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# Constructing a probability distribution for random variable

AP.STATS:
VAR‑5 (EU)
,
VAR‑5.A (LO)
,
VAR‑5.A.1 (EK)
,
VAR‑5.A.2 (EK)
,
VAR‑5.A.3 (EK)

## Video transcript

let's say we define the random variable capital X as the number of heads we get after three flips of a fair coin so given that definition of a random variable we're going to try to do in this video is think about the probability distribution so what's the probability of the different of the different possible outcomes or the different possible values for this random variable it will plot them to see how that that distribution is spread out amongst those possible outcomes so let's let's think about all of the different values that you could get when you flip through it when you flip a fair coin three times so you could get all heads heads heads heads you could get heads heads tails you could get heads tails heads you could get heads tails tails you could have tails heads head you could have tails head tails you could have tails tails heads and then you could have all tails so there's eight equally when you do the the actual experiment there's eight equally likely outcomes here but which of them how would these relate to the value of this random variable so let's think about what's the probability there is a situation where you have a zero heads so we could say what's the probability that our random variable X is equal to zero well and that's this situation right over here where you have zero heads it is one out of the eight equally likely outcomes so that is going to be that's going to be one over eight what's the probability what's the probability that our random variable capital X is equal to one well let's see what which of these outcomes do get gets us exactly one head we have this one right over here we have that one right over there we have this one right over there and I think that's all of them so three out of the eight equally likely outcomes are provide us get us to get us to one head which is the same thing as saying that our variable equals one so this has a 3/8 probability now what's the probability I think you're getting maybe getting a hang for it at this point what's the probability that a random variable X is going to be equal to two well for X to be equally two we must we that means we got two heads when we flipped the coin three times so that's this outcome meets that constraint this outcome would get our random variable to be equal to 2 and this outcome would make our random variable equal to two and this is three out of the eight equally likely outcomes so this is a 3/8 probability and then finally we could say what is the probability that our random variable X is equal to 3 well how does our random variable X equal 3 well we would have to get 3 heads when we flip the coin so there's only one out of the eight equally likely outcomes that meets that constraint so it's a 1/8 probability so now we just have to think about how we plot this to really see how it's distributed so let me draw so over here on the vertical axis oh this will be the probability probability probability and it's going to be between 0 & 1 you can have a probability larger than 1 so just like this so let's see if this is if this is 1 right over here and let's see everything here it looks like it's 1/8 so let's put everything in terms of eighths so that's 1/2 this is 1/4 that's 1/4 I can that's not quite a fourth this is 1/4 right over here and then we can do it in terms of eighths so that's a pretty good rough approximation and then over here we could have the out comes out comes and so outcomes I'll say outcomes for all right let's let's write this so value so value for X so X could be 0 1 actually we do those same colors X could be 0 X could be one X could be 2 X could be equal to 2 and X could be equal to 3 X could be equal to 3 these are the possible values for X and now we're just going to plot the probability the probability that X has a value of 0 is 1/8 is 1/8 that's I'll make a little bit of a little bar right over here that goes up to 1/8 so I just let me draw it like this so those up - so this is 1/8 right over here the probability that x equals 1 is 3/8 so that's 2/8 3/8 gets us right over let me do that in that purple color so probability of 1 that's 3/8 that's right over there that's 3/8 so let me draw that bar draw that bar and just like that the probability that x equals to probability x equals 2 is also 3/8 so that's going to be that same level just like that and then the probability that x equals 3 well that's 1/8 1/8 so is going to be the same height as this thing right over here so actually I'm using the wrong color so it's going to look like this it's going to look like it's going to look like this and actually let me just write this a little bit neater that can move that 3 so cut and paste remove that 3 a little bit closer in just so it looks a little bit neater and I can move that 2 in actually as well so cut and paste so I can move that 2 and there you have it we have made a probability distribution for the random variable X and the random variable X can only take on these discrete values it can't take on the value 1/2 or the value PI or anything like that and so this what we've just what we've just done here is we've just constructed a discrete probability distribution let me write that down so this is a this right over here is a discrete it only takes the random variable only takes on discrete values it can't take on any value in between these things so discrete probability probability distribution distribution for the our random variable X