# Formula of Sample Space in Math

Instructor: Meredith Fontana
In this lesson, we explore what is sample space, how it is related to probability and the formula to use to find it when given probability problems. We will break down the formula P = Specific event/Sample Space in order to solve for the value of possible outcomes in an experiment.

## Introducing...Sample Space!

Life is a game of chance and choice. We live in a day and age where choices bombard us on things that were not 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 are not 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. In general terms, sample space is the number of possible outcomes of an experiment (or study).

## How do we find sample space?

Unlike many math concepts, there is not a specific formula for how to find sample space (unless you are given other related values that we will discuss later). What do I mean by this? Let us look at a few examples to show you what I 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 twenty-one possible outcomes for selecting a piece of paper with a consonant on it.

Notice in both of our examples, that a mathematical formula was not 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.

## Sample space, probability, and 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/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

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