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Fundamental Counting Principle

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Hi. This tutorial covers the fundamental counting principle. So let's just start with an example to motivate this idea.

So supposing Annie is running late in the morning, and she needs to get dressed. She randomly grabs a shirt from her top drawer and a pair of shorts from her bottom drawer. There are three shirts in the top drawer and four pairs of shorts in the bottom drawer. How many shirt and shorts outfits could she wear? How many different shirt and shorts outfits could she wear?

OK. So there's a bunch of different ways of figuring this out. Let's just start with the shirt. So let's say we have shirt 1, shirt 2, and shirt 3. So if we think about if she picked shirt 1 first, each of the different shorts would form a different outfit with shirt 1. So because there are 4 pairs of shorts that she could choose after she chose shirt 1, really, there are 4 outfits here.

OK. Shirt 2-- she could either-- she could pick any of these 4 pairs of shorts, and that would also make 4 outfits. OK? And same thing with shirt 3. She could end up with 4 outfits again. OK? So since there are 4 outfits in each of these three categories, we would end up with 12 total outfits. OK?

So again, there are a bunch of different ways of figuring this out, but this was the way that we used to come up with that 12 different outfits. OK? So like I said, this is an example to motivate the fundamental counting principle, which is a principle that states that there are-- if there are m different possible outcomes for a trial of 1 experiment and n different possible outcomes for a trial of another experiment, then there are m times n different outcomes if both experiments are done together.

And the fundamental counting principle only applies if the outcome of a-- if the outcome of a trial of one experiment does not have an effect on an outcome of a trial of another-- of the other experiment. OK. So if we think about going back to this example again, one experiment would just be picking a shirt. Another experiment would be picking a pair of shorts. OK?

So there are three outcomes. If we can-- three outcomes of the first experiment, four possible outcomes of the second experiment. So if we wanted to know, well, how many outcomes total if the two experiments are done together, what we could do is just take 4 times 5 and also get 12. OK? And again, that's assuming that whatever she picks as a shirt has no effect on which pair of shorts she picks. And if she's doing this randomly or haphazardly, the experiments will be done independently of each other.

OK. So let's-- let me also talk a little bit about a type of diagram called a tree diagram. So a tree diagram is a diagram that shows the branching possibilities of one or more chance experiments. OK. Let's take a look at another example here. So suppose a couple plans to have three children. How many different boy-girl outcomes are possible?

OK. So what we're going to do is approach this problem in two ways. We're going to start first by figuring out how many outcomes there are for these three chance experiments using a tree diagram, and then we're also going to use the fundamental counting principle. OK. So if we start with a tree diagram and look at the first experiments, the first experiment will simply just be the first child. All right. And the outcomes there are either boy or girl. So there are two possible outcomes of that first experiment.

OK. Now, if we look at our second experiment, the second child being born, if the first child was a boy, then the second child then, again, could either be a boy or a girl. And then these two outcomes will represent the outcomes where the first child is a girl, and then these would be the outcomes of the second child or the second experiment.

OK. Now, if we look for now the third child-- again, boy, girl, boy, girl, boy, girl, boy, girl-- so if we look at these outcomes now-- let's say we wanted to look at this specific outcome. What this would represent is that the first child, the oldest child, would be a girl, the middle child would be a boy, and the youngest child would be a boy. So that would represent one of the possible outcomes.

Another possible outcome would be this one. It would be girl, girl, girl. So the couple would have three girls. OK? And maybe let's go to this one. In this case, the oldest child would be a boy, the middle child would be a girl, and the third child would be a boy. So if we look at the possible outcomes here, we can see that there are 1, 2, 3, 4, 5, 6, 7, 8 possible outcomes. OK?

So let's now confirm that result using the fundamental counting principle. So remember, the fundamental counting principle says that all you need to do-- as long as your outcomes here are independent, you can multiply the number of outcomes in each experiment by each other. So if we do that, there are two possible outcomes in the first experiment-- boy or girl-- two possible outcomes in the second experiment, and two possible outcomes in the third experiment. So if we multiply those together, 2 times 2 times 2 is equal to 8. OK? So we can confirm the result that we used by displaying the sample space using a tree diagram by using the fundamental counting principle.

So that has been the tutorial on the fundamental counting principle. Thanks for watching.