Basic Probability An introduction to probability
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- 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 just a little bit familiar with it
- Hopefully this will get you a little deeper understanding.
- So let's say that I have a fair coin over here.
- 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,
- the 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.
- trying to draw George Washington
- I'll assume it's a quarter of some kind
- And then the other side, of course, is the tails
- So that is heads,
- and the other side right over here is tails.
- And so if I were to ask you,
- "What is the probability?"
- I am 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 in 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 likely possibilities
- And of the number of equally possibilities,
- I care about the number that contains 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 condition of heads
- So it'll be one over two
- So one way to think about it is the probability
- of getting heads is equal to one over two
- is equal to one half.
- If I wanted to write that as a percentage,
- we know that one half is the same thing as fifty percent.
- 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 only think
- an experiment is doing something with,
- in chemistry or physics or all the rest,
- but an experiment is every time 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 a thousand 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 fifty percent probability of getting heads,
- is as if I were to run this experiment tons of times,
- if I were to run this forever,
- or an infinite number of times,
- what percentage of those would be heads,
- you would get this fifty percent
- 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 a hundred or two hundred quarters or pennies,
- take 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 are going to see
- that the larger the number that you are doing,
- the more likely
- you're going to get something close to fifty percent
- There's always some chance,
- even if you flip the coin a million times,
- there's some super duper small chance that you get all tails.
- But the more you do,
- the more likely that you're going to get
- that things are going to trend towards
- fifty percent 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 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,
- you have different sides of the die
- So that's the one, that's the two, that's the three
- and what I want to do, and we know of course
- I'm assuming this is a fair die,
- and so there are six equally likely possibilities.
- When you roll this,
- you can get a one, a two, a three, a four
- a five, or a six. And they are 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 one?
- 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 conditions.
- That right there.
- So there is a one sixth probability of rolling a one.
- What is the probability of rolling a one or a six?
- Well once again,
- there are six equally likely possibilities for what I can get,
- and there are now two possibilities that meet my conditions.
- So I could roll a one, or I could roll a six.
- So now there are two possibilities that meet my constraints,
- my conditions,
- so there is a one third probability of rolling a one or a six
- Now what is the probability
- 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 two and a three?
- And I'm just talking about one roll of the die.
- Well, in any roll of the die I can only get a two or a three.
- I'm not talking about taking two rolls of this die
- So in this situation, there are six possibilities
- but none of these possibilities are two AND a three.
- On one trial,
- you cannot get a two and a three in the same experiment
- Getting a two and a three are mutually exclusive events,
- they cannot happen at the same time.
- So the probability of this is actually zero.
- There's no way to roll this normal die and all of a sudden
- you have a two and a three
- 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, two is even, four is even, and six is even.
- So three of the possibilities meet my conditions,
- meet my constraints.
- So this is one half. If I roll a die,
- I have a one half chance of getting an even number.
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