Algebra of Events

The algebra of events is the set of rules and regulations that shows how the event has occurred. These operations on the events show us how the values are related to each other. Fundamental operations are union, intersection, and complement. These operations can help analysts to predict the likelihood of an event.

Algebra of Events

The set of outcomes from an experiment is known as an Event.

When a die is thrown, sample space $S = \{1, 2, 3, 4, 5, 6\}.$

Let $A = \{2, 3, 5\},$ $B = \{1, 3, 5\},$ $C = \{2, 4, 6\} $

Here, $A$ is the event of the occurrence of prime numbers, $B$ is the event of the occurrence of odd numbers and $C$ is the event of the occurrence of even numbers.

Also, observe that $A, B,$ and $C$ are subsets of $S$.

Now, What is the Occurrence of an event?

From the above example, the experiment of throwing a die. Let $E$ denote the event “ a number less than $4$ appears”. If any of $' 1’$ or $ '2'$ or $'3'$ had appeared on the die then we say that event $E$ has occurred.

Thus, the event $E$ of a sample space $S$ is said to have occurred if the outcome $ω$ of the experiment is such that $ω ∈ E$. If the outcome $ω$ is such that $ω ∉ E$, we say that the event $E$ has not occurred.

Complimentary Event

For every event $A$, there corresponds to another event $A$' which contains all outcomes in sample space that are not covered in $A$. Such event is called the complementary event to $A$. It is also called the event ‘not $A$’.

For example, take the experiment ‘of tossing two coins’. The sample space is

$S = \{HH, HT, TH, TT\}$

Let $A=\{HT, TH\}$ be the event ‘only one tail appears’

Thus the complementary event ‘not $ A$’ to the event $A$ is

$A’ = \{HH, TT\}$

or $ A’ = \{ω : ω ∈ S$ and $ω ∉A\} = S - A$

The Event $'A$ or $B’$

As we have studied in the first chapter, ‘Sets’, the union of two sets $A$ and $B$ denoted by $A ∪ B$ contains all those elements which are either in $A$ or in $B$ or in both.

When the sets $A$ and $B$ are two events associated with a sample space, then $A ∪ B$ is the event ‘either $A$ or $B$ or both’. This event $A ∪ B$ is also called $'A$ or $B’.$

Therefore

Event
$
\begin{aligned}
\mathbf{A} \text { or } \mathbf{B} & =\mathbf{A} \cup \mathbf{B} \\
& =\{\omega: \omega \in \mathbf{A} \text { or } \omega \in \mathbf{B}\}
\end{aligned}
$

The Event $'A$ and $B’$

The intersection of two sets $A ∩ B$ is the set of those elements which are common to both $A$ and $B$. i.e., which belong to both $'A$ and $B’.$

If $A$ and $B$ are two events, then the set $A ∩ B$ denotes the event $'A$ and $B’.$

The Event $'A$ but not $B’$

The $A – B$ is the set of all those elements which are in $A$ but not in $B$. Therefore, the set $A – B$ may denote the event $'A$ but not $B’.$

Also, $A - B = A ∩ B´$ or $A - ( A ∩ B).$

Equally Likely Events

Equally likely means that each outcome of an experiment occurs with equal probability. For example, if you toss a fair, six-sided die, each face $(1, 2, 3, 4, 5, or 6)$ is as likely to occur as any other face. If you toss a fair coin, a Head $(H)$ and a Tail $(T)$ are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.

Exhaustive events

Consider the experiment of rolling a die. The associated sample space is

$S = \{1, 2, 3, 4, 5, 6\}$

Let us define the following events

$ A:$ ‘a prime number number less than $6$ appears’,

$B:$ ‘a number less than $2$ appears’

and $C:$ ‘a number greater than $3$ appears’.

Then $A = \{2, 3, 5\}, B = \{1\}$ and $C = \{4, 5, 6\}.$

Observe that, $A ∪ B ∪ C = \{2, 3, 5\} ∪ \{1\} ∪ \{4, 5, 6\} = \{1, 2, 3, 4, 5, 6\} = S$

So if union of given events equals sample space, then these events are called a system of exhaustive events. Thus events A, B and C are called exhaustive events in this case.

In general, $E_1, E_2, \ldots, E_n$ are $n$ events of a sample space $S$ and if

$\mathrm{E}_1 \cup \mathrm{E}_2 \cup \mathrm{E}_3 \cup \ldots \cup \mathrm{E}_n=\bigcup_{i=1}^n \mathrm{E}_i=\mathrm{S}$

then $E_1, E_2, \ldots, E_n$ are called exhaustive events.

Mutually exclusive events

Consider the experiment of rolling a die. The associated sample space is

$ S = \{1, 2, 3, 4, 5, 6\}$

Consider events, $A$ ‘an odd number appears’ and $B$ ‘an even number appears’

$A = \{1, 3, 5\}$ and $B = \{2, 4, 6\}$

Clearly $A ∩ B = φ$, i.e., $A$ and $B$ are disjoint sets.

In general, two events $A$ and $B$ are called mutually exclusive events if the occurrence of any one of them excludes the occurrence of the other event, i.e., if they can not occur simultaneously. In this case the sets $A$ and $B$ are disjoint.

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