Probability is a statistical measure used to express the chances and risks in real life happenings. The study of statistics starts with the probabilities involved in simple experiments like tossing a coin or throwing a die.
Advanced methods involving probability are used in predicting Poll outcomes, future job requirements, testing the validity or effectiveness of new medicines and in production planning.

## Probability Definition

The probability of an event is a number between $0$ to $1$ which measures the chance of that event occurring. The likelihood of the occurrence of the event increases as the probability varies from $0$ to $1$. The probability of the occurrence of an impossible event is $0$ and that of a certain to occur event is $1$.

Probabilities can be expressed as decimals, fractions or percents.

## Probability Formulas

Theoretical Probability of an event:

$P(Event\ A)$ = $\frac{(Number\ of\ outcomes\ in\ event\ A)}{(Total\ number\ of\ outcomes\ in\ the\ sample\ space)}$

Where the outcomes in the sample space are equally likely to occur.

If $n(A)$ and $n(S)$ denote the number of outcomes in the event $A$ and sample space correspondingly, then

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

Experimental Probability of an event:

$P(A)$ = $\frac{Number\ of\ trials\ where\ A\ occurs}{Total\ number\ of\ trials}$.

## Laws of Probability

The general addition law of Probability of two events $A$ and $B$ is

$P(A\ \cup\ B)$ = $P(A) + P(B) - P(A\ \cap\ B)$

If $A$ and $B$ are mutually exclusive events, then the addition law simplifies to

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

Probability of Compliment:

$P(A^{c})$ = $1 - P(A)$

Multiplication Law of Probability:

If $A$ and $B$ are independent events, then

$P(A\ \cap\ B)$ = $P(A)\ .\ P(B)$

## Probability Theory

Probability theory provides a mathematical approach to the study of randomness. Probability Theory is studied using algebraical methods and Calculus Based analysis.

Random variables, discrete and continuous probability distributions and Markov chains are dealt in detail in Probability theory.

## Probability Examples

### Solved Examples

Question 1: In a game played with tiles with alphabets. If 60 of these tiles have consonants and 45 with vowels. Find the probability that a player picks 5 consonants and 2 vowels.
Solution:

7 tiles can be picked from 105 tiles in C(105, 7) ways.

The number of ways picking 5 consonants = C(60, 5)

The number of ways of picking 2 vowels = C(45, 2)

P(Picking 5 consonants and 2 vowels) = $\frac{C(60, 5).C(45, 2)}{C(105, 7)}$ ≈ 0.2376.

Question 2: Events A and B are mutually exclusive and P(A) = 0.11 and P(B) = 0.2.
find P(A or B)
Solution:

P(A or B) = P(A) + P(B)                  Addition law of Probabilities
= 0.11 + 02 = 0.31.

## Probability Practice Problems

### Practice Problems

Question 1: Alex has 8 books, 6 fictions and 2 Biographies. When arranged in a shelf, what is the probability the two Biographies are placed together?     (Ans: $\frac{1}{8}$)
Question 2: A school has 1200 students out of which 525 are females. 500 students took active participation in Sports of which 300 are female. Are the events, "Selecting a student participating in Sports" and "Selecting a female students" mutually exclusive. Justify your answer.