Binomial Distribution (Introduction) | ExamSolutions

hi welcome to this tutorial on the binomial distribution now the aim of this tutorial is to show you what a binomial distribution is and the notation that we use now the binomial distribution has four properties and the best way of demonstrating this is by way of tree diagrams and the first tree diagram I'm going to show you is that of throwing a dart till you get a bull's-eye you can throw the dart into the dartboard you either get a bull's-eye or you don't and if you get a bull's-eye I'm going to stop throwing the dart so I carry on to my second throw and I either get a bull's-eye or I don't get a bull's-eye and so on now I could keep throwing the dart at the dartboard and never in theory get a bull's-eye so this tree diagram would go on and on and on forever and ever it would have what we call an infinite number of trials trials are these sections in between the dotted lines here they represent in this particular example the first so the second throw third throw and so on so what is the first property for a binomial distribution well it is that we have a fixed number of trials n and so this particular model would not fit a binomial distribution because as I say it's got an infinite number of trials so let's have a look at another tree diagram which has a finite number of trials a fixed number of trials and here's one we've got the playing of two football matches I've got my fixed trials each one represents a match this is the first match from the second match and then you'll notice that what I've got here is that in any match you can either win it draw or lose and then if you've won say the first match you could go on to win the can match draw the second match or lose the second match and so on now what is the second property for a binomial distribution well the second property is that we should have two outcomes in a trial success or failure now in this particular tree diagram in say this trial here I haven't got two outcomes the outcomes are winning drawing or losing three outcomes and you'll see in this trial here as well I haven't got two outcomes I've got three outcomes winning drawing or losing so this type of tree diagram is not suitable for a binomial distribution so remove that particular diagram and we'll have a look at another one playing two games of tennis I've got a fixed number of trials n I've got two trials my first match my second match I've got two outcomes in a trial success or failure because in any match I can either win the match or lose the match now what is the third property for a binomial distribution well the third property is that trials are independent what does that mean well if I look at this trial here my second match I've written this particular notation and if you're not familiar with it what it means is that assuming that I won the first match this is about winning the next match given that I won the first match and the same down here we've got for this one if I won the first match this is about losing it given that I won the first match and what I have is that this trial is not independent of the first trial winning or losing in the can match depended on how I performed in the first match so this particular type of tree diagram does not fit the properties for a binomial distribution I need independent trials so let's see what else we can come up with what about this tree diagram where I throw a die and I spin a coin I've got a fixed number of trials two trials again first trial is where I'm looking at throwing a six and the second trial is where I spin the coin and I'm looking at getting ahead so I have two outcomes in the trial success or failure I have either getting a six or not getting a six I either get a head well I don't get a head and so on so I've got two outcomes success or failure success or failure are the trials independent well of course they are because when I spin a coin it is unaffected by throwing a die so what is the fourth property of a binomial distribution well the fourth property is that the probability of a success P remains constant now the success here was throwing a six and it was 1/6 and the success in this trial of getting ahead the probability of getting a head was 1/2 and you'll notice then that the probabilities of getting a success is not constant it has changed from 1/6 to 1/2 so therefore we see that the probability of success is not constant so I haven't met this property for a binomial distribution so I have to rule out this type of tree diagram okay so what are we going to have next well we could have this one and that is where I throw a die three times first row second row third throw and a success is if I score a six so there are two outcomes I either get a six or I don't get a six the trials are independent because when I throw a die the score that I get on the die is unaffected by the previous throw so trials are independent and you'll notice the probability of success P remains constant the success being that I get a six and you can see it's remains at one-sixth all the way through the tree diagram so I've met all the conditions so therefore I have a binomial distribution now when you have a binomial distribution what I want to do is look at what we call a random variable X I've let X be the random variable RV for short which is the number of sixes that I score in these three throws and that random variable X could be 0 I don't score any sixes as if I would down this path here not a six in the first row not a six in the second row not a six in the third throw or it could be that I score one six and there's many ways of scoring one six I could score it on the first throw and then not on the second throw but on the third throw and so on I could also score two sixes and I could also score three sixes I certainly couldn't score four sixes so the random variable X is going to take on the values not one two and three for this particular type of problem and we write this as X is distributed we write this kind of squiggly line here binomially and we write a capital B here and to describe this particular distribution we have two parameters two values that we put in here and the first value is n the number of trials so for this particular binomial distribution there are three trials and the second parameter is P the probability of success and the probability of success is this one here one-sixth so we mark that in as 1/6 so this is the notation that we would use in describing a binomial distribution in general then if X is the random variable that represents the number of successes in n trials we describe this as X is distributed binomial e and we write in brackets two parameters N and P n is the number of trials and P is the probability of success so this brings us now to the end of this tutorial but in the next tutorial I'm going to show you how we can develop this further and do calculations using this particular model you

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Keywords: Binomial distribution, Binomial, distribution, Statistics, examsolutions, exam solutions, A-level, probability, independent trials, trials, binomial distrubution, binominal distribution, binomial distribution calculator, binomial distribution in probability, binomial distribution statistics, binomial distribution examples, binomial distribution formula

Id: NaDZ0zVTyXQ

Length: 10min 30sec (630 seconds)

Published: Wed Oct 28 2009