This tutorial covers binomial distributions. To start, let's talk about a binomial experiment. A binomial experiment has a fixed number, n, of repeated trials. Each trial has just two possible outcomes, success and failure. The probability of success, p, is the same in every trial, and the trials are independent. The variable of interest is the number of successes.
Successes could be something like flipping a coin, where heads is the success and tails is the failure. Or it could be something like finding a defect. Even though a defect is thought of as a bad thing, finding it could still be considered the success in a binomial experiment.
Now furthermore, we have binomial random variables, and that's the number of successes, x, in n repeated trials of a binomial experiment. So now the binomial distribution, the thing that we're talking about, is the probability distribution of that binomial random variable, x. One example is flipping a coin. Here we have the possibilities of the number of heads when you flip a coin. You can get 0, 1, or 2, and here are the probabilities shown.
I've calculated these theoretical probabilities by using a tree. Now here for the number of heads, the probability is 0.25. For 1, it's 0.5, And for 2, it's 0.25. So this here shows the probability distribution for the binomial random variable, x.
Here is the probability formula for the binomial distribution. Now the formula looks quite long and complicated, but once you break it down, it's not so bad. So first, x, like we talked about before, stands for the number of successes from the experiment. And we've also seen already in this tutorial and that is the number of trials in the experiment. P is the probability of success on any individual trial. 1 minus p is the probability of failure on an individual trial.
This last part, nCx refers to the number of combinations of n things taken x at a time. If you've studied combinations before, then you already know that the formula is n factorial divided by x factorial, times n minus x factorial. If you don't understand this notation, then you can easily find an online combination calculator, where you just enter in n and x, and it will complete the calculations for you. Let's look at an example.
And typically, we use a calculator or some sort of technological assistance in order to do this. But for a simple example like this, we can go through it ourselves. However, for most of them, you can search online to find a binomial distribution calculator. And then you just have to input the number of successes, the number of trials, and the probability, and it will compute these calculations for you.
So here, let's first start off by writing what we have. We have b of-- and then our x is going to be that 2. the number of successes that we're looking for. n is the number of trials. That's our 5. And p, our probability, is 1/6.
And now we're just going to keep filling in across, and I'm just going to label this as x and as n so that I remember as I go forward. And that equals n so 5-- C 2, times the probability. So that is our 1/6. So times 1/6 to the x. So that's squared right there. Times 1 minus our probability, so 1 minus 1/6, which is 5/6 to the n minus m so n minus x is 5 minus 2, which is 3.
When we put this part in, we get 10 times 1 over 36, times 125, over 216. And then when we multiply all those three pieces together, then we end up with approximately 0.161. So I rounded this here. And that's telling us that we have a 0.161, or about a 16.1% chance of getting exactly two fours when we toss a die five times.
So this here is an example of the binomial distribution. But then again, we more commonly would use some sort of technological aid or online calculator to do this for us for the more complicated problems. This has been your tutorial on binomial distributions.