Sample space formula

Sample space refers to all possible outcomes of an experiment. It is denoted by S. A sample space may have number of possible outcomes. The number of outcomes depends upon the experiment. If a sample space has the finite number of outcomes, it is called as the discrete or finite sample space.

For random experiments, the sample space is written within "{}" curly braces.

Here, we are defining some events with all possible outcomes or sample space.

• When a dice is thrown, there are six possible outcomes, i.e., Sample space (S) = (1, 2, 3, 4, 5, and 6).
• When a coin is tossed, the possible outcomes are Head and Tail. So, in this case, the sample space (S) will be = (H, T).
• When two coins are tossed, there are four possible outcomes, i.e., S = (HH, HT, TH, TT).

The elements of a sample space may be letters, words, numbers, symbols, etc. A sample space can be finite, countably infinite, or uncountably infinite.

There is a difference between a sample space and an event. Let's see a brief description of an event.

Event: The subset of sample space is called an event. Event is generally denoted by the letter 'E'. We can understand the difference between the event and a sample space by the below example -

Suppose a dice is thrown, so the sample space (S) for this dice is = {1, 2, 3, 4, 5, 6} but the event can be {1, 3, 5} representing the set of odd numbers, and {2, 4, 6} representing the set of even numbers.

Now, let's solve some questions of finding out the sample space.

Ques - 1) What will be the sample space of the interval [3, 9]?

Ans - 1) The sample space (S) for the given interval is = {3, 4, 5, 6, 7, 8, 9}

Ques - 2) What will be sample space when two dice are thrown together?

Ans - 2) On rolling two dice together, we will get 36 outcomes. So the possible sample space will be -

S = {(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)}