only get a 2 or a 3. I'm not talking about taking
two rolls of this die. So in this situation,
there's six possibilities, but none of these
possibilities are 2 and a 3. None of these are 2 and a 3. 2 and a 3 cannot exist. On one trial, you cannot get a 2
and a 3 in the same experiment. Getting a 2 and a 3 are
mutually exclusive events. They cannot happen
at the same time. So the probability of
this is actually 0. There's no way to roll this
normal die and all of a sudden, you get a 2 and a 3, in fact. And I don't want to confuse
you with that, because it's kind of abstract and impossible. So let's cross this
out right over here. Now, what is the probability
of getting an even number? So once again, you have six
equally likely possibilities when I roll that die. And which of these possibilities
meet my conditions, the condition of being even? Well, 2 is even, 4 is
even, and 6 is even. So 3 of the possibilities
meet my conditions, meet my constraints. So this is 1/2. If I roll a die, I
have a 1/2 chance of getting an even number.