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Binomial Distribution
Common Core: S.MD.1

Binomial Distribution

Author: Sophia Tutorial

This lesson will explain the binomial distribution.

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What's Covered

In this tutorial, you're going to learn about the binomial distribution. Specifically you will focus on:

  1. Binomial Distibution


Binomial distribution is a probability distribution that follows the binomial setting. A scenario can be considered binomial if it fits these four criteria:

Term to Know

Binomial Distribution

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.

  1. The chance experiment has a fixed number of trials that you'll call n, and that every chance experiment that you do has only two outcomes for every trial. And you're calling those success and failure. 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. 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. 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.
  2. Each trial then has to be independent of any others. So die rolling, coin flipping, spinning a spinner, all of those are independent events.
  3. There also has to be a fixed probability of success on every trial. You're going to call that p.
  4. And finally, what you’re interested in measuring is the number of times you achieve success out of those n trials. Typically, this is denoted as either one of two variables, either k or r. You will use the letter k in this tutorial.

One way that you could look at this is following the betting of black on a roulette wheel.

Assume the gambler bets black every time. They might win every time. That would be pretty lucky. But they might lose every time. Follow all the different scenarios. They might lose all four times, and that would be no fun for them. There's only one way to do that-- lose, lose, lose, and lose.

They might win exactly one time as well. If you take a look below, there are four ways to do that.

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.

They could also win twice.

There are six ways to do that. They could win twice in a row. Then lose twice in a row or they could lose-win, lose-win. Notice regardless of which yellow branch they travel, 18/38 appears twice, and 20/38 appears twice, regardless of which yellow branch you follow.

They might also win three times.

If you can see what you're going for-- 18/38 appears three times. 20/38 appears once. That happens on four branches of the tree diagram.

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.

Summarize all this in a probability distribution.

Zero wins, that happened once. When that happened, 20/38 was the probability four times in a row. When you had one win, that happened on four of the branches. 20/38 appeared three times and 18/38 appeared once on each of those four branches. On the six branches that had them winning twice, 20/38 appeared twice, 18/38 appeared twice. On the four branches that had them winning three times, there were three 18/38 on them and only one 20/38. And on the one time, where they won all four times, 18/38 appeared four times.

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.

The 4 has 18/38 to the 0 power. The 3 has 18/38 to the first power, and so on. 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.

Three losses means one win. And so they lose 20/38 probability and they do that every time of the four that they don't win. 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.

There are some similarities. There is a way to calculate the probability of winning exactly a certain number of games.

The formula is denoted as n choose k times the probability of success to the power of k, which means you 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.

It's notated sometimes 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.

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.

This is the k successes, probability of success k time. And the rest, n minus k are the failures.

What's the probability the gambler breaks even? That means he wins twice and loses twice.

In 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.

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.

This is a little bit different. What you need to actually do this time is find the probability that you get no 5s, and add it to the probability that you get exactly one 5.

q is the probability of failure. Sometimes it is written out with p and q. What you find is 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.


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.

Source: This work adapted from Sophia Author Jonathan Osters.

Terms to Know
Binomial Distribution

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.