Sample space - Probability and probability Distributions, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Sample space is the collection of all certain outcomes of an experiment. A process which produce a set of data is known as an experiment and a set of all possible outcomes of an experiment is called the sample space. It is denoted by S. Sample points are the element of sample space which simulate the experiment in terms of the sample space. To know the probability of an event to occur it is needed to know the the total number of events in the sample space, that is, all the possible events should be known.

Definition

A sample space is a collection of all possible outcomes of a random experiment. A random variable is a function defined on a sample space. A sample space may be finite or infinite. Sample Space is a set of all possible outcomes. It is mainly denoted as 'S'. Infinite sample spaces may be discrete or continuous.

Formula

If we talk about formula or equation for the sample space, we would say that there is no particular formula for finding or estimating sample space for an experiment. We learnt that the sample space is defined as the collection of all the possible outcomes of a trial. In order to write correct sample space, one needs to focus on the experiment carefully and think about what the possible outcomes it could have. Thus, list all the outcomes and we obtain required sample space.

Sample Space Probability

In probability, there are few quite-commonly seen sample spaces in probability theory. For example - the sample space when two coins are tossed together, is {(HH), (TT), (HT), (TH)}. 

Also, the sample space for two dice thrown simultaneously is given below :

{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

We may also often want to calculate the probability of getting an even or odd number on a roll of a die. In this case, sample space would be {2, 4, 6} or {1, 3, 5}.

In this way, sample spaces for different probabilities can be found.

Multiple Sample Spaces

There may be many experiments in which there are multiple sample spaces. This actually depends upon the result the experimenter is interested to obtain. For instance - when a card is drawn from a deck of 52 playing cards. There arise two possibilities - one is that there may be four different suits (hearts, clubs, spades and diamonds) and another is that there may be different ranks ((Ace, two, three, ..., king). However, a complete description of results would specify both rank as well as suit. There are several such examples possible.

Infinitely Large Sample Spaces

We are aware that each subset of our sample space is known as an event. This concept gives rise to the situations where there are infinite members in the subspace, i.e. the subspace is infinitely large. Therefore for an event, a more precise definition is required. According to this definition, an event is considered to be only measurable subsets of a sample space, constituting σ-algebra over the sample space itself. However, this shows only theoretical significance because usually σ-algebra includes all the subsets of interest in experimental applications.