Sample Space in Probability

Last Updated : 15 Oct, 2026

A sample space in probability is the set of all possible outcomes of a random experiment. It is usually denoted by S or \Omega. Every possible outcome of the experiment must be included in the sample space.

Sample-Space-of-Rolling-a-Die-and-Tossing-a-Coin


Another example of a sample space for some random experiment:

  • Experiment: Spinning a European roulette wheel
  • Sample space: {0, 1, 2, 3, 4, 5, ..., 36}

Types

There are three types of Sample space:

1. Finite: The number of outcomes is countable and limited (e.g., rolling a die).
Sample Space: S = {1, 2, 3, 4, 5, 6}

2. Infinite: The outcomes go on forever (e.g., the Number of coin tosses needed until the first head appears).
Sample Space: S = {1, 2, 3, 4, 5, 6, ... }

3. Continuous: The outcomes can take any real value within a given range (e.g., A real number between 0 and 1).
Sample space: S = {x ∈ R ∣ 0 ≤ x ≤ 1}

Finding Sample Space in Probability

To find the sample space in Probability, follow the steps below:

  1. Identify all possible outcomes of the experiment.
  2. List these outcomes in a set, ensuring each one is unique.

Combining sample spaces when multiple events occur helps calculate complex probabilities.

Tossing Coins

The sample space for tossing coins is given below:

1. When Two Coins are Tossed: Each coin can result in two possible events: either a head or a tail. In the case of flipping two coins, There are 4 outcomes in the sample space:

{HH, HT, TH, TT}

2. When Three Coins are Tossed: Sample Space for rolling 3 coins can be calculated keeping in mind the following:

When flipping three coins, the sample space encompasses all the possible combinations of heads and tails for the three coins.

It can contain 23 = 8 different outcomes, each with varying numbers of heads and tails in different orders, given as

Sample-Space-on-Tossing-Three-Coins

Rolling Dice

Sample space for rolling a die and two dices are given below:

1. When One Die is Rolled: On rolling a die, we can have 6 outcomes. So the sample space for rolling a die will be,

S = {1, 2, 3, 4, 5, 6}.

2. When Two Dice are Rolled: When rolling two dice, the sample space represents all the combinations of outcomes that can occur. It consists of pairs of numbers ranging from (1, 1) to (6, 6), consisting of 62= 36 pairs. It helps in calculating probabilities for various sums or events involving two dice.

The following is a table of the sample space of rolling two dice.

Sample-Space-of-two-die

Practice Questions on Sample Space

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