Binomial variables | Random variables | AP Statistics | Khan Academy

What we're going to do in this video is talk about a special class of random variables known as binomial variables. 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; it 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 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 10 flips of my coin. Now, what makes this a binomial variable? Well, one of the first conditions that's often given for a binomial variable is that it's made up of a finite number of independent trials.

So, it's made up of 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 and heads or tails on some previous flip.

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 as a trial, can be 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 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 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.

So you might immediately say, "Well, this feels like it could be binomial; we have each trial can be classified as either success or failure." For each trial, 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 a king on the 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 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—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; it 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 card 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 that 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.