So, in recent videos, we've talked a lot about discrete random variables, numbers that can't be broken down any further, like the number of prizes in a raffle or the number of kids in a household. But there are many possible outcomes in those situations, like zero, one, two, three, four, five, and so on and so forth. Now, a lot of times, we will be looking at situations where there are only two possible outcomes, like heads or tails in a coin flip. So one of the most common examples of a discrete random variable is the outcomes in what we call a binomial experiment. We'll talk a lot about probabilities, means, and standard deviations in a later video.
But in this video, I just want to introduce you to what it means for an experiment to be binomial. There are basically these four criteria that we're going to look for. So let's go ahead and take a look, and we'll do some examples together. All right? So, a binomial experiment.
Basically, the word binomial just means that there are two. These are events with two possible outcomes, either you get a success or a failure. Those don't necessarily mean something good or bad. So, for example, heads could be a success, and tails could be a failure, not necessarily good or bad. Alright?
The variable little x that we'll be using in these problems is basically the number of successes. So in other words, this is your discrete random variable. So what is the word experiments? Basically, what is happening is you're just going to be doing a bunch of trials, a series of independent trials. So if I flip a coin four times, that's just four independent trials.
The word independence we've actually seen before. This just means that the outcome of one trial doesn't affect the next. If I flip a coin, it's going to land on heads or tails, and it has no effect on what the next coin flip is going to be. Alright? Each trial has a probability of success.
For example, the probability I land on heads, and the variable we use for that is the letter little p. Each trial also has a probability of failure, and these successes and failures are not always going to be fifty-fifty. The letter that we use for this is q. And, basically, we've seen the relationship between complements before, where the probability of something not happening, if there are only two possible outcomes, is one minus the probability that it does. So, basically, what happens here is if failure is the opposite of success, then it means that the probability of failure is just one minus p.
Usually, what's going to happen in these problems is they'll give you what the probability of success is, and you can always figure out what q is using this relationship. Alright? So that's all the variables you need to know. Let's take a look at our example here. So we're going to take a look at these following experiments and figure out if they are binomial.
And if they are, we're going to figure out the values of n, p, q, and little x. Alright? So let's go ahead and get started with our first example here. You're going to flip a coin four times. You count the number of times the coin lands on heads, which is three times.
Okay? So is this binomial? There are basically four questions to ask here, which have to do with the information that we just talked about up here. So let's get started. The first thing you have to answer is, are there only two possible outcomes?
Well, if you flip a coin, you either get heads or tails. There's no other possible outcome. Right? So there's only two outcomes in this experiment. So are there a fixed number of trials?
Are you doing this a set number of times, or are you flipping the coin until something happens? Well, basically, what we're told here is that you're flipping the coin exactly four times, so that is a fixed number of trials. Are the trials independent, which means that the outcome of one affects the other? We've already talked about this. One coin flip doesn't affect the next one, so these are definitely independent.
And is there an equal probability of success per trial? Oh, in other words, if success is the number of heads that I get, then it's always fifty-fifty because they're independent. So usually, what happens is that these last two questions are sort of tied together. Alright? So this is definitely a binomial experiment.
Let's take a look at the values of n, p, q, and little x. Number, n is basically just the number of trials. And so in this case, what happens is we're flipping the coin four times. So that is our n. Right?
So n equals four. So what about p, the probability of success? Well, essentially, with a coin flip, it's fifty-fifty, but we usually write these as decimals. So in other words, p is 0.5, and that just means that q is equal to one minus 0.5, which is also 0.5. Now just be really careful here.
Just because this coin flip is fifty-fifty, doesn't mean that all binomial experiments are going to be fifty-fifty like this. We're going to see other types of probabilities. And finally, our last variable is x, which is essentially the number of successes that we got. Well, the desired outcome that we wanted is we wanted heads, and we landed on that three times. So that is basically what little x is.
Alright? That's just the number of desired outcomes. So that is for this first example. Let's take a look at the second one over here. So we're going to pull four marbles out of a bag of red and blue marbles, but we're not going to replace them.
So we're not replacing them. The chance of getting a blue marble is 60%, and you're going to get one blue marble. Alright? So are there only two possible outcomes? You reach into a little bag of red and blue marbles, and you can only pull out either red or blue.
So there's definitely only two outcomes. Are there a fixed number of trials? Well, what's happening here is you're pulling four marbles out of the bag, so there's definitely a fixed number of trials. Now, are the trials independent? Does the outcome of one affect the next?
Well, what's going to happen here is you're basically going to reach into the bag, and once you pull a marble out, you're not going to replace them. So in other words, what happens is if I reach into this bag and I take this marble and I pull it out, all of a sudden now that affects the probability of me pulling out another blue marble on the next pull. So this idea here of not replacing the marble actually has to do with the independence of the trial. Because I am not replacing this marble, and I'm going to write this in black here like this, because I'm not replacing, that means that this fails. This is not an independent trial because the outcome of one affects the next one.
Essentially what happens is there's not an equal probability of success per trial because you're removing the number of marbles. Alright? So one thing I want to point out by the way is the chance of getting a blue marble with 60%. And again, just because you have two possible outcomes doesn't necessarily mean it's fifty-fifty, and this is an example of that. Alright? So this fails. This is not a binomial experiment.
Alright, folks. So now that we understand the basics of what a binomial experiment is, let's take a look at some practice.