# Algebra of Events

Events are the outcomes of an experiment. The probability of an event to occur is the ratio of number of favorable events to total number of events. For two events happening, sometimes they can happen together or it may happen that only one of them will happen. Algebra of events will give an event where some operations are performed over two given events. The operations are union, intersection, complement and difference of two events. As events are the subset of sample space, these operations are performed as set operations.

## Complementary Events

For an event $A$, there is a complimentary event $B$ such that $B$ represent the set of events which are not in the set $A$. For example, if two coins are tossed together then the sample space will be $\{HT, TH, HH, TT \}$. Let $A$ be the event of getting one head, then the set $A$ = $\{HT, TH \}$. The complementary events of $A,\ B$ = $\{HH, TT \}$.

## Events with AND

AND stands for the intersection of two sets. An event is the intersection of two events if it has got the members present in both the event. For example, if a pair of dice is rolled then the sample space will have $36$ members. Suppose $A$ is the event of getting both dice having same members and $B$ is the event having the sum as $6$.

$A$ = $\{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)\}$

$B$ = $\{(3, 3), (1, 5), (5, 1), (2, 4), (4, 2)\}$

$A$ AND $B$ = $\{(3, 3)\}$

## Events with OR

OR stands for union of two sets. An event is called union of two events if it has got members present in either of the sets. For example, if two coins are tossed together the sample space, $S$ = $\{HT, TH, TT, HH\}$. Let event $A$ be the event having only one head and event $B$ be the event having two heads.

$A$ = $\{HT\}$

$B$ = $\{HH\}$

Union of $A$ and $B$, $A$ OR $B$ = $\{HT, HH\}$

## Events with BUT NOT

For two events $A$ and $B$, $A$ but not $B$ is the event having all the elements of $A$ but excluding the elements of $B$. This can also be represented as $A$ - $B$. Suppose, there is an experiment of choosing $4$ cards from a deck of $52$ cards. The event $A$ is having all cards as red cards and event $B$ is having all cards as king. Then the event $A$ but not $B$ will have all red cards excluding the two kings.