Introduction of Probability

INTRODUCTION

The theory of probability had its origin in gambling and games of chance. It owes much to the curiosity of gamblers who prestered their friends in the mathematical world with all sorts of questions. Unfortunately, this association with gambling leads to very slow and sporadic growth of probability theory as a mathematical discipline.

The first attempt at some mathematical rigor is credited to Laplace. Laplace gave the classical definition of the probability of an event that can occur only in a finite number of ways as the proportion of the number of favourable outcomes to the total number of all possible outcomes, provided that all the outcomes are equally likely.

Laplace said "We see that the theory of probability is at bottom only common sense reduced to calculation, it makes us appreciate with exactitude what reasonable minds fell by a sort of instinct, often without being able to account for it... It is remarkable that this science, which originated in the consideration of games of chance, should have become the most important object of human knowledge... The most important questions of life are, for the most part, really only problems of probability".

SOME IMPORTANT TERMS OF PROBABILITY

i. Deterministic experiments

There are experiments, which always produce the same result, i.e., unique outcome or unique events, Such experiments are known as deterministic.

ii. Random experiments

The experiments which do not produce the same result or outcome on every trial are called Random experiments.

Example a. Throw an unbiased die b. If we toss a uniform unbiased coin.

Note: Outcome

In the study of probability, any process of observation is referred to as an experiment. The results of an observation are called the outcomes of the experiment.

iii. Trial and Event

The performance of a random experiment is called a Trial and the outcome is called an event.

Example: Throwing of a coin is a trial and getting H or T is an event.

iv. Sample space

The totality of the possible outcomes of a random experiment is called the sample space of the experiment and it will be denoted by O (Greek alphabet) or S (English alphabet)

Each outcome or element of this sample space is known as the sample point or event and it is denoted by Greek alphabet ?.

The number of sample points in a sample space is generally denoted by n (s)

Example:

a. Tossing a coin S = {H, T}; n(s) = 2

b. Tossing two coins simultaneously

S = {HH, HT, TH, TT}; n(s) = 4

c. Rolling a die S = {1, 2, 3, 4, 5, 6}; n(s) = 6

v. Finite sample space

If the set of all possible outcomes of the experiment is finite, then the associated sample space is a finite sample space.

Example:

a. One dimensional sample space

b. Two dimensional sample space

vi. Countably infinite

A sample space, where the set of all outcomes can be put into a one-to-one correspondence with the natural numbers, is said to be countably infinite.

vii. Countable or a discrete sample space.

If a sample space is either finite or countably infinite, we say that it is a countable or a discrete sample space.

viii. Continuous

If the elements (points) of a sample space constitute a continuem, such as all the points on a line, all the points on a line segment, all the points in a plane, then the sample space is said to be continuous.

ix. Equally likely events

The possibilities or events are said to be equally likely when we have no reason to expect any one rather than the other.

Example

In tossing an unbiased coin, the head or tail are equally likely.

x. Mutually exclusive events [Disjoint events]

Two events A and B are said to be mutually exclusive events or disjoint events provided A ∩ B is the null set.

Note: If A and B are mutually exclusive, then it is not possible for both events to occur on the same trial.

Example: In the throw of a single dice, the events of getting 1, 2, 3, ... 6 are mutually exclusive.

xi. Exhaustive events

Events are said to be exhaustive when they include all possibilities.

Example: In tossing a coin, either the head or tail turns up. There is no other possibility and therefore these are the exhaustive events.

xii. Favourable events

The trials which entail the happening of an event are said to be favourable to the event.

Example: In the tossing of a die, the number of favourable events to the appearance of a multiple of 3 are two (i.e.,) 3 and 6.

xiii.

xiv. Mathematical (or a priori) definition of probability.

If there are n exhaustive, mutually exclusive and equally likely events, probability of the happening of A is defined as the ratio m/n, m is favourable to A.

Thus probability is a concept which measures numerically the degree of certainty or uncertainty of the occurrence of an event.

Notation: p(A) = p = m/n = Favourable number of cases/Exhaustive number of cases

This gives the numerical measure of probability. Clearly, p is a positive number not greater than unity.

So that 0 ≤ p ≤ 1.

Since the number of cases in which the event A will not happen is n - m, the probability q that the event A will not happen is given by

Note: An event A is certain to happen if all the trials are favourable to it and then the probability of its happening is united, while if it is certain not to happen, its probability is zero. Thus if p = 0, then the event is an impossible event while if p = 1, the event is certain.

xv. Permutation

Permutation means selection and arrangement of factors.

Notation: np_r (or) p(n, r) (or) P_n,r (or)  (or) (n)_r.

The value of npr is equal to the number of ways of filling 'r places with 'n' things.

xvi. Permutations with Repetitions

Let p (n ; n₁, n₂, ... n_r) denotes the number of permutations of n objects of which n₁ are alike, n₂ are alike, n_r are alike,

Example: The number of permutations of the word 'RADAR' is 5!/2! 2! = 30. Since there are five letters of which two are 'R' and the two are 'A'.

xvii. Rule of Sum

If an event can occur in m ways and another event can occur in 'n' ways there are m + n ways in which exactly one event can occur.

Example: Consider a box containing 8 red balls and 5 green balls, then are 8 + 5 ways to choose either red ball or a green ball.

xviii. Rule of product

If there are m outcomes for event E, and n possible outcomes for event E₂ then there are mn outcomes for the composite event E₁ E₂.

Example: When a pair of dice is thrown once, the number of possible outcomes are 6 x 6 = 36. Since each dice has 6 possible outcomes.

xix. Combination

Combinations means selection of factors.

Notation: 

Note: 

xx. Statistical or Empirical probability

If in n trials an event E happen m times, then the probability p of the happening of E is given by