A generalisation

Considering an event \(A\) and its complement, \(\overline{A}\). These two events are called a partition of the sample space since they cannot happen together and one of the two events must happen.

This can be generalised to situations where there are several events, in which one must happen, but where more than one is impossible.

Definition

A collection of events, \(A_1, A_2, \dots, A_k\), is called a partition of the sample space if the events are mutually exclusive and their union is the complete sample space,

\[ A_1 \cup A_2 \cup ... \cup A_k = S \]

From the axioms of probability, it can be proved that

\[ P(A_1) + P(A_2) + ... + P(A_k) = 1 \]

The following examples show different partitions of a sample space.

Marriage in Australia

This contingency table describes all 121,690 couples who were married in Australia in 2011.

Previous marital status of couple Both partners
born in Australia
Both partners
born in same
overseas country
Born partners in
different countries
First marriage both partners 51,663 10,522 24,630
First marriage one partner 9,450 2,910 8,050
Remarriage both partners 6,769 1,889 5,807

Consider a randomly selected marriage from 2011. The following three events are a partition of the sample space.

since any marriage must be in one, and only one, of these categories. Similarly, the following three events are another partition of the sample space.

World population by age and region

The table below shows the world population in 2013, categorised by region and by age group.

World population (millions)
  Age
  0-14 15-64 65+
Africa 0,435.6 0,619.5 039.6
Asia 1,071.3 2,917.7 303.8
America, Europe and Oceania 0,357.2 0,1162.3 223.0

Consider randomly selecting one person in the world. The three age groups form a partition of the sample space, as do the three regions of the world.

Proportional Venn diagrams

Proportional Venn diagrams can be generalised to situations in which there are more than two events in each partition of the sample space.

World population by age and region

This proportional Venn diagram below displays the probabilities of randomly selected person in the world being in different age groups, and from different regions of the world.

The diagram initially splits the unit square horizontally using the marginal probabilities for the regions (\(Y\)) — the probabilities of a random person being from each of the three regions. Each row is split according to the conditional probabilities for age group within that region. From the diagram, we can easily see that:

Click on any rectangle in the diagram to observe how its area equals the product of a marginal and conditional probability and therefore is the joint probability for the corresponding categories.

Click the rightmost formula under the diagram. The rectangles change in shape but retain the same areas to make them arranged into vertical columns corresponding to the marginal probabilities for age groups. Each column is split in proportion to the conditional probabilities of region given age group. From this version of the diagram, observe that