Definitions of Probability

Definitions of Probability

Probability is defined by four different approaches:

1. Mathematical (Classical or prior) definition
2. Statistical (Relative frequency or Empirical) definition
3. Subjective definition
4. Axiomatic (Set-theoretic Modern) definition

Mathematical ( Classical or Prior) Definition of Probability

This approach to probability was first defined by JamesBernoulli to obtain a quantitative measure of uncertainty of an event.

Definition:

If random experiment results in an exhaustive,equally likely and mutually exclusive outcomes,out of which m case are favourable to an event A,then probability of happening of the event A,denoted by (A),is defined as the ratio \(\frac{m}{n}\) i.e.,

$$P(A)=\frac{Number of case favourable to A}{Total Number of all possible (exhaustive) case}=\frac{m}{n}$$

The probability of happening of an event A;P(A) is also known as the probability of success and it is usually denoted by P.

Since both m and n are non-negative integers i.e. m\(\geqslant\)0 and n\(\geqslant\)0,we have P(A)\(\geqslant\)0.Again since m\(\leqslant\)n,we have P(A)\(\leqslant\)1.Hence, we get

0\(\leqslant\)P(A)\(\leqslant\)1

This means, the probability of happening of any event is a number lying between 0 and 1.This is a special perporty of probability.

If P(A)=0,then the event A is said to be an impossible event or null event.If P(A)=1,then the event A is called a certain or sure event. If P(A) lies between 0 and 1 i.e. if 0 \(\leqslant\) P(A) \(\leqslant\) 1, then the event A is called an uncertain event.If P(A)=0.5,then it indicates that the event A has 50:50 chance of occurring and non-occurring.

Hence, the probability is a numerical measure of the likelihood of an event occurring.

Merits and Demerits of Classical Definition of Probability

Merits:

• The classical definition of probability provides a numerical value of the likelihood of occurrence of an event and someone can make a decision easily on the basis of the probability value.
• The classical probability can be calculated without obtaining any experimental data,

Demerits:

• If the exhaustive events n of the random experiment is infinite.
• If the outcomes of the experiment are not equally likely.For example,if a person jumps from the top of a storied building,then events of his survival and death,through exhaustive and mutually exclusive,are not equally likely.
• If the actual value of n is not known.For example,if the total number of balls contained in an urn is unknown,no one can compute the classical probability of drawing a green ball or red ball or any ball from the urn.

Statistical (Relative Frequency or Empirical) Definition of Probability

The statistical definition of probability was given by Richard Von Mises (1931).

Definition:

If an event A happens 'm' times in 'n' repetitions of a random experiment,performed under a set of identical conditions,then the probability of happening of the event A is given by

$$P(A)=\lim_{n\to \infty}\frac{m}{n}$$

The ratio or relative frequency \(\frac{m}{n}\) approaches to a constant as the number of trials n becomes sufficiently large i.e. as \(n \rightarrow\infty\). The statistical definition of probability is based on experimental data and is a result of the law of statistical regularity.The statistical law of regularity states that if a random experiment is repeated a large number of times under the identical conditions,average result of a long sequence of the experiment exhibits a certain visible regularly.So, this definition is the modification of the mathematical(classical) definition of probability.

Merits and Demerits of Statistical Definition

Merits:

• To find the limiting value, a large number of experiments has to perform under identical conditions.
• The showing of a statistical regularity is very important and greatest merit of the statistical definition of probability.

Demerits:

• The experimental conditions may not remain same or identical when an experiment is conducted a large number of times.
• The relative frequency \(\frac{m}{n}\) may not tend to a unique value even if \(n\rightarrow\infty\).
• The empirical probability provides only a close estimate of the probability of an event.

Subjective Definition of Probability

Subjective definition of probability was given by Frank Ramsey in his book 'The Foundation of Mathematics and Othe Logical Essays " in 1926.This approach was further developed by Bernard Koopman,Leonard Savage, and Richard Good.

Subjective probability is defined as the probability assigned to the occurrence of an event by personal judgment, belief, intuition, and expertise.This probability assigned by top level of authorities on the basis of their discretion.For example,a member of National Planning Commission (NPC) says that the chance of surviving any hydropower project in Nepal is 90%.In this statement ,the probability of surviving any hydropower project in Nepal is assigned by personal belief and wisdom of the member of the NPC.Such probabilities assigned by a different person to the same event may vary from person to person.

Subjective definition of probability is applied in situations in which the above two definitions namely mathematical and statistical definition cannot be applied.This definition is useful to explain the concept of probability for a Layman.However,it is insufficient to solve the various problems of our everyday life.

Axiomatic ( Set Theoretic or Modern) Definition of Probability

Axiomatic definition of probability is the modern approach to probability and this was originated by Von Mises and A.N. Kolmogorov in the book "Foundations of Probability" published in 1933.

Definition:

Let A be an event on a finite sample space S associated with a random experiment.Then the probability of happening of the event A is defined as a real-valued function or set function P(A) which satisfies the following three axioms:

1. 0\(\leqslant\)P(A)\(\leqslant\)1
2. P(S)=1
3. If A₁,A₂,..........A_n are n mutually exclusive(or disjoint) events in S, then the probability that at least one these events occur is the sum of their respective probabilities, i.e.

P(A₁\(\cup\)A₂\(\cup\)...........\(\cup\)A_n)=P(A₁)+P(A₂).........+P(A_n)

or. P(\(\bigcup_{i=1}^{n}\)A_i)=\(\sum_{i=1}^{n}\)P(A_i)

The Axiom (1) is an axiom of non-negativity, (2) is an axiom of certainty and (3) is an axiom of additivity or union.The set function P(A) is,in fact, a probability function and is called a probability measure.A value of the probability function P(A) is called a probability of the event.The sample space S along with the probability measure i.e. the pair {S,P(A)} is called the probability space.The nature of the probability space depends on upon the nature of the sample space.

In general,If S={e₁,e₂,........e_n} is a sample space consisting of n equally likely events (or sample points)e₁,e₂,........e_n and if A={e₁,e₂,........e_m} is an event containing m sample points of S, then as probability of each event is P(e_i)=\(\frac{1}{n}\); (i=1,2,..........n) the probability of the event A is given by the sum of the probabilities of the m events.Thus,

P(A)=P(e₁)+P(e₂)+................+P(e_m)

=\(\frac{1}{n}\)+\(\frac{1}{n}\)+.....................m times =\(\frac{m}{n}\)

=\(\frac{Number of sample points in A}{Total Number of sample points in S}\)=\(\frac{n(A)}{n(S)}\)

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