# Equally Likely Events

When the likelihood of happening of two events are same they are known as equally likely events. If we toss a coin, there are equal chances of getting a head or a tail. Hence, getting a head or a tail by tossing a coin are equally likely events.

If a dice is rolled, then getting an odd number and getting an even number are equally likely events, whereas getting an even number and getting a 1 are not equally likely events.

## Formula

To find the probability of equally likely events, the given formula is used.

$P(E)$ = $\frac{n(E)}{n(S)}$

where, $n(E)$ is total number of favorable events
$n(S)$ is total number of events in sample space

Example:

Find the probability of getting an odd number when a dice is rolled.

Solution:

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

$n(S)$ = $6$

$E$ = $\{1, 3, 5\}$

$n(E)$ = $3$

$P(E)$ = $\frac{3}{6}$ = $\frac{1}{2}$

## Independent Events

Two events are known as independent events if occurrence of one event will not affect the probability of the other to occur. For example, if a toss a coin and roll a dice simultaneously then the probability of getting a head will not be affected by the event of getting any number in the dice. If $A$ and $B$ are independent events, the probability of both the events happening together, $P(A\ and\ B)$ = $P(A)\ \times\ P(B)$
Example:

If a dice is thrown twice, then what is the probability of getting a $2$ and $4$ in first and second throw.

Solution:

Probability of getting a $2$ = $\frac{1}{6}$

Probability of getting a $4$ = $\frac{1}{6}$

Probability of getting a $2$ and $4$ = $\frac{1}{6}$ $\times$ $\frac{1}{6}$ = $\frac{1}{36}$

Hence, the probability of getting $2$ in first throw and a $4$ in second throw is $\frac{1}{36}$

## Exhaustive Events

Two events $A$ and $B$ are known as exhaustive events if the union of $A$ and $B$ gives the sample space. For example, if we have deck of $52$ cards then the event of getting a black card and the event of getting a red card together will give the total number of cards. So, both these events are exhaustive events.
Example:

If a die is rolled, then the event of getting a prime number and an odd number are exhaustive events or not?

Solution:

Event of getting a prime number = $\{1, 2, 3, 5\}$

Event of getting an odd number = $\{1, 3, 5\}$

Union of both events = $\{1, 2, 3, 5\}$

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

Both events together do not equal to the sample space. Hence, these events are not exhaustive.

## Mutually Exclusive Events

Two events $A$ and $B$ are known as mutually exclusive if they cannot happen simultaneously. They are also known as disjoint events. The probability of $A$ or$B$ happening will be, $P(A\ or\ B)$ = $P(A)\ +\ P(B)$
Example:

The probability of getting a king from deck of $52$ cards in $0.6$ and the probability of getting a queen is $0.2$. Find the probability that if a card is picked randomly, a king or  a queen will come.

Solution:

Probability of getting king = $0.6$

Probability of getting a queen = $0.2$

Both are mutually exclusive events. Hence, the probability of getting a king or a queen = $0.6\ +\ 0.2$ = $0.8$.