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

For any two events $A$ and $B$, they can be classified as exclusive, exhaustive, independent and dependent events. Based on the types of events they are, the probability of $A,\ P(A)$ and the probability of $B,\ P(B)$ will have different rules of probability for their addition, subtraction and multiplication.

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

For two events $A$ and $B$, if their probabilities are written as $P(A)$ and $P(B)$ then the rules of addition can be written as,
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

For two events $A$ and $B$, if their probabilities are written as $P(A)$ and $P(B)$ then the rules of addition can be written as,
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

For an event $A$, if the probability of occurrence of $A$ is $P(A)$ then the probability that $A$ will not occur will be,

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

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

Two events are known to be independent events if occurrence of one event will have no effect on the probability of the other one. If $A$ and $B$ are two independent events, and $P(A)$ and $P(B)$ be their probabilities then independent probability rule of multiplication is,

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 = $P(A)\ \times\ P(B)$ = $\frac{1}{13}$ $\times$ $\frac{1}{6}$ = $\frac{1}{78}$