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Fundamental Counting Principle
Common Core: 7.SP.8b

Fundamental Counting Principle

Author: Katherine Williams

Calculate the number of possible outcomes for a given situation using the fundamental counting principle.

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This tutorial discusses the fundamental counting principle. Now the fundamental counting principle says-- it's bit wordy and it'll make a lot more sense when we do examples, but stick with me-- if there are m different outcomes for one trial and n different ways for another trial, and the trials are independent, they don't affect each other, then there are m times n ways for the two trials to both occur. That's the key part right there, that m times n ways for the two trials to occur. So when you have two separate trials, and you're combining them together into one thing, and you want to know how many different outcomes there are altogether, you're going to multiply those two pieces together.

Here's our example. If you're rolling two dice, there are six different possibilities in the first die, six outcomes, 1, 2, 3, 4, 5, 6. And on the other die, there are another six outcomes, 1, 2, 3, 4, 5, 6. What you get on the first one doesn't change what can happen on the second one, so they're independent. So if you wanted to know how many different possible things could happen when you roll two die, then you would multiply these two numbers together, m times n. And in this case, they're the same thing, 6 times 6.

So there are 36 different possible things that can happen when you roll two die. You could get a 1 and a 1, a 1 and a 2, 1 and a 3, a 1 and a 4. And that's all different from getting a 4 and a 1, a 4 and a 2, a 4 and a 3.

So one way of helping you to organize all these possibilities out if you need to see it is by using a tree diagram. So for example, if we were talking about a family, and they have three children, or they're going to have three children. And then, you could either be a boy or a girl.

For that first child, that first plot, they could have a boy or they could have a girl. And I'm simplifying with B and G. Now when they go to have their second child, if the first one was a boy, they could still have a boy or a girl. And if it was a girl first, they could have a boy or a girl. So this chart is showing us all the possibilities. And then again, no matter what their second child was, their options are still boys and girls for each.

So now here, this is showing us all of the possible things that could happen. They could have three boys, so they could have boy, boy, boy. They could have two boys and then a girl. They could have boy, girl, boy. They could have boy, girl, girl. And then we're going to run through the same set on the other side. I'll do that one more quickly, since you're starting to get the hang of it.

So this is showing us all of the possibilities that could happen. There's eight different things that they could have. And again, we're just doing that m times n, but kind of in the background. So here, there's three different times they're having kids, so there's actually three trials. So for that first trial, there's two different chances that they have. And then we're going to multiply for that second trial another two. And in that third trial, another two. So there's eight outcomes possible.

So both the tree diagram and the fundamental counting principle are going to show us what the possible outcomes are. If I were to try to do this tree diagram for the dice, it would start to get messy because I'd be trying to do strings of six off of each other. So sometimes a tree diagram isn't going to be as clean or as helpful as just using the fundamental counting principle. But if I needed to list out all the possibilities, then a tree diagram is a really great way of organizing that because you can see the different pathways give us the different outcomes. This has been your tutorial on the fundamental counting principle and tree diagrams.

Terms to Know
Fundamental Counting Principle

If chance experiment A has m possible outcomes for one trial, and chance experiment B has n possible outcomes for its trial (independent of the first trial), then there are mxn potential outcomes when A and B are done together.

Tree Diagram

A way to visualize the different "paths" that a sequence of chance experiments could take.