# Formula of Sample Space in Math

## What Is Sample Space?

Life is a game of chance and choice. We live in a day and age where choices bombard us on things that weren't available to us years ago. And because of this reality, the concept of probability and sample space have become a part of our lives even if we aren't always aware of it. Take for instance, cell phones. Today there are hundreds of different phones available for consumers to purchase. What if we wanted to find out how many different cell phones are available for purchase or lease? The number of cell phones available can be seen as the sample space of an experiment or study involving cell phones. **Sample space** is the number of possible outcomes of an experiment or study.

## Finding Sample Space

Unlike many math concepts, there isn't a specific formula for how to find sample space, unless you are given other related values that we will discuss later. What do we mean by this? Well, let's look at a few examples to show you what we mean.

#### Example 1

What is the sample space of rolling a six-sided die?

The sample space would be the possible outcomes from rolling the dice {1, 2, 3, 4, 5, 6}. In other words, there are six possible outcomes for this problem.

#### Example 2

What is the sample space of picking a consonant from the alphabet if each letter is written on an individual piece of paper?

The sample space would be the possible outcomes from picking a piece of paper with a consonant written on it: {B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, W, X, Y, Z}. In other words, there are 21 possible outcomes for selecting a piece of paper with a consonant on it.

Notice in both of our examples, that a mathematical formula wasn't used to derive our answer. We had to obtain our answer based on information given in the problem. When no other related pieces of information are given other than what is being studied, we must carefully read the problem to come up with our correct sample space. A deck of cards, the numbers on a pair of dice, and the sides of a coin are examples of cases where we can easily figure out the sample size.

## Probability & Specific Events

Staying with the example of cell phones, let's say you wanted to do a study on a specific brand of cell phones. In order to find the probability of that brand being owned or leased compared to the number of all brands, you would need to know the sample space and the specific event. A **specific event** is a certain outcome that you are wanting to find the likelihood of occurring. For example, let's say you wanted to find the probability of a cell phone user currently possessing the latest hot cell phone. The specific event in this case is the number of people who possess that phone, and the sample space would be the possible cell phone brands possessed. The probability would then be found by dividing the specific event from the sample space:

*P* = *Specific Event* / *Sample Space*

Using the formula, *P* = *Specific Event* / *Sample Space*, we can calculate the sample space if we are given the values (or ability to obtain) the probability and specific event. For example, if you know that the probability (*P*) of owning or leasing an iPhone is 1/3 and that there are 30 people that own an iPhone (the specific event, or *SE*), we can find the sample space (*SS*) of possible outcomes in the study:

*P* = *Specific Event* / *Sample Space*

We know that *P* = 1/3 and that the specific event (*SE*) = 30, so we need to insert those values in the formula. Because we don't know the value for sample space (*SS*), we'll use the variable *S* to represent it, like you can see here:

To find the value of *S* we need to cross multiply, like this:

So our sample space in this problem is equal to 90, meaning that there are 90 possible outcomes for our study on cell phone brands.

## Lesson Summary

All right, let's take a moment to review what we've learned in this lesson. As we learned, **sample space** is the number of possible outcomes of an experiment or study and is used to help find the probability of a specific event (with a **specific event** being a certain outcome that you are wanting to find the likelihood of occurring). There isn't a set formula for finding the sample space unless you are given (or can solve for) the probability and specific event values. You then use the formula *P* = *Specific Event* / *Sample Space*, plug in the *P* and *SE* values, and cross multiply to find the *SS*.

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## Formula of Sample Space in Math: Practice Problems

In this activity, you will check your knowledge of how to find sample space for experiments, studies, or specific events.

### Directions

For this activity, print or copy this page on a blank piece of paper. Then, carefully read the given word problems and provide a solution.

1.) In a business study conducted for an up-and-coming cereal brand, it was found out that the probability of people buying Cheerios is 3/5 and that the number of people who buy Cheerios is sixty. Find the sample space of possible outcomes on the study on cereal brands.

2.) A school conducts a study on milk brands that students prefer to include in school meals. They found out that the probability of students preferring brand ZY milk is 2/3 and that the number of students who actually prefer the said brand is eighty-two. Find the sample space of possible outcomes on milk brands.

### Answer Key

Use the formula * P = Specific Event / Sample Space* to find the sample space.

Use the variable *S* to represent sample space.

1.) plug P = 3/5, SE = 60

3/5 = 60 / *S*

Cross multiply

There are **100 possible outcomes** on this study on cereal brands.

2.) plug P = 2/3 , SE = 82

2/3 = 82 / *S*

Cross multiply

There are **123 possible outcomes** on this study on milk brands.

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