What is a Sample Space?

When dealing with any type of probability question, the sample space represents the set or collection of all possible outcomes. In other words, it is a list of every possible result when running the experiment just once. For example, in one roll of a die, a 1, 2, 3, 4, 5, or 6 could come up. Each of these are considered outcomes and together they make up a sample space. In the following lesson, we will look at the notation used for a sample space as well as some examples of finding the sample space for a probability experiment.

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Notation

Sample spaces are usually written using set notation. This means that each possibility is listed only once within curly brackets. For example, if there were three possibilities we would write the sample space as:

\(S = \{ \text{outcome 1}, \text{outcome 2}, \text{outcome 3} \}\)

Many times, the outcomes will be written in some kind of shorthand. It is not necessary to use shorthand when writing out the possibilities, but if you do, make sure that the shorthand is easy to follow. For example, suppose we flipped a coin. The possibilities are “heads” and “tails” which could be written as “H” and “T”. Using this, the sample space would be:

\(S = \{ \text{H}, \text{T} \}\)

Examples of finding the sample space

With any type of probability experiment, describing the sample space just requires thinking carefully about all of the possibilities and making sure none are missed. You can see this type of thinking in the examples below.

Example

A single 12 sided die has the whole numbers 1 through 12 written on each face. The die is rolled once and the number that appears is noted. Describe the sample space of this experiment.

Solution

Any of the 12 sides could have come up on a single roll. Therefore the sample space would be

\(S= \left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}\)

Example

Suppose that two coins are flipped and the side which they land on is noted. Describe the sample space for this experiment.

Solution

One possibility is that both coins land on the same side. If this was the case, that side could be either heads or tails. These two outcomes could be represented by HH and TT. If the coins don’t land on the same sides then what are the other possibilities? Well, it could be the first coin landed on heads and the second tails (HT) or the other way around (TH). Notice that when we deal with sample spaces, the order is important!

\(S = \left\{ HH, TT, HT, TH \right\}\)

Example

A person is asked to guess a random number between 1 and 10 and it is noted whether or not he guessed correctly. Describe the sample space for this experiment.

Solution

This one is much trickier than it looks! Notice that the number being guessed doesn’t matter here – all that is being noted is if the number was guessed correctly (possibility 1) or if the number was not guessed correctly (possibility 2). There is nothing in between him guessing right or wrong, so this covers all of our possibilities!

\(S = \left\{ \text{correct}, \text{incorrect} \right\}\)

This last example shows how important the context is. If we were interested in the possible numbers he could have guessed, then our sample space would have looked completely different. Paying attention to details like this is important in all probability calculations.

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Summary

The sample space of an experiment is just a listing of all the possible outcomes (results) from that experiment. To find the sample space, you need to make sure you think of all the possible results. Be sure to pay close attention to the context and what aspect of the probability experiment is of interest.

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