Sample Space

A complete list of all possible outcomes of a random experiment is called sample space or possibility space and is denoted by $$S$$. Each outcome is called an element of the sample space. A sample space may contain any number of outcomes. If it contains a finite number of outcomes, it is called a finite or discrete sample space. When two bulbs are selected from a lot, the possible outcomes are four, which can be counted as:

  1. Both bulbs are defective
  2. The first is defective and the second is good
  3. The first is good and the second is defective
  4. Both are good

Here the sample space is discrete.

When the possibilities of the sample space cannot be contained, it is called continuous. The number of possible readings of temperature from $${45^ \circ }C$$ to $${46^ \circ }C$$ will make a continuous sample space.

The sample space is a basic term in the theory of probability. We shall discuss some sample spaces in this tutorial. It is not always possible to make a sample space. If it contains a very large number of points, we cannot register all the outcomes, but we must understand how to make the sample space. The outcomes of the sample space are written within the $$\left\{ {} \right\}$$. Some simple sample spaces are discussed below.

A coin is tossed

When a coin is tossed, it has two possible outcomes: heads and tails. To be brief, heads is denoted by $$H$$ and tails is denoted by $$T$$. Thus the sample space consists of heads and tails. In set theory notation, we can write $$S$$ as:

\[S = \left\{ {{\text{head, tail}}} \right\}or\,\,S = \left\{ {H,T} \right\}\]


Two coins are tossed

When two coins are tossed, there are four possible outcomes. Let $${H_1}$$and $${T_1}$$ denote the head and tail on the first coin and $${H_2}$$ and $${T_2}$$ denote the head and tail on the second coin. The sample space $$S$$ can be written as

\[S = \left\{ {\left( {{H_1},{H_2}} \right),\left( {{H_1},{T_2}} \right),\left( {{T_1},{H_2}} \right),\left( {{T_1},{T_2}} \right)} \right\}\]

It may be noted that the sample space of the throw of two coins has $$4$$ possible points. A sample space of $$3$$ coins will have $${2^3} = 8$$ possible points and for $$n$$ coins, the number of possible points will be$${2^n}$$.


A die is thrown

An ordinary die has six faces. These six faces contain $$1,2,3,4,5,6$$ dots on them. Thus for a single throw of a die, the sample space has $$6$$ possible outcomes, which are:

\[S = \left\{ {1,2,3,4,5,6} \right\}\]

Two dice are thrown

A die has six faces. Each face of the first die can occur with all six faces of the second die. Thus there are $$6 \times 6 = 36$$ possible pairs or points when two dice are tossed together. These $$36$$ pairs are written as:

\[S = \left\{ {\begin{array}{*{20}{c}} {\left( {1,1} \right)}&{\left( {1,2} \right)}&{\left( {1,3} \right)}&{\left( {1,4} \right)}&{\left( {1,5} \right)}&{\left( {1,6} \right)}\\ {\left( {2,1} \right)}&{\left( {2,2} \right)}&{\left( {2,3} \right)}&{\left( {2,4} \right)}&{\left( {2,5} \right)}&{\left( {2,6} \right)}\\ {\left( {3,1} \right)}&{\left( {3,2} \right)}&{\left( {3,3} \right)}&{\left( {3,4} \right)}&{\left( {3,5} \right)}&{\left( {3,6} \right)}\\ {\left( {4,1} \right)}&{\left( {4,2} \right)}&{\left( {4,3} \right)}&{\left( {4,4} \right)}&{\left( {4,5} \right)}&{\left( {4,6} \right)}\\ {\left( {5,1} \right)}&{\left( {5,2} \right)}&{\left( {5,3} \right)}&{\left( {5,4} \right)}&{\left( {5,5} \right)}&{\left( {5,6} \right)}\\ {\left( {6,1} \right)}&{\left( {6,2} \right)}&{\left( {6,3} \right)}&{\left( {6,4} \right)}&{\left( {6,5} \right)}&{\left( {6,6} \right)} \end{array}} \right\}\]

If $$3$$ dice are thrown, the sample space will have $${6^3} = 216$$ possible points, with each point being a triplet of $$3$$ digits.