Binomial Distribution 2 More on the binomial distribution
Binomial Distribution 2
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- In the last video we defined our random variable x as the
- number of heads we get after flipping a coin five times,
- and it's a fair coin.
- And then we figured out the different probabilities that
- the random variable could take on different values.
- Let me rewrite them all here.
- The probability that we get exactly one head or
- zero heads-- actually, let's start there.
- The probability we get zero heads, and that's the same
- thing as getting five tails-- we figured out was 1 out of 32.
- The probability of getting 1 head we figured
- out was 5 over 32.
- The probability of getting 2 heads-- probability that our
- random variable is equal to 2, what was that?
- I think it was 10/32.
- Yeah, it was 10/32 or 5/16.
- 10/32 which is equal to 5/16.
- The probability that we get three heads, that was actually
- equal to the same thing, 10/32 which is equal to 5/16 .
- And that made sense because the probability of getting 3
- heads is the same thing as a probability of getting 2 tails.
- And the probability of getting 2 tails should really be the
- same thing as the probability of getting 2 heads.
- And then finally, or almost finally, the probability that
- we get 4 heads, that was equal to 5/32, which also makes sense
- because the probability of getting 4 heads is the same
- thing as the probability of getting 1 tail, right?
- And 1 tail should be the same thing as 1 head, so that's why
- these numbers are the same.
- And then finally, the probability that we get all 5
- heads, which is the same thing as getting no tails, is 1/32.
- The probability of no tails is the same thing as the
- probability as no heads.
- So let's draw this.
- Let's draw our probability distribution.
- OK, let me draw the x-axis.
- That's not thick enough.
- I'll do it a different color.
- All right, and just let me try draw the y-axis.
- Almost there.
- All right, so what are the different values that we care
- about for our random variable?
- Notice, these are all discreet values.
- They're all particular finite numbers that we care that our
- random variable can take on.
- And that was actually a function of our definition of
- the random variable, right?
- This random variable, number of heads after 5 flips-- you can't
- have an infinite number of values here.
- Your value can either be 0 through 5, and it can't
- be a non-integer even.
- But the important thing to realize that there's a finite
- number of values that this random variable can take on and
- that's why we have a discreet probability distribution.
- All right, so the probability that it takes on-- let me just
- draw 0, 1, 2, 3, 4, and 5.
- This is for our random variable, x.
- So this is the number of heads in 5 trials.
- Let me draw the high points first.
- So let's see, the high points are-- they're all right there.
- That's the probability of 2 and 3 and they're the same.
- Let me draw little bar graph.
- So 2 and 3; let's just say that this is 10/32 right there.
- I'll draw them the highest because those are
- the highest values.
- 2 and 3 both hit 10/32.
- Actually, let me draw it a little differently.
- Let me make it so they touch each other.
- So let's say 2 is there, and then we draw 3.
- And 3 is also there.
- There you go.
- All right, what's the probability that you get 1?
- That was 5/32, so that's 1/2 of these.
- These two should be the same height.
- So that's 5/32.
- Should be like that.
- That's the same thing as getting 4 heads,
- so that's like that.
- Let's assume that this 4 is down here.
- The probability of getting either 0 or
- 5 is the same thing.
- That's 1.
- So that's going to be 1/5 of this height.
- So it's going to look something like that.
- Let's say this last one is for 5.
- The drawing is the hardest part.
- That's going to look something like that.
- All right, and so this is for 5 and this is for 4 right here.
- And this right here is 10/32 or 5/16.
- This is 5/32.
- And this right here is 1/32.
- What I have just drawn is a binomial probability
- It's a particular instance of the binomial
- probability distribution.
- And actually, as you get to kind of an infinite number of
- values, and once we get to the continuous, this'll approach
- the famous bell curve that you hear about where
- things actually start looking like a bell.
- Or it'll start looking like that.
- And we'll do some-- maybe I'll get Excel out and I'll do it
- where I start saying-- instead of saying, the number
- of heads in 5 trials.
- If I did this as the number of heads in 5 million trials,
- these bars would get very, very close together and you'd start
- seeing them approach this thing that looks like a bell curve.
- And this is a very important distribution.
- One, because it actually describes a lot of
- random processes.
- But it's actually very important to statistics
- generally because a lot of times in statistics you don't
- know the underlying mechanism that's generating your results
- and you just assume that there's just a bunch of
- random stuff going on.
