Basic probability An introduction to probability
- What I want to do in this video is give you at least a basic overview of probability.
- Prob-ability. A word that you've probably heard a lot of,
- and you are probably a little familiar with it,
- but hopefully this will give you a little deeper understanding.
- So let's say I have a fair coin over here
- And when I say a fair coin, what I mean is
- that it has an equal chance of landing on one side or another.
- So maybe you can maybe view it as the sides are equal,
- their weight is the same one 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.
- So you have one side of this coin,
- (so this would be the heads, I guess. I'm trying to draw George Washington; I'll assume it's a quarter of some kind.)
- and on the other side, of course, is the tails.
- So that is heads, and the other side over there is tails.
- So, if I were to ask you, "if I flip this coin, what is the probability of getting heads?"
- And I could write that like this.
- And you probably (just based on that question) have a sense of what probability is asking.
- It's asking for some way of getting your hands around an event that is 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 probability tends to be introduced in textbooks),
- is you say, "Well, look. How many different equally likely possibilities are there?"
- So, number of equally likely possibilities...
- and of the number of equal possibilities, I care about the number that contain my event.
- So, the number of possibilities that meet my conditions...
- So, in the case of the probability of figuring heads, what is the number of equally likely possibilities?
- Well, there's only two possibilities.
- We're assuming that the coin can't stand on it's corner and stand straight up. We're assuming it lands flat.
- So, there's two equally likely possibilities.
- You could get heads, or you could get tails.
- Now, what's the number of possiblities that meets my conditions?
- Well, there's only one: the condition of heads.
- So, it'll be 1 over 2.
- So, one way to think about is that the probability of getting heads is 1 over 2 (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 the 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.
- Normally, you view an experiment as doing something in chemistry or physics, etc.
- 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, many, many, many times.
- Maybe I run the experiment a thousand times, or a million times, or a billion, or a trillion, and the more the better,
- what percentage of those would give me what I care about; what percentage would give me heads.
- And so, another way to think about this 50% probability of getting heads is:
- if I were to run the experiment tons of times, (maybe forever, or an infinite number of times), what percentage of those would be heads.
- You would get this 50%.
- And you can run that simulation. You could 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, and you stick them in a big box,
- shake the box, so you are simultaneously flipping all the coins,
- and count how many of those are going to be heads
- And you're going to see that the larger the number of coins that you are doing,
- the more likely you are 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 get all tails.
- But the more you do, the more likely that things are going to trend toward 50% of them being heads.
- Now, lets just apply these same ideas.
- And while we're starting with probability, at least the kind of basics,
- this is probably an easier thing to conceptualize,
- but a lot of times, this is actually a helpful one too:
- the idea that if you run the experiment many, many times,
- what percentage of those trials are going to give you what you were asking for (in this case heads).
- Now lets do another very typical example when you are first learning 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 one, and that's the two, and the three.
- We know of course that there are (assuming a fair die) six equally likely possibilities.
- You could get a 1, 2, 3, 4, 5, or a 6.
- 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 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 1 of them meet my conditions.
- 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,
- and now there are now two possibilities that meet my condition:
- I could roll a 1, OR I could roll a 6.
- So, now there are two possibilities that meet my constrains (my conditions).
- So there is a 1/3 probability of rolling a 1 or a 6.
- Now, this might seem silly to even ask it, but I'll ask it to make it clear.
- What is the probability of rolling a 2, AND a 3 (on 1 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 are 6 possibilities, but none of these possibilities are a 2 AND a 3.
- On one trial, you can't 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 zero.
- There's no way to roll this normal die, and all of a sudden you get a 2 AND a 3.
- Now, I don't want to confuse you with that, because it's abstract and impossible, so let's cross this out.
- Now, what is the probability of getting an even number?
- So, once again, you have 6 equally likely possibilities, and which of these possibilities meet my conditions (being even)?
- Well, 2 is even, 4 is even, and 6 is even, so three of the possibilities meet my conditions (or constraints).
- So this is 1/2. If I roll a die, I have a 1/2 chance of getting an even number.
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