Sample Space

Scientists and product testers love to run all kinds of experiments, and sample space becomes a useful concept to understand as you run your own experiments. You'll soon see that while this concept involves subjects like probability, it might be much more straightforward than you think. Let's figure out sample space together:

What is sample space?

A sample space is a set of values that represent all of the possible outcomes of an experiment. As you might expect, a sample space for a very complicated experiment can be almost endless. The more variables in an experiment, the more possible outcomes. If you choose a card from a deck, there is a "sample space" of 52 outcomes. But if you flip a coin, the sample space is only 2 possible outcomes. Rolling a six-sided die has a sample space of 6 outcomes. As you can see, the concept is quite straightforward.

How to write sample space values

To properly represent sample spaces in mathematics, we need to communicate the possible outcomes of an event using a specific notation, rather than just stating the number of outcomes.

For a coin, the sample space includes two possible outcomes: heads and tails. We can represent this sample space as $\left\{H,T\right\}$ , where $H=\mathrm{heads}$ and $T=\mathrm{tails}$ .

For a six-sided die, there are six possible outcomes, each corresponding to the numbers on the faces of the die. We can represent the sample space as $\left\{1,2,3,4,5,6\right\}$ .

Events and simple events

There are a few other concepts we need to cover before we start running our own experiments:

• Event: In the world of mathematics, an "event" is a subset of possible outcomes for an experiment. We might also define this as a subset of the sample space. Why? Because an event exists within a sample space, and it may have a defined set of possibilities. If we're talking about all possible outcomes for an experiment, we call this a sample space instead of an event.
• Simple Event: A simple event is something that can only happen one way. It has a single possible outcome. For example, there is only one "way" to get heads when you flip a coin.

Examples of sample spaces

We know that the sample space for a six-sided die is $\left\{1,2,3,4,5,6\right\}$ . But what is the "event" that we get a number less than four?

We would write this as $\left\{1,2,3\right\}$ .

What about the event that we get an even number?

We would write this as $\left\{2,4,6\right\}$ .

But what might happen if we flipped two coins at the same time? We would have to consider all of the possibilities, and we would write this as:

$\left\{\mathrm{HH},\mathrm{HT},\mathrm{TH},\mathrm{TT}\right\}$

$\mathrm{HH}=\mathrm{heads}$ and then heads, $\mathrm{TT}=\mathrm{tails}$ and then tails, $\mathrm{HT}=\mathrm{heads}$ and then tails, $\mathrm{TH}=\mathrm{tails}$ and then heads.

Probability can become quite complex

Although these foundational concepts might seem simple, they provide us with a logical system that helps us map out probability with accuracy. But why might we want to run random experiments? Random experiments help to eliminate potential biases by ensuring that each member of the population has an equal chance of being selected for the sample. This helps to ensure that the sample accurately represents the population as a whole and that any differences between the sample and the population are due to chance, rather than systematic biases.

Topics related to the Sample Space

Geometric Probability

Independent Events

Binomial Probability

Flashcards covering the Sample Space

Statistics Flashcards

Common Core: High School - Statistics and Probability Flashcards

Practice tests covering the Sample Space

Probability Theory Practice Tests

Common Core: High School - Statistics and Probability Diagnostic Tests

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