Venn diagrams

Venn diagrams allow visualization of union, intersection, and complement probabilities. They can also be used calculate simple conditional probabilities.

On Venn diagrams, probabilities are the relative sizes of different regions. However they are just for illustrative purposes and are not to scale.

Contents

• Probability review
• Components
• Intersection and union
• Inclusion-exclusion principle
• Special cases of intersection
  • independence
  • mutual exclusivity
• Conditional probability
  • special cases

Probability review

For a countable number of events, probability is

This is highly based on the assumption of all items are equally likely. Nevertheless it is useful to think of probability as a fraction.

A closely related formula says

An event is an outcome of the measurement or trial. The probability of event $A$ happening is $\text{P}(A)$. Events are typically labelled using capital letters such as $A$ and $B$

Components

A venn diagram consists of

• a rectangle labelled $U$ for the sample space (universe) of a single measurement or trial, such that $\text{P}(U) = 1$
• one or more circles representing events.

Event $A$ has a $\text{P}(A)$ chance of occurring, and $1 - \text{P}(A)$ chance of not occurring. The latter is called the complement probability and is denoted as

In addition, any region on the venn diagram can be considered an event; it does not have to be circular.

Intersection and union

In intersection, both $A$ and $B$ occur, and it is represented by the overlap of the two circles, and denoted by $A \cap B$ and is equivalent to $B \cap A$. The intersection characterizes part of the relationship between $A$ and $B$. In absence of additional information, we cannot deduce what $\text{P}(A \cap B)$ is, other than it’s no greater than the minimum of $A$ and $B$.

One very useful equation relating intersection and complements is

which states that when $A$ occurs, $B$ either occurs or does not occur.

In union, at least one of $A$ and $B$ occur, and it is represented by the two circles including their overlap, and denoted by $A \cup B$ and is equivalent to $B \cup A$.

Inclusion-exclusion principle

For two events, we have

In general

where the signs alternate with increasing number of events in the intersection, and all coefficients are $1$.

On exams, only inclusion-exclusion of two events will be required, though in certain situations when you fail see the faster method, you may opt to use inclusion-exclusion if it applies.

Special cases of intersection

independence

Two events are independent if and only if

Another way to view them is that they are unrelated and uncorrelated. The probability is unchanged whether or not the other event happens. For example, day of the week is independent with weather of the day.

mutual exclusivity

Two events are mutually exclusive if and only if

In other words, both events cannot occur at the same time. They are often cases or different scenarios. For example, watching sports is mutually exclusive with sleeping.

Conditional probability

On venn diagrams, conditional probabilities are viewed as ratios of areas or ratios of probabilities. In a way, it is about narrowing the sample space as information becomes available.

For events that are not independent, event $A$ impacts the chances of $B$ occurring, and vice versa.

The conditional probability $\text{P}(B \,\vert\, A)$ of $B$ given $A$ is calculated by

and should be seen as the fraction of ${\color{purple} A \cap B}$ relative to the circle of $\color{blue} A$.

From this and

we obtain

and

Combining both forms lead to Bayes’ theorem, as explored in Tree diagrams.

special cases

Independent events $A$, $B$ satisfy

Mutual exclusive events $A$, $B$ satisfy