Binomial variables

what we're going to do in this video is talk about a special class of random variables known as binomial variables and as we will see as we build up our understanding of them not only are they interesting in their own right but there's a lot of very powerful probability and statistics that we can do based on our understanding of binomial variables so to make things concrete as quickly as possible I'll start with a very tangible example of a binomial variable and then we'll think a little bit more abstractly about what makes it binomial so let's say that I have a coin so this is my coin here doesn't even have to be a fair coin let me just draw this really fast so that's my coin and let's say on a given flip of that coin the probability that I get heads is 0.6 and the probability that I get tails well would be 1 minus 0.6 or 0.4 and what I'm going to do is I'm going to define a random variable X as being equal to the number of heads after after 10 flips of my coin now what makes this a binomial variable well one of the first conditions that's often given for binomial variable is that it's made up of a finite number of independent trials so it's made up made up of independent independent trials now what do I mean by independent trials well a trial is each flip of my coin so a flip is equal to a trial in the language of this statement that I just made and what do I mean by each flip or each trial being independent well the probability of whether I get heads or tails on each flip are independent of whether I just got heads into heads or tails on some previous lip so in this case we are made up of independent trials now another condition is each trial can be clearly classified as either a success or failure or another way of thinking about it each trial clearly has one of two discrete outcomes so each trial and in the example I'm giving the flip is a trial can be classified classified as either success or failure so in the context of this random variable X we could define heads as a success because that's what we are happening to count up and so you're either going to have success or failure you're either going to have heads or tails on each of these trials now another condition for being a binomial variable is that you have a fixed number of trials fixed number of trials so in this case we're saying that we have 10 trials 10 flips of our coin and then the last condition is the probability of success and in this context successes of heads on each trial each trial is constant and we've already talked about it on each trial on each flip the probability of heads is going to stay at 0.6 if for some reason that were to change from trial to trial maybe if you were to swap the coin and that each coin had a different probability then this would no longer be a binomial variable and so you might say okay that's reasonable I get why this is a binomial variable can you give me an example of something that is not a binomial variable well let's say that I were to define the variable Y and it's equal to the number of kings after taking two cards from a standard deck of cards the standard deck without replacement without replacement so you might immediately say well this feels like it could be binomial we have each trial can be classified as either a success or failure for each trials when I take a card out if I get a king that looks like that would be a success if I don't get a king that would be a failure so it seems to meet that right over there it has a fixed number of trials I'm taking two cards out of the deck so it seems to meet that but what about these conditions that it's made up of independent trials or that the probability of success on each trial is constant well if I get a king the probability of King on the first trial probability I say King on first trial would be equal to well out of a deck of 52 cards you're going to have four kings in it so the probability of a king on the first trial would be 4 out of 52 but what about the probability of getting a king on the second on the second trial what would this be equal to well it depends on what happened on the first trial if the first trial you had a king well then you would have so let's see this would be the situation given first trial first King well now there would be three Kings left in a deck of 51 cards but if you did not get a king on the first trial now you have four kings in a deck of 51 cards because remember we're doing it without replacement you're just taking that first card whatever you did and you're taking it aside so what's interesting here is this is not made up of independent trials that does not meet this condition the probability on your second trial is dependent on what happens on your first trial and another way to think about it is because we aren't replacing each car that we're picking the probability of success on each trial also is not constant and so that's why this right over here is not a binomial variable now if y if we got rid of if without replacement and if we said we did replace every card after we picked it then things would be different then we actually would be looking at a binomial variable so instead of without replacement if I just said with replacement well then your probability of a king on each trial is going to be 4 out of 52 you have a finite number of trials your probability of success is going to stay constant and they would be independent and obviously each trial could easily be classified as either a success or a failure