Probability Theory and Statistics, Schemes and Mind Maps of Law

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Probability & Statistics

Dr. Santosh Kumar Yadav

Department of Mathematics

Lovely Professional University, Phagwara,

Punjab.

Motivation

Statistics is fundamentally based on the theory of probability. There are two types of statistics: Descriptive Statistics and In- ferential Statistics. Descriptive Statistics belongs to the data analysis where the data set size is manageable and can be analysed ana- lytically or graphically.

Inferential Statistics is applied where the entire data set (population) can not be analysed as a whole. So we draw a sample (a small or manageable portion) from the popula- tion. Then we analyse the sample for the characteristic of interest and try to infer the same about the population.

Inferential Statistics is applied where the entire data set (population) can not be analysed as a whole. So we draw a sample (a small or manageable portion) from the popula- tion. Then we analyse the sample for the characteristic of interest and try to infer the same about the population.

For example: when you cook rice, you take out few grains and crush them to see whether the rice is properly cooked. Similarly, survey polls prior to voting in elections, TRP ratings of TV channel, shows etc are samples based and therefore belong to the inferential statistics.

Why first the theory of Probability?

The discipline of probability forms a bridge between the de- scriptive and inferential techniques. The knowledge of probability leads to a better understanding of how inferential procedures are developed and used, how statistical conclusions can be translated into everyday language and interpreted. As probability deals with the uncertainty , thus before un- derstanding what a particular sample can tell us about the population, we should first understand the uncertainty asso- ciated with taking a sample from population. This is why we study theory of probability before statistics.

Definition of Probability

If a random experiment results in n mutually exclusive, and equally likely outcomes, where m are favourable to an event A. Then

P(A) =

m n

number of cases favorable to A total number of possible cases

since m ≥ 0 and n ≥ 0 thus, we always have 0 ≤ P(A) ≤ 1.

Definition of Probability

If a random experiment results in n mutually exclusive, and equally likely outcomes, where m are favourable to an event A. Then

P(A) =

m n

number of cases favorable to A total number of possible cases

since m ≥ 0 and n ≥ 0 thus, we always have 0 ≤ P(A) ≤ 1. The probability of non-happening of event A is denoted by A¯ and is given by

P(A¯) =

n − m n

m n

= 1 − P(A)

Thus, for any event A, we have

P(A) + P(A¯) = 1

Some Basic Terminology

Random experiment: An experiment whose outcome or re- sult is random, that is, is not known before the experiment, is called random experiment.

Some Basic Terminology

Random experiment: An experiment whose outcome or re- sult is random, that is, is not known before the experiment, is called random experiment. Examples: Tossing a fair coin Rolling an unbaised die Drawing a card from a well-shuffled pack of cards.

Sample space:

Some Basic Terminology

Random experiment: An experiment whose outcome or re- sult is random, that is, is not known before the experiment, is called random experiment. Examples: Tossing a fair coin Rolling an unbaised die Drawing a card from a well-shuffled pack of cards.

Sample space: Set of all possible outcomes of a random experiment is called sample space, usually denoted by S. For example, Tossing of a fair coin, S= {H, T} Throwing an unbaised die, S = {1,2,3,4,5,6}

Conti...

Equally Likely Events:

Conti...

Equally Likely Events: Any two elementary events of a sam- ple space are called equally likely if both of them have equal chance of occurrence, i.e., there is no reason to preffered one in comparison to other event. Example: for S = {H, T }, the events {H} and {T } are equally likely.

Conti...

Equally Likely Events: Any two elementary events of a sam- ple space are called equally likely if both of them have equal chance of occurrence, i.e., there is no reason to preffered one in comparison to other event. Example: for S = {H, T }, the events {H} and {T } are equally likely. Mutually Exclusive and Exhastive Events: Two events are said to be mutually exclusive if happening of one event precludes the happening of the other. For exam- ple, the events {H} and {T } in the sample space of the toss of a fair coin are mutually exclusive. The events {H} and {T} in the sample space of the toss of a fair coin are mutually exclusive and exhaustive.

Some Problems

Q1: Find the probability of getting exactly two heads in toss- ing of two fair coins? Sol: P(A) = 1 /4.

Q2: If three fair coin are tossed simultaneously, find the prob- ability that at least two tails occur. Sol: P(A) = 4 /8.