Binomial Distribution EXPLAINED!

welcome to this series on probability distributions where I'm going to be cracking open a whole bunch of distributions that are commonly used in statistics the first cab off the ranks today is the binomial distribution and we're gonna start here with the cliched but quite useful coin toss example think about tossing a coin 10 times how many heads do you think you're gonna get from those 10 coin tosses do you think you're likely to get 10 heads well it's certainly possible but it's quite unlikely you have to get heads every time what do you think it's more likely to get somewhere around four five or six heads well that is a binomial distribution peeps and each of those heights each of those probabilities associated with these outcomes can be calculated using this scary-lookin formula now if you've watched this channel at all you'd know that I am NOT one for just applying formulas blindly so we're gonna dig a little deeper into what each part of this formula represents and get you to appreciate that it's really quite intuitive so let's start with something real no no this coin toss malarkey studies show colorblindness affects about 8% of men and we have a random sample of ten men taken so the first question is what makes this a binomial distribution so the prerequisites of a binomial distribution are that there are two potential outcomes per trial now it's a general word trial here each trial is a single man in a sample but there are two potential outcomes for each man in the sample either they have color blindness or they don't the second prerequisite is that the probability of success which we're going to call P here is the same across all trials now again success is a general term you'll hear when talking about binomial distributions technically success here means having colorblindness which doesn't sound like a success to celebrate but it's a statistical success nonetheless so in this case is going to be 0.08 or 8% the number of trials is fixed so that's ten men and each trial is independent so whether or not the first man has color blindness doesn't affect the second man's chance of having color blindness so it seems that at least in theory this setup follows a binomial distribution now we're going to learn by doing in this example so I've got four questions for you straight up knowing that this is a binomial distribution where the parameter P is zero point zero eight and the N is ten I'm gonna ask you to find the probability that all ten men are colorblind the no men are colorblind but exactly two men are colorblind and at least two men are colorblind now I'm sending you out in the dark here to start with because we haven't really discussed too much about how to calculate this but I reckon you're gonna be able to do the first few even without my help what's the probability that all ten men are colorblind well here are 10 men for them all to be colorblind the first one obviously has to be colorblind and that's got an 8% chance of happening so 0.08 of course the second man also has to be colorblind the third the fourth and in fact all of them have to be colorblind so you're just gonna multiply all those together to get a very very small number so I've written down here the probability of X equaling 10 is simply equal to 0.08 to the power of 10 you certainly don't need any help from something called a binomial distribution to answer that question and that's a very very small number one point oh seven times ten to the minus eleven very unlikely so that's your answer to a already what's the probability that no men are colorblind well this should be equally simple we know that the first man second man third man in fact all the men have to not be colorblind and the chance of that happening is 0.92 on each occasion so that's 0.92 to the power of 10 and we get zero point 4 3 4 4 how about this next question what's the probability that two men are colorblind now in this case we know that two of the men have be colorblind so that's 0.08 and 0.08 there and we know that the other eight men have to not be colorblind so that's 0.92 times 0.92 et cetera et cetera so you might think well that's just simple we're just gonna raise 0.08 to the power of two and multiply that by 0.92 to the power of eight and that should be our answer but wait just a second appreciate that there's numerous ways that this outcome can occur the first two men could have color blindness maybe the first and the third man has color blindness maybe the first and fourth man or maybe the last two are color blind so we know that there's quite a few potential combinations where two men out of ten could be color blind so that's where this combinations function comes in this 10c to function which you can use your calculator for you can also check it out on Google and it's just a mix of various factorial functions but all that will do will provide for you the number of ways that you can choose two items out of ten you can see why we need to multiply that into this product so the probability of X being two here is 0.14 eight so that's the probability of two men out of the ten being colorblind so what we've effectively done in the last few examples is used this formula here to calculate the heights or the probabilities of each of the discreet outcomes of number of colorblind men out of ten in a sample so reading this formula it says the probability of X which is our random variable is equal to a certain value is equal to that combination function choosing X colorblind men out of ten and is going to be 10 in this case and P is our probability of being colorblind which is going to be 0.08 so we raise point o8 ^ however many colorblind people there are in a sample and then we raise the probability of not being colorblind to the number of people in the sample that are not colorblind so when we found the probability of X being two that's two colorblind people in the sample we just subbed in two for our value of x recognize that from few slides ago and we found that was 0.148 and that's the height of this bar here for the discrete outcome 2 and notice we actually did the same for when X was equal to 0 as well so that's a zero colorblind people out of 10 you might recall we just did this final part of the equation which was raised point nine two to the power of ten but realistically these first