Probability

The numerical evaluation of degree of uncertainty is called probability

The word probability derives from the Latin probabilitas, 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

1. It consist of more than one outcome
2. The possible outcomes of experiment are known in advance
3. 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]

=  {(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]