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Algebra II: High School23 chapters | 203 lessons

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Lesson Transcript

Instructor:
*Joseph Vigil*

There are many situations in which you will have to make several decisions simultaneously. The fundamental counting principle will help you determine how many different possible outcomes there are when you have to make multiple simultaneous decisions.

Choices, choices, choices; it seems we're bombarded with choices daily. It can be hard enough making up your mind about a single decision, but what about when you have to make multiple decisions at once?

Restaurants these days, for example, have great lunch specials. Let's say we go to your favorite restaurant for lunch. Their special consists of a drink (soda, tea or lemonade), a salad (garden or Caesar) and a small entree (pasta, chicken or meatloaf). Talk about decisions!

Let's slow down and take a look at all of our options. Just how many different possible lunch combinations are there? Well, let's look at the possibilities. For any drink you choose, you could then go with one of two salads. So each drink option branches out into two further decisions.

From each of those two salad options, you then have three entree options. So those two possibilities now become six. In other words, when we start with one drink, we then have six possible combinations that branch out from there. Since we have three drinks, we can multiply three times six to find the total number of lunch combinations. 3 * 6 = 18, so there are 18 possible lunch combinations.

We've used the fundamental counting principle without even knowing it. But before we examine that principle more closely, let's look at some fancy vocabulary you might find in these situations.

Remember how you had to make three separate decisions for lunch: drink, salad and entree? Each of these single decisions is called an event. An **event**, in the world of probability, is a single occurrence or decision with a distinct set of possible outcomes. For example, if you flip a coin, that single coin flip is an event, because it's one occurrence that has a defined set of possible outcomes.

Every event has a **sample space**, which is simply the complete set of possible outcomes for a single event. For a coin flip, the sample space includes heads and tails because those are the possible outcomes in the event of a coin flip.

And, finally, **sample points** are the individual possible outcomes in a sample space. So those two possible outcomes in our coin flip - heads and tails - are our two sample points. Some sample spaces are larger than others. It just depends on how many possible outcomes there are in a single event.

For example, if we're determining whether an item from a garden is a fruit or vegetable, that's a small sample space because there are only two possible outcomes, or sample points. But if we then determine what type of fruit or vegetable that item is, our sample space could potentially become much larger because it consists of every kind of fruit and vegetable contained in that garden.

According to **the fundamental counting principle**, when we're dealing with multiple events, we can multiply together the number of sample points in each event to determine the total number of possible combinations.

This is essentially what we did when we determined the total number of possible lunch combinations. We had three events: drink choice, salad choice and entree choice.

In the drink selection event, we have three sample points because there are three drinks to choose from. Likewise, we have two sample points in the salad selection event and three sample points in the entree selection event. So all we need to do is multiply together those numbers of sample choices. 3 * 2 * 3 = 18.

We were right! There are a total of 18 possible lunch combinations! This counting principle works in any situation with multiple events.

For a final project, your history teacher has given the class some options. You can cover either the American Revolution, the French Revolution or the Chinese Cultural Revolution. You can write a research paper or create a digital presentation. Finally, you can record a presentation and play it for the class or give a live presentation. How many different possible projects can you choose from?

According to the fundamental counting principle, we need to determine how many options, or sample points, there are in each individual event.

You can write about any of three revolutions, you have two ways that you can format your research and you have two presentation options. Now, all we have to do is multiply together the numbers of sample points for each event. 3 * 2 * 2 = 12. So, there are 12 combinations that you can choose from given all the options your teacher has given you.

The **fundamental counting principle** helps us find the total number of possible combinations that come from multiple events, whether we're putting an outfit together with different shirts, pants and shoes; selecting a meal with different drink, salad and entree options; or putting together any combination of choices.

Each of these single decisions is an **event**. Each event has a **sample space**, which is the group of all possible outcomes, or **sample points**. All you need to do to determine the total number of possible combinations is multiply together the number of sample points for each event.

You should have the ability to do the following after watching this video lesson:

- Define event, sample space and sample points
- Explain what the fundamental counting principle is and how to use it

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Algebra II: High School23 chapters | 203 lessons

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