Good morning or evening or whatever it is where you are,
wherever you're happening to watch this movie.
Anyway, I've been requested to do a playlist on probability,
and I think that's an excellent idea, so I will start doing
a playlist on probability.
So let's do a playlist on probability.
It's a good place to start probability.
I don't do videos on spelling.
Probability: so what is it?
And I think all of us have kind of a sense of
it, very informally.
And as far as I can tell there actually isn't a formal
definition of what a probability is.
There are several almost formal competing definitions.
So just in our everyday life, you know if the weather man
says there's a 50% percent chance of rain the next
day, he's essentially giving a probability.
He's saying that-- well, there's a couple of ways
that you could interpret the 50% probability.
It could be that if 100% was that he is sure there's rain
tomorrow and 0% is that he is not sure that there's is rain
tomorrow, that 50% kind of means well, he's kind of
neutral between those two possibilities.
So one definition could be how strongly you believe.
Actually, there's a whole school of probability where
they view probability like this, and it's called the
Baysian, and we'll go into that more.
Actually, when we do easy problems, all of these things
kind of are the same thing.
But later on we'll see what the difference is.
Another way of interpreting this and this is kind of the
frequentist school of thought, is if I were to have the data
that this-- or if the weather man had this data that he has
right now as far as where the clouds are and what the
barometer reads and where the moon is and all of the data.
Given all of the data that he has, when he has that same
exact data hundred times, fifty times or 50% of those
times there will be rain.
So you can almost view it as, given the data that he has, if
you had that data a hundred times or if you were able to
run this experiment a hundred times-- although that's very
unlikely that you would have that exact same number
of data points.
You know, whether Mars is in the right place and the
sun is flaring and all.
It's very unlikely you have those exact same-- you know,
the butterfly effect.
One butterfly can effect the wind patterns across the ocean.
So it's very unlikely that you could perform that experiment a
hundred times, but what the weather man could be saying is
well, if I did have data identical to this, a hundred
times, 50 of those times or 50% of the time we would
have rain the next day.
That's 50% of experiments with same-- I guess you could say
measurable initial conditions-- I'm kind of doing this on the
fly so don't take this as gospel.
But it I think it'll give you the sense.
With same initial conditions, would result in rain.
They're almost the same thing, but we'll see later that this
frequentist-- I tend to view the world kind of like this,
but there are a lot of circumstances where you
really-- it's hard to say that you could perform that same
exact experiment over again.
For example, if someone said in 2003 there's a 50% chance or
there's an 80% chance that Saddam Hussein has weapons of
mass destruction, that I think would-- and that would
be a probability.
You know, you'd have these CIA analysts who aren't being
influenced by their bosses saying hey, after all the data
we see, we can't be sure, but we think there's an 80% chance.
They would be in this camp, right?
Because you really couldn't perform that experiment
a hundred times.
There haven't been a hundred times or a thousand times or a
large number of times where you had that exact same set of
circumstances where you a guy with the big mustache in the
Middle East kind of giving the run around for
weapons inspectors.
Anyway, so let's move on.
This is very subtle, but it gives you the difference
between these two things.
It's quite subtle, but I think it gives you nice frame work
for what probability is.
So let's just do a little bit of notation.
I actually looked it up on Wikipedia and they had one
definition-- maybe it wasn't Wikipedia, it was
maybe another website.
And actually, I think you do see this definition of
lot where they say the probability-- sometimes it's
written as probability of a.
Sometimes it's just written as P of a.
So the probability of a occurring is equal to the
events in which a is true over total number of events.
And this, for the most part, can be a good definition, but
I'll show you one place where I think it's a little
bit more squirmy.
So if I told you that I'm going to flip a coin and--
actually, even better.
Let's say, let's roll a dice.
And let's say I say the probability-- I'm going
straight to more difficult things.
So say the probability of an even number.
Well, let's use this definition that they gave.
Well, what's the probability that this event is true?
Well, let's see, what are all the numbers I could get?
I could get a 1, 2, 3, 4, 5, 6.
This is just a normal die, it's not one of those Dungeons
and Dragon's dice.
So what are the number of events where we get an even
number, where this is true, where even is true?
Let's see.
2, 4, 6.
Those are all the situations where we get even as true.
So there are 3 where even is true.
And then, what is the total number of events?
Well, we could get 1 of 6 numbers, so there are 6 total.
And that equals 1/2.
And that also equals 50%, right?
We know how to convert fractions to percentages.
And this is right, this is completely right.
But the only time where you can really apply this and most of
what you'll do in school and things you can apply this, but
this assumes that all of the events are equally
likely to occur.
You could have had a dice or a die-- I forgot how to say
the plural or the singular.
You could have that situation where maybe the six-sided is
weighted a little bit more.
You know, someone's handed it down so it's more likely
to have a 3 or something.
And in that case, you wouldn't be able to use this definition.
So I'm going to modify this definition, although
I don't know if it's traditionally modified.
This is one possible, events in which a is true divided
by-- well, let's say, equally probable.
Equally probable events in which a is true divided by
equally probable total events.
So in order for this to hold true each of these six
circumstances have to have exact equal chance
of occurring.
And we're going to do maybe in this video, actually
probably not in this, I only have 3 minutes left.
But in this series I'll show you situations where you we'll
have an unfair dice or die or we'll have a set of
circumstances where all of-- each of the total number events
they're not equally probable.
So that's why I want you become a little bit
weary of this situation.
So with that said, let's do a couple of probability problems
that maybe give you a little bit more intuition for--
whoops-- for what's going on here in the world
of probability.
So if I'm flipping a dice and I said, well, what's the
probability of heads?
That's pretty easy.
We could use that definition and it's a completely
fair dice.
We could use that definition and say, well, what are the
total number of events where I could get heads
or tails, right?
So there's 2 total events.
And the probability the getting heads, that's
one of the events.
So there's a 1/2 probability.
The way I like to think of it so we don't have to use that
previous definition is, if I were to conduct this experiment
a hundred times, what percentage of those times am
I likely to get heads?
And then I would say, well, there's 50% of the time
I would get heads.
And the reason why, you know, I could make a symmetry argument
that it's just as likely to go on heads as it is to tails.
There's no reason why I would expect 51 heads or
49 nine tails, although that could happen.
But there's no reason I could expect it.
Heads and tails are equally likely.
They're just different words for different sides of a
coin that's equally likely to fall on either side.
Anyway, let's say I'm going to now flip a coin twice.
And it's the same coin.
So I'm going to flip it, and then I'm going to pick it up,
and I'm going to flip it again.
And so what's the probability that I get-- I'll
call it heads, heads.
So that's the probability that I get heads on the first flip
and then heads on the second flip.
Well, look at it this way.
If on the first flip we already know that we have a 5% chance
or 1/2 chance on the first flip, right?
So let's think of it of the frequentist philosophy.
So if I were to do this a hundred times, 50 of the
times I would get heads.
Let's call that on the first flip.
Then of course, 50 of the times I would get tails
on the first flip, right?
Now we're at this state of the universe and now we do the
experiment over again.
So of these 50 times, what percentage of the times is
the next flip going to be heads again?
Well, we could say it's going to be another 50% chance, or
you could say, well, in 50 tries the first one was heads,
and then of those 50, 50% are going to be heads again.
So we get 25%.
I just multiplied these two numbers.
And of course, to get heads and then tails would be 25% chance.
Heads Heads, heads tails, and then this is tails
heads, tails heads.
I'm getting confused.
Tail heads is 25%.
And then tails tails is 25%.
Anyway, I'm rushing it because I'm 25 seconds over.
I'll continue this in the next video.