Probability of an event lies between $0$ and $1$. If an event is certain to occur it will have a probability of $1$, and if there is no chance of the occurrence of an event then the probability will be $0$. Depending on various types of events, there are different rules of probability. For example, if $A$ and $B$ are two independent events, then the probability of $A$ and $B$ happening together will be product of their individual probabilities. Again if $A$ and $B$ are mutually exclusive events then the probability of $A$ and $B$ happening together is zero.

## Definition

Given are the assumptions made to get the probability rules:

**1)**The rules are for finite group of events.

**2)**If $A$ and $B$ are events, then $(A\ and\ B),\ (A\ or\ B)$ and $A'$ are also events.

## Rules of Addition

**Rule 1**

**(Mutually Exclusive Events):**

$P(A\ or\ B)$ = $P(A)\ +\ P(B)$

$P(A\ +\ B)$ = $1$

**Rule 2**

**(Non-mutually Exclusive Events):**

$P(A\ or\ B)$ = $P(A)\ +\ P(B)\ -\ P(A\ and\ B)$

## Rules of Multiplication

**Rule 1**

**(Mutually Exclusive Events):**

$P(A\ and\ B)$ = $0$

**Rule 2 (Independent Events):**

$P(A\ and\ B)$ = $P(A) \times\ P(B)$

**Rule 3**

**(Dependent Events):**

$P(A\ and\ B)$ = $P(A) \times\ P$$(\frac{B}{A})$

## Rules of Subtraction

$P'(A) = 1 - P(A)$

If for an event, only two outcomes are possible such as tossing a coin where either head and tail will come then,

$P(B) = 1 - P(A)$, where $A$ and $B$ are two possible outcomes.

**For example:**If in a biased coin the probability of getting a head is 0.8 then what will be the probability of getting a tail?

The probability of getting a tail will be, $1 - 0.8 = 0.2$.

## Independent Probability Rules

Probability of both $A$ and $B$ occurring, $P(A\ and\ B)$ = $P(A)\ \times\ P(B)$

Event of getting a king from the deck of $52$ cards and getting a $6$ if a dice is rolled are independent events. Find the probability of both these events happening together.

Probability of getting a king, $P(A)$ = $\frac{4}{52}$ = $\frac{1}{13}$

Probability of getting a $6$, $P(B)$ = $\frac{1}{6}$

Probability of both the events happening = $P(A)\ \times\ P(B)$ = $\frac{1}{13}$ $\times$ $\frac{1}{6}$ = $\frac{1}{78}$