# Sample space

In a probabilistic experiment, the sample space is the set of all possible outcomes of the experiment.

## Symbol

The sample space is often denoted by the Greek letter (Omega).

A possible outcome, which can also be called a sample point, is indicated as , where (omega) is the lowercase version of .

The letter is the last letter of the Greek alphabet and it is often linked to the idea of something omnicomprehensive (in our case the set of all things that can possibly happen). See also the related Wikipedia article.

## Examples

Here are some simple examples.

Example Suppose that the probabilistic experiment is the toss of a die. The six numbers that can appear face up, from 1 to 6, are the six possible outcomes of the experiment. Hence, the sample space is

Example When the experiment is the flip of a coin, there are two sample points, heads and tails . The sample space is

Example When we flip two coins, there are four possible outcomes and the sample space is

## Product space

The sample space in the last example is a product space. It is obtained by combining the sample spaces of two experiments (two individual flips of a coin).

In general, if we repeat a probabilistic experiment two times, then the product space is the Cartesian product of the sample spaces of the two experiments.

The Cartesian product is the set of all possible couples of sample points, where the first point is taken from the sample space of the first experiment and the second point is taken from the sample space of the second experiment.

Example In a first experiment, we randomly pick one of the first two letters of the alphabet. The sample space isIn a second experiment, the sample space isThen, the product space is

If there are more than two experiments, we follow the same logic. For three experiments, we list all possible triplets; for four, we list the quadruplets, and so on.

## Infinite sample space

In probability theory and statistics, it is quite common to deal with infinite sample spaces, made up of an infinite number of sample points.

This happens, for example, when a probabilistic experiment is repeated indefinitely until a certain condition is met.

Example We flip a coin until we get heads. It is quite likely that we will obtain heads after few throws. However, there is no upper bound on the number of tosses required to meet the stopping criterion. For example, it is very likely that we will get heads at least once in the first one hundred tosses, but there is also a very small chance of observing one hundred tails in a row. Similarly for one thousand, one million, and so on. Therefore, if we want to properly enumerate all the possible outcomes, we need to consider an infinite sample space, equal to the product of infinite individual experiments:

## Different kinds of infinity

As you probably know, infinite sets come in different sizes:

• countably infinite sets, whose elements can be put in one-to-one correspondence with the set of natural numbers;

• uncountably infinite sets, for which such a correspondence cannot be found.

Sample spaces can be not only countably infinite, as in the previous example, but also uncountable. Actually, both types are very common in statistics.

If you are not familiar with these two kinds of infinity, we suggest that you read two marvellous Scientific American articles on the topic:

A longer discussion can be found in a chapter of the Handbook of the Philosophy of Science by Peter Fletcher.

## Uncountable sample space

Frequently used uncountable sample spaces are the set of real numbers and its sub-intervals, such as the interval .

How do these infinite spaces arise? How could they possibly describe a real-world probabilistic experiment?

Although admittedly unrealistic, they are typically used because of their mathematical convenience.

Example Consider a physics experiment in which the proportion of atoms exhibiting a certain behavior is seen as random. A proportion is a number in the interval . If we knew exactly the total number of atoms involved in the experiment (easily a huge number!), we could enumerate all the possible values of the proportion, and we would have a finite sample space. However, it is much more convenient to assume that the sample space is the whole interval . For more details about this example, read our lecture on continuous random variables.

## More details

A detailed mathematical presentation of the concept of sample space - including a discussion of the properties that a sample space needs to satisfy - can be found in the lecture on Probability.

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