PROBABILITY: AXIOMS AND RULES

INTRODUCTION OF PROBABILITY:

Probability theory is a very fascinating subject which can be studied at different mathematical levels. Probability is the foundation of statistical theory and applications. Perhaps there is no branch of mathematics that is more intimately connected with everyday experiences than the theory of probability.

The theory of probability deals with the averages of mass phenomena occurs in sequentially or simultaneously. The purpose of theory is to describe and predict the averages in terms of probabilities of events.

Probability is a concept which is numerically measures the degree of certainty or uncertainty of occurrence or non-occurrence of events. The study of probability provides a mathematical framework for numerical measurements, it always dealing with the uncertainty of a random experiment.

For Example : 1. THE INSURANCE BUSINESS,2. INDUSTRIAL QUALITY CONTROL,3. ALL GAMES OF CHANCE,4. BIRTH AND DEATH RATES,5. QUEUING THEORY etc..

OBJECTIVES OF PROBABILITY :

One should be able to:

1: Define experiment, outcome, event, probability and equally likely.

2: Restate the formula for finding the probability of an event.

3: Determine the outcomes and probabilities for experiments.

4:Interact with die rolls and spinners to help predict the outcome of experiments.

5:Distinguish between an event and an outcome for an experiment.

6: Recognize the difference between outcomes that are equally likely and not equally likely to occur.

7:Apply probability concepts to complete five interactive exercises.

METHODOLOGIES OF PROBABILITY:

1: The sum of the probability is always unity.

2: The probability of an event always lies between 0 and 1, where’ 0’ is

Lower limit and ‘1’ is upper limit.

3: If P(A)= 0, A is an impossible event.

4: If P(A)= 1, A is sure or certain event.

5: The events A, B are said to be Mutually Exclusive or disjoint

if A∩B=∅

n ( A∩B )=n (∅ )=0

i.e., none of the sample point is common in both the events.

6: The events are said to be “Equally likely”, if there is no reason to expect one is preference to others.

7: The events A,B are said to be “Exhaustive” if A∪B=Ω .

AXIOMS OF PROBABILITY:

Let Ω be the sample space of a random experiment. Probability is a

Real valued set function, it satisfies the following axioms

Axiom 1: P(A)≥ 0 (Non negative axiom).

Axiom 2: P(Ω) =1 (Normality).

Axiom 3: If A1, A2,……….An are mutually exclusive in Ω, then P(A1∪ A2∪……∪An) = P(A1)+P(A2)……P(An).

APPLICATIONS OF PROBABILITY:

There are different applications of probability in different areas.

Probability is used informally in day-to-day life. We daily come across the sentences like:

1. Possibly, it will rain to night.2. There is a high chance of my getting the job next month.

3. This year’s demand for the product is likely to exceed that of the last years.4. The odds are 3:2 in favor of getting the contract applied for.

All the above sentences with words like ‘possibly’, ‘high chance’, ‘likely’ and ‘odds’ are expressions indicating a degree of uncertainty about the happening of the event.

PROBLEM:

In the above problem the probability of opinions of students for the subject maths is totally equal to ‘1’ (where probability of likeness is 0.675 and dislikeness is 0.325).

CONCLUSION: