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 element of the sample space. A sample space may be containing any number of outcomes. If it contains finite number of outcomes, it is called 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. first is defective and second is good
  3. first is good and 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.

Sample space is the basic term in the theory of probability. We shall discuss some sample spaces in this tutorial. It is not always possible to make the sample space. If it contain very large number of points, we cannot register all the outcomes but we must understand as to how we can 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. One is called head and the other is called tail. Anyone of the two faces may be called head. To be brief, head is denoted by H and tail is denoted by T. Thus the sample space consists of head and tail. In set theory notation, we can write S as:

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

Two coins 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 respectively. The sample space S can be written in the form 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 a sample space of 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 which is used in games of chances has six faces. These six faces contain 1,2,3,4,5,6dots 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 thrown:

A die has six faces. Each face of the first die can occur with all the 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 below 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, each point being a triplet of 3 digits.