Source: Tables and charts created by the author
In this tutorial, you're going to learn about the binomial distribution. Binomial distribution is a probability distribution that follows the binomial setting. So a scenario can be considered binomial if it fits these four criteria. The first is that the chance experiment has a fixed number of trials that we'll call n, and that every chance experiment that you do has only two outcomes for every trial. And we're calling those success and failure.
Now a couple things to note. Success is a fairly arbitrary term. It doesn't necessarily have to be a good thing. It could be that losing a gambling bet could be considered a success and winning could be considered a failure for instance, for the casino. So it doesn't really matter what you call success and failure, so long as they're complementary events.
Also, you can rig the scenario to have two outcomes, even if there aren't technically two outcomes. So for instance, on a die there are six outcomes, but maybe you could call rolling a five success and anything but a five failure. In that case, you've rigged it so you get two outcomes per trial. Each trial then has to be independent of any others.
So die rolling, coin flipping, spinning a spinner, all of those are independent events. There also has to be a fixed probability of success on every trial. We're going to call that p. And finally, what we're interested in measuring is the number of times we achieve success out of those n trials. Typically, this is denoted-- depending on textbooks and convention-- as either one of two variables, either k or r. I'm going to use the letter k in this tutorial.
So one way that we could look at this is following the betting of black on a roulette wheel. So suppose that the gambler bets black every time. He might win every time. That would be pretty lucky. That would be fun. But he might lose every time, and that wouldn't be so fun.
Let's follow all the different scenarios. He might lose all four times, and that would be no fun for him. But there's only one way to do that-- lose, lose, lose, and lose. He might win exactly one time as well. And if you take a look, there are four ways to do that. Now notice, all of these four branches on the tree diagram have the same probabilities on them. 18/38 appears once and 20/38 appears three times on each of these four branches.
He could also win twice. And there are six ways to do that. he could win twice in a row. Then lose twice in a row or he could lose-win, lose-win. Notice regardless of which yellow branch he travels, 18/38 appears twice, and 20/38 appears twice, regardless of which yellow branch you follow. He might also win three times. That's fun. And if you can see what we're going for-- 18/38 appears three times. 20/38 appears once. And that happens on four branches of the tree diagram. And finally, the most fun scenario for the gambler, the four wins and 18/38 appears four times, and 20/38 doesn't appear at all, but that only happens one way.
So let's summarize all this in a probability distribution. So zero wins, that happened once. And when that happened, 20/38 was the probability four times in a row. When we had one win, that happened on four of the branches. And 20/38 appeared three times and 18/38 appeared once on each of those four branches. On the six branches that had him winning twice, 20/38 appeared twice, 18/38 appeared twice. On the four branches that him winning three times, there were three 18/38 on them and only one 20/38. And on the one time, where he won all four times, 18/38 appeared four times.
So now let's look at the similarities and differences within these boxes. You might notice something. They all have the number of ways that these events happened. They all have an 18/38, even this one. This one has 18/38 to the 0 power. This one has 18/38 to the first power, second power, third power, fourth power. Notice that's the same 0, 1, 2, 3, and 4 as the number of wins. So if we're calling it k wins, it's 18/38 to the power of k, so like to the power of one or power of two.
And if you take a look, three losses means one win. And so he loses 20/38 probability and he does that every time of the four that he doesn't win. So zero wins means four losses. One win means three losses. Two wins means two losses. Three wins means one. And four wins means the 20/38 doesn't appear at all.
So there's some similarities here. There's a way to calculate the probability of winning exactly a certain number of games. So the formula that follows from this is denoted as n choose k times the probability of success to the power of k, which means we want to succeed k times out of n times, times the probability of failure, 1 minus p, to the rest of the trials that weren't successes, n minus k.
So this number out here, it's pronounced n, choose k. It's also notated sometimes this way, with subscripted n, big C, subscripted k, n, choose k. It's also sometimes written like that. It's a number of ways to achieve k successes out of n trials. If you're using a calculator most calculators use the command, nCr. Remember, I mentioned that r or k are the most common choices for this variable here. So a lot of calculators use the command nCr, requiring you to put in something like 5nCr2. Then, you get an answer when you hit Enter.
And then this is the k successes, probability of success k time. And the rest, n minus k are the failures. So what's the probability the gambler breaks even? Well, that means he wins twice and loses twice. So four trials, 18/38 is the probability of winning. Two successes go through. Put all those values in, and you get about a 37% chance of breaking even.
Let's try another one. Five dice are rolled. What's the probability of obtaining no more than one 5. Now, no more than one, could be zero 5s or one 5. So this is a little bit different. What we need to actually do this time is find the probability that we get no 5s, and add it to the probability that we get exactly one 5.
By the way, q is the probability of failure. And so sometimes we write it out with p and q. So what we find is we find that this is the probability of getting exactly zero 5s, and this is the probability of getting exactly one 5. As it turns out, just by sheer coincidence, they're the same number. And so the combined probability of getting not more than one 5 is 0.804.
And so to recap, binomial probability arises from the binomial setting. There are four parts, a fixed number of trials, two outcomes for trial, a fixed probability of success, and independent trials. You can find the probability of a given number of successes using the formula. You could also use your calculator. So we talked about the binomial distribution. That was the probability distribution for the binomial situation. Good luck and we'll see you next time.
The distribution of the number of successes that occur within n independent trials of a chance experiment with two outcomes per trial and p probability of success per trial.