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# Geometric random variables introduction

AP.STATS:
UNC‑3 (EU)
,
UNC‑3.E (LO)
,
UNC‑3.E.1 (EK)

## Video transcript

so I have two different random variables here and what I want to do is think about what type of random variables they are so this first random variable X it's equal to the number of sixes after twelve roles of a fair die well this looks pretty much like a binomial random variable in fact I'm pretty confident it is a binomial random variable we could just go down the checklist the outcome of each trial can be a success or failure so trial outcome success or failure it's either gonna go either way the result of each trial is independent from the other ones whether I get a six on the third trial is independent on whether I got a six on the first or the second trial so result let me write this trial I'll just do a shorthand trial results results independent independent that's an important condition let's see there are a fixed number of trials fixed number of trials in this case we're going to have twelve trials and then the last one is we have the same probability on each trial same probability of success probability on each trial so yes indeed this met all the conditions for being a binomial binomial random random variable and this was all just a little bit of review about things that we have talked about in other videos but what about this thing in the salmon color the random variable Y so this says the number of rolls until we get a six on a fair die so this one strikes us is a little bit different but let's see where it is actually different so doesn't meet that the trial outcomes that there's a clear success or failure for each trial well yeah we're just gonna keep rolling so each time we roll it's a try and success is when we get a six failures when we don't get a six so the outcome of each trial can be classified as either a success or failure so it meets and Lumi up with the checks right over here it meets this first constraint our the results of each trial independent well whether I get a six on the first roll or the second roll or the third roll or the first fourth roll or the third roll it the probabilities shouldn't be dependent on whether I did or didn't get a six on a previous roll so we have the independence and we also have the same probability of success on each trial in every case it's a 1/6 probability that I get a 6 so this stays constant and I skipped this third condition for a reason because we clearly don't have a fixed number of trials over here we could just we could roll 50 times until we get a 6 the probability that we'd have to roll 50 times is very low but we might have to roll 500 times in order to get a 6 in fact think about what the minimum value of Y is and what the maximum value of Y is so the minimum value that this random variable can take I'll just call it min Y is equal to what well it's gonna take at least one roll so that's the minimum value but what is the maximum value for Y and I'll let you think about that all assumed you thought about it if you pause the video well there is no Max value you can't say oh it's a billion because there's some probability that it might take a billion in one rolls it is a very very very very very very small probability but there's some probability it could take a googol roles or Googleplex rolls so you can imagine where this is going so this type of random variable where it meets a lot of the constraints of a binomial random variable each trial has a clear success or failure outcome the probability of success on each trial is constant the trials the trial results are independent of each other but we don't have a fixed number of trials in fact it's a situation we're saying how many trials do we need to get do we need to have until we get success maybe that's a general way of framing this type of random very how many trials until success while the binomial random variable was how many trials or how many successes I should say how many successes in finite number of trials so if you see this general form and it meets these conditions you can feel good it's a binomial random variable but if you're meeting this condition clear success or failure outcome independent trials constant probability but we're not talking about the successes in a finite number of trials we're talking about how many trials until success then this type of random variable is called a geometric geometric random random variable and we will see why in future videos it is called geometric because the math that involves the probabilities of various outcomes looks a lot like geometric growth or geometric sequences and series that we look at in other types of mathematics and in case I forgot to mention the reason why we call binomial random variables is because when you think about the probabilities of different outcomes you have these things called binomial coefficients based on combinatorics and those come out of things like Pascal's triangle and when you take a binomial to ever-increasing powers so that's where those words come from but in the next few videos the important thing is to recognize the difference between the two and then we're gonna start thinking about how do we deal with geometric random variables
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