Approaches and Laws of Probability

Approaches and Laws of Probability

Published by: sadikshya

Published date: 13 Jun 2021

Approaches and Laws of Probability photo

Approaches to Probability

The following are the approaches to probability:

  1. Classical Approach
  2. Empirical Approach
  3. Subjective Approach
  4. Axiomatic Approach

 Classical Approach

If there are ‘n’ mutually exclusive and mutually likely cases and ‘m’ of them are favorable to an event ‘E’. Then, the probability of the happening of an event ‘E’ denoted by probability is given by:

 

Classical Approach

Remarks:

  1. The probability takes a value between 0 and 1.
  2. If ‘E’ is an impossible event then, the probability is 0.
  3. The ‘E’ is sure then P(E)=1.
  4. The sum of the probabilities of occurrence and non-occurrence is 0 and 1

ie. Sum of Probabilities

p + q = 1

Laws (Rules) of Probability

1. Additive Law

If A and B are two events with their respective probabilities P(A) and P(B) then the probability of occurrence Occurance at least one  of these two events given by:

 

Additive Law

where P(A U B) is the probability of the simultaneous occurrence of events A and B.

Remarks:

1. If A & B are mutually exclusive event then, Mutually Exclusive

 

Exclusive

2. If A & B & C are 3 events then the ‘P’ of occurrence of at least one of these events is given by:

 

Events

If A & B & C are mutually exclusive then, the probability is

Exclusive

 

2. Multiplicative Law

If two events A & B are independent then the probability of their simultaneous occurrence is equal to their individual probabilities. If P(A) and P(B) be the probabilities of occurrence of events A & B respectively then the probability of their simultaneous occurrence is given by:

 

Mutiplicative Law

Remarks:

  1. If A, B, & C are three independent events then the probabilities of their simultaneous occurrence is given by:

 

Three independent Events

  1. If A & B are two dependent events then:

 

Two dependent Events

where,

B Given Ais the probability of occurrence of event B given that A has already occurred.

Marginal and Joint Probability

  A B Total
C N1 N2 N1 + N2
D N3 N4 N3 + N4
Total N1 + N2 N2 + N4 N

The probability of each event given by a row or column in the contingency table is called marginal probability.

The marginal probabilities are:

 

P(B)

 

P(C)

P(D)

 

The probability of the joint event whose frequency is given in the cells in the table is called joint probability.

The joint probabilities are:

 

P(A)

 

P(B)

 

P(C)

 

P(D)