The numerical evaluation of degree of uncertainty is called probability

The wordprobabilityderives from the Latinprobabilitas, which can also mean “probity”

Chances of occurrences lies between two things possible and impossible

## Introduction

- The
**probability**and combinatorics have common origins. - It was first developed when
**gambling games**were analysed more than 300 years ago. - While probability theory has originally been invented to learn gambling, it is now a key Role in a wide range of fields.
**For example**, the theory of probability is widely used**Genetics Study**where the inheritance of characteristics can be understood.- The likelihood of applicability of mathematics is still a very popular element, which is still a very popular human effort.
- The theory of probability plays an important role in studying
**computer science algorithms** - The average case complexity of algorithms is determined by ideas and techniques from
**probability theory**. - Probabilistic algorithms can be used to solve many problems which deterministic algorithms cannot easily or practically solve.
- In a probabilistic algorithm, instead of always taking the same steps as a deterministic algorithm when given the same input, the algorithm makes one or more random choices that can lead to
**different outputs**. - Probability theory can help us answer questions that involve uncertainty

To calculate the probability, note that there are nine possible outcomes, and four of

these possible outcomes produce a blue ball. Hence, the probability that a blue ball is chosen

is 4/9.

## Theory of probability

- Like other theories, the theory of probability is a representation of its concepts in formal terms—that is, in terms that can be considered separately from their meaning.
- These formal terms are manipulated by the rules of mathematics and logic, and any results are interpreted or translated back into the problem domain.
- The scientific study of probability is a modern development of mathematics. [Wikipedia]

**Example:**

An bag contains four blue balls and five red balls. What is the probability that a ball chosen at random from the bag is blue?

**Solution:**

To calculate the probability, note that there are nine possible outcomes, and four of

these possible outcomes produce a blue ball. Hence, the probability that a blue ball is chosen

is 4/9.

## Random experiment

An experiment which produces different results even when it is repeated under similar conditions is called random experiment

**For Example:**

Tossing a coin is a simple random experiment

**Properties:**

The random experiment has following three properties

- It consist of more than one outcome
- The possible outcomes of experiment are known in advance
- The experiment can be repeated many/any number of times

**Sample Space:**

The set containing all possible outcomes of the random experiment is named as sample space

It is denoted by ‘S’ . Each member of the sample space is named as sample point.

## Trail

The single repetition of random experiment is called trail

If trails are more than one then:

[latex] No. of possible outcomes= n( S ) =m^n [/latex]

**m** = Outcomes of single trail

**n** = No. of outcomes

**Examples:**

**Random Experiment= Tossing 3 coins**

**(1) **[latex]Total no. of possible outcomes= n(S) =m^n=2^3=8[/latex]

{ HHH, HHT , HTH , HTT , THH ,THT ,TTH ,TTT }

**(2) **[latex] Total no. of possible outcomes= n(S) =m^n=2^4=16[/latex]

{ HHHH , HHHT , HHTH , HHTT , HTHH , HTHT , HTTH , HTTT ,

THHH , THHT , THTH , THTT , TTHH , TTHT , TTTH , TTTT }

**(3) **[latex] Total no. of possible outcomes= n(S) =m^n=6^1=6[/latex]

S={ 1, 2, 3, 4, 5, 6 }

**(4) **[latex]Total no. of possible outcomes= n(S) =m^n=6^2=36 [/latex]

*S
*=
{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

## Event

The particular output of sample space is called event

**Example:**

**(1)** S={ H , T }

**Possible Events**[latex]= 2^n = 2^2 = 4[/latex]

**Solution:**

**Possible Events**[latex]= 2^n = 2^6 = 64[/latex]