- You just add up a bunch of random events.
- When you're adding up random events that's like counting
- the number of heads.
- So you assume that a lot of things have-- well,
- in the discreet case, a binomial distribution.
- And then, in a few videos from now I'll show you the
- continuous cases, which is the normal distribution.
- And this is really important to realize because sometimes
- people make that assumption about certain things and the
- biggest one-- and this is probably the most negative
- repercussions, especially in this type of financial
- Well, they'll assume that some process has a binomial or a
- normal distribution where it really doesn't.
- Because if you assume that something has a distribution
- like this you start saying, oh, well the probabilities at
- these ends are really low.
- But what if the distribution is something like this?
- And I'll do more work on this.
- I don't want to get too advanced too fast.
- But the general idea, it's really important to understand
- the assumption you're making when someone says, oh, we
- assume it's a normal distribution.
- We assume it's a binomial distribution.
- But with that said, that's a little bit of a nugget
- of why this is important going into the future.
- But just to kind of review a little more what we've
- already studied, let's think about this.
- Why is this called a binomial distribution?
- Well, if you think about it, when you flipped 5 coins or
- when you had 5 flips of the same coin-- each arrangement,
- I'll pick an arbitrary arrangement of heads and tails.
- Each circumstance of heads and tails, I guess I could call it.
- You know, that's one of them, 5 flips.
- Each of these have exactly a 1 in 32 probability.
- In fact, there's exactly 32 ways I can draw some
- combination of heads and tails here.
- So each of those have a 1 in 32 probability.
- And what we essentially did in the last video when we said,
- OK, what was the probability of getting exactly two heads?
- We essentially said, OK, how many of these circumstances, or
- how many of these-- I guess we could call it-- how many of
- these permutations had 2 heads?
- And we counted them up and we said, 10 of them have 2 heads
- and that's why the probability was 10 out of 32.
- And the way we figured that out is for example,
- if we said, 2 heads.
- We said, OK, 1 of 5 flips could be the first head, and then 1
- of 4 flips could be the second head.
- And then we don't care about order, so we don't care if flip
- 1 was a first head and flip two was a second head or was
- the other way around.
- So we divide by 2, and we said that this was the
- same thing as 5 factorial.
- Actually, let me do it color coded.
- This part is the same thing as 5 factorial over 3 factorial.
- And then, that's actually the same thing as 2 factorial
- because 2 factorial is 2 times 1.
- So in general, let me write this down.
- The probability that x equaled n-- and this is
- for the x that I defined.
- The random variable, the number of heads I get in 5 flips of
- the coin-- it was equal to 5 factorial divided
- by n factorial.
- In this case it was 2, 2 was n.
- We said was the probability that we get exactly 2 heads.
- Times 5 minus 2 factorial.
- And I encourage you to review because it really is important
- to get an intuitive feel of this.
- I encourage you to review the videos I made in the
- probability playlist on binomial coefficients, and on
- this flipping of coins type of probability problems because I
- go a little more in detail in the actual intuition of this.
- But if you watch those or if you have a little bit of
- experience with this, you would recognize that this is
- the binomial coefficient.
- And actually, I even have a video why it's called the
- binomial because these coefficients show up when you
- actually multiply binomials.
- Binomials are just things like if I just have x plus why and I
- start multiplying it by itself, take it to different powers,
- the coefficients of these tend to be-- well, they are
- the same as these.
- And I give you a whole video on why that-- well,
- that's why that works.
- And so this is why it's even called the
- binomial coefficient.
- And why this is called the binomial distribution.
- But another way of writing this, the shorthand
- is 5 choose n.
- And they say that because what were we doing?
- We said we had 5 flips and we were choosing 2 of
- the flips to be heads.
- Or, in this case, we have 5 flips and we're choosing n
- of the flips to be heads.
- So this will tell you how many of the different permutations
- satisfy our condition.
- And then we say, well, what's the probability
- of each of those?
- It's times 1 out of 32.
- So this is why it's called a binomial distribution.
- Each of the values of probabilities for each of the
- random variable values-- you can figure them out by using
- your binomial coefficients.
- Anyway, I just realized I'm out of time again.
- And don't worry, I'm going to do a couple of more examples
- with this because I really want you to get the hang of the
- binomial distribution before we move into the normal
- See you soon.
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