Algebra of events talks about the operations which can be carried upon the subsets of the sample space known as events. Events are the outcomes of a random experiment. The algebra of events are:

**1)** $AND$: The intersection of two events is represented by $AND$ operator.

**2)** $OR$: The union of two events is represented by $AND$ operator.

**3)** $NOT$: The complementary event of an event is obtained using $NOT$ operator.

**4)** $BUT\ NOT$: For two events $A$ and $B,\ A\ BUT\ NOT\ B$ represents the events in which are in $A$ but not in $B$.

## Word Problems

Let us see few examples for better understanding of concept.

**Problem 1:**

Two dices are rolled simultaneously. Find the event where sum of both the throws is $8$ and at least one throw is an odd number.

**Solution: **

The events where sum of throws is $8,\ A$ = $\{(2,\ 6),(3,\ 5),(4,\ 4),(5,\ 3),(6,\ 2)\}$

Event that at least one throw is an odd number, $B$ = $\{(1,\ 1)...(1,\ 6),\ (3,\ 1),\ (3,\ 2)...,(3,\ 6),\ (5,\ 1)...(5,\ 6),\ (2,\ 1),\ (2,\ 3),\ (2,\ 5)\}$

**Problem 2:**

Three coins are tossed simultaneously. Find the complement of the event where at least one tail appears.

**Solution: **

The sample space of throwing three coins is, $S$ = $\{(H, H, H),\ (H, H, T),\ (H, T H),\ (T, H, H),\ (T, T, H),\ (H, T, T),\ (T, H, T),\ (T, T, T)\}$

The event where at least one tail appears, $E$ = $\{(H, H, T),\ (H, T H),\ (T, H, H),\ (T, T, H),\ (H, T, T),\ (T, H, T),\ (T, T, T)\}$

The complement of event $E,\ E'$ = $\{(H, H, H)\}$

**Problem 3: **

From a deck of $52$ cards, $2$ cards are chosen. What will be the set of events if the card is ace but not black.

**Solution: **

There are $26$ black and $26$ red cards in a deck out of which $13$ club and $13$ spades are black, and $13$ diamond and $13$ hearts are red.

Event of getting a black card, $A$ = $13$ club and $13$ spade

Event of getting an ace, $B$ = One ace from club, spade, diamond and heart

Event of getting an ace but not black, $A - B$ = $\{diamond\ ace,\ heart\ ace\}$

**Problem 4: **

A game is such that it generates a random number from $1$ to $9$ every time red button is clicked. Find the event that the number is an odd number less than $7$ or an even number greater than $6$.

**Solution: **

The sample space, $S$ = $\{1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10\}$

Event of getting an odd number less than $7,\ A$ = $\{1,\ 3,\ 5\}$

Event of getting an even number greater than $6,\ B$ = $\{8,\ 10\}$

Event of getting an odd number less than $7$ or an even number greater than $6,\ A$ or $B$ = $\{1,\ 3,\ 5\}\ \cup\ \{8,\ 10\}$ = $\{1,\ 3,\ 5,\ 8,\ 10\}$