Probability (part 1) What probability is.
Probability (part 1)
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- 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
- Bayesian, 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 affect 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.
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