two factors are going to both become one so technically this formula still holds to get zero point four three four as the height of that outcome here zero and we also found the probability of X being ten that's ten colorblind people out of ten and that was a very very small number indeed again two of these factors of this formula will become 110 see 10 the number of ways you can choose 10 items out of 10 is 1 and also 0.92 raised to the power of 0 is also 1 so even though we didn't use this formula at the time we just used our intuition for this value this formula still holds so what have you asked to find the probability of at least 2 men being colorblind well the best way to think about this is to shade the particular region of interest on your column chart above so the probability of X being greater than or equal to 2 is just all of those blue shaded columns and don't forget that keeps going on until 5 6 7 8 9 and 10 these are not 0 here they're just very very small and not showing up on a graph to this scale so what we could do is add up all of the probabilities from 2 to 10 but a simpler way of thinking about it would be to subtract those two probabilities at 0 and 1 because if we subtract those from the number 1 we're going to find that blue shaded region how does that work well we know that all of these probabilities must sum to 1 because it's a probability distribution so the probability of X being greater than or equal to 2 is just 1 minus the probability of X being equal to 1 and X being equal to 0 so if you use that formula to find those two values we can simply find that the probability of X being greater than or equal to 2 is 0.188 and that's the region of interest right there now you might be thinking look is there a simple way of doing this and yes there is using Excel you can take advantage of the function which is the Benham disk function so if I'm trying to find the probability of X being 0 I can use equals B na missed and what that'll do is provide for me the height or the probability of each discrete outcome from the binomial distribution so the probability of getting zero men out of 10 can be found by putting in that first argument being 0 which is our value that we're interested in the next argument of this function that's required by Excel is the total number in our sample which in this case is 10 the third argument requires the probability of success which is 0.08 and the fourth argument requires us to tell Excel whether we want a cumulative distribution at this point or not now I'll explain what a cumulative distribution is in a second but for the moment let's just put false meaning that we're just trying to find the individual probability of that one outcome so we could use the exact same formula where x equals 1 to find the probability of having one man out of ten with colorblindness an Excel will calculate that for us to be zero point three seven eight and that's this bar here here it is for x equals two it's a zero point one four eight and all I've done is just change this first argument of the Benham dysfunction to two or we can do where x equals 10 and again I've got 10 in there now and it'll provide for us that small probability of X being 10 and E in it now if you're curious as to what happens when you write true instead of false in that final argument well that'll give for you the cumulative distribution function at that point and when I say that I mean the probability of getting that value or less so if I'm trying to find the probability of X being less than or equal to 1 I can write equals B Nam and dist and I'll put a 1 in there again make it 10 and 0.08 because they're the parameters of this binomial distribution and by writing true it's going to give me the probability of JA these first two outcomes that's X being 0 & 1 if I wrote false it just gives me the probability of X being 1 so again if I'm trying to find the probability of X being greater than or equal to 2 we've found this out before we knew it was zero point one eight eight and it's all of these values added up together but you can use this cumulative distribution function again but appreciate that the cumulative distribution function we're interested in to calculate this is the point from one because when we calculate the cumulative distribution function from one it's going to give us these two yellow bars because don't forget cumulative functions only go in one direction it's always the probability of that value or lower we technically want the value of two or higher so we have to think to ourselves well that's just one minus the probability of one or lower so that's what I've done here one minus the probability of one or lower so the final thing we can do with the binomial distribution is assess the expected number and standard deviation of the number of colorblind men in our sample so here I've got two questions that ask exactly that so the first one is asking us to find the expected number of colorblind men in the sample and to do that we can utilize the formula here which is the expected value is n times P but it's a pretty boring formula because you can kind of figure this out yourself you know there are 10 men in the sample each one of these men has an 8 percent chance of being colorblind so you're just gonna times them together anyway and find that the expected value is 0.8 it's quite intuitive so out of 10 men you're expecting 0.8 of them to have colorblindness that would be the mean of the distribution what's the standard deviation of a number of colorblind men in the sample well that utilizes this other feature of a binomial distribution that the variance is equal to n times P times 1 minus P and when you multiply all that together you get zero point seven three six and then all we need to do is take the square root of that to get our standard deviation all right so there you go this is the first and what will likely be a long series on that distributions and you can check it out once I put them up here's some links so you can stay in contact to do just that adios [Music]

Channel: zedstatistics

Views: 127,405

Rating: 4.9279819 out of 5

Keywords: Binomial distribution, zedstatistics, zstatistics, justin zeltzer, probability distributions, binomial, what is a binomial distribution?

Id: e04_wUoscBU

Length: 13min 29sec (809 seconds)

Published: Tue Apr 11 2017