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Probability - Rules, Types, Application, Uses, Characteristics

Probability

What is Probability ?


Computation of the chance of an occurrence of a certain event is called probability. Hence, probability is the study of random happening of an event and uncertainty, e.g. "Probability of rain tomorrow is only 35%", and "Probability of winning a lottery is 1 out of 40,406,353".
Before conducting an experiment, the absolute certainty of the outcome is unpredictable, therefore it is termed as probability, which actually depends on the idea of a random experiment'.
The theory of probability originates from the 'game of chance' associated with gambling, e.g., consider a game like gambling, tossing of a coin, throwing of a die, and random drawing of cards from a pack of cards. Thus, in statistics, probability (of an event) is defined as the 'proportion of time' in which an event occurs and is observed, even if the experiment is run an infinite number of times.

Contents :                                  

  1. Meaning and Definition of Probability.
  2. General Rule for Probability.
  3. Application of Probability.
  4. Characteristics of Probability Function P(A).
  5. Uses of Probability in Decision Making.
  6. Types of Probability : Objective Probability & Subjective Probability Theory.

Meaning and Definition of Probability :


Probability is the mode of expression of knowledge or belief whether the event has occured or will occur. Thus, in the theory of probability this concept has been given an exact mathematical significance for being used widely in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy, so that conclusions can be drawn regarding the probability of potential events and the mechanics of complex systems being hidden.

According to Levin :
"We live in a world in which we are unable to forecast the future with complete certainty. Our need to cope with uncertainty leads us to the study and use of probability theory".

According to American Heritage Dictionary :
"Probability is the branch of mathematics that studies the likelihood of occurrence of random events in order to predict the behavior of a defined system".

General Rule for Probability :


The probability of an event (say A) is defined as follows :

If an event happens m times and it fails n times, then the probability (p) of event A is defined as :
              m
P(A) = --------
            m+n

        The number of cases favorable to
         the occurence of the event
P(A) = ------------------------------------------------
        Total number of mutually exclusive
         and exhaustive cases

So the total number of trials favorable to the event (A) divided by the total number of ways in which the event (A) can happen.

         The number of cases favorable to 
           the occurence of the event
P(A) = -------------------------------------------------
       Total number of mutually exclusive
         and exhaustive cases

Probability in favor of the occurrence of the event :

        The number of cases favorable to
m      the occurence of the event
--- = ---------------------------------------------------
n     The number of cases against
         the occurrence of the event

Probability against the occurrence of the event :
        The number of cases against 
n      the occurence of the event
--- = ------------------------------------------------
m     The number of cases favorable to
         the occurrence of the event

Applications of Probability :


  1. It plays a great role in making predictions.
  2. It is used to solve scientific investigations as well as day-to-day problems.
  3. It is very useful when one wants to predict the uncertainties of betting.and chances of success.
  4. It is extensively used in business and economics.
  5. It is the basis of the law of statistical regularity and law of inertia of a large number.

Characteristics of Probability Function P(A) :


  1. The limit of probability of an event A is between 0 and 1, i.e. 0<P(A)<1.
  2. When probability of two events A and B which are mutually exclusive, is combined, the probability is the sum of individual probabilities of A and B, i.e., P(AOB) will be P (A) + P (B).
  3. Two events which are equivalent are assigned the same probability.

Uses of Probability in Decision-Making :


Following are the business situations where managers use probability to take a decision :

1) Investment Problem : 
A company's manager has two different projects (with different initial costs) for investment. The decision taken by the manager is based on the choice that the outcome of which is dependent on the level of demand.

2) Introducing a New Product : 
While introducing a new product there arises a problem in deciding about the introduction of the product. The decision maker is not sure whether the product would be acceptable or not. Hence after conducting test, marketing in three different regions the results being contradictory, the idea of introducing a new product should be dropped ? Thus, it is necessary to answer this question : what would be the chances of success of introducing a new product ?.

3) Stocking Decisions :
The demand of a perishable commodity is unknown to its dealer in advance. Thus if the commodity is not sold by the end of the day it will be spoiled. Therefore, he is not sure about the demand pattern and stock amount (that how much commodity is to be stored in advance).

4) The Individual Investor : 
Maximisatiorn of returns is the motive of investors who are engaged in buying and selling of equities. The behavior of equities/ security prices is uncertain and it depends upon various factors. In this situation, managers take decisions on the basis of their prediction about future prices of the securities. This prediction helps the investor to take a decision as to which.securities he/she should select for the investment.

PROBABILITY THEORY


Introduction :

The branch of mathematics that is concerned with random phenomena analysis is known as probability theory. The occurrence of a random event cannot be determined and it can be any single outcome from various possible outcomes. The actual outcome can be determined only by chance.

Types of Probability :


There are different definitions of probability which are shown below :

Objective Probability :

The probability of the occurrence of an event is entirely based on the analysis where each measure depends on the documented observation, in place of a subjective estimate. A more accurate way of determining probabilities is by way of objective probabilities in contrast to the observations based on subjective measures, like personal estimates.

For example :
The objective probability is determined for tossing a coin 100 times and recording each observation that the outcome of the coin after landing will be "beads". In statistical analysis, it is very important that each observation must be an independent event without manipulations. The lesser the observation being biased, the lesser will be the end probability.

Objective probabilities are of two types :
  • Classical Probability.
  • Empirical Probability.

Classical Probability :

Classical Probability is the first approach to the theory of probability.

According to Laplace : 
"Probability is the ratio of the number of favorable. cases to the total number of equally likely cases".

The fundamental assumption to this theory is that the various outcomes of an event are 'equally likely'. Thus the probability of happening of these events is also equal. In this theory, the probability of happening of an event is determined prior to the happening of the event. Therefore these probabilities are known as prior probability.
The classical theory is suitable for determination of probability of those events that are possible though games of chance, where various outcomes are equally likely to happen. In the tossing of a dice the happening of the event can be in six possible ways. Thus the probability of success of an event is described as p and the probability of the failure of an event is termed as q where there is no third event.
          Number of favorable cases
P = -------------------------------------------------------
     Total number of equally likely cases

Let there be x ways for the occurrence of an event and y ways for its failure and they are equally likely to occure, then the probability of the happening of the event
  x
------- is donated by p.
 x+y
These probabilities are also known as unitary, theoretical or mathematical probability. p is the probability of the occurrence of the event and q is the probability of its not occurrence.
         x
p = ------- and
       x+y
         y
q = -------
       x+y
So,
              x           y         x+y
p+q = ------- + ------ = ------- = 1
          (x+y)    (x+y)      x+y

Therefore, p+q = 1, or 1-q= p, or 1-p= q.

The probability of occurrence of an event E is the ratio between the number of cases in its favour and the local number of cases (equally likely).
Therefore,
            n(E)
P(E) = --------- =
            n(S)
Number of cases favourable to event E
---------------------------------------------------------------
Total number of cases

If x is the number of cases favourable to the event E', y is the number of cases favourable to the event E, the odds in favour of E are x:y and odds against of E are y:x.
In this case,
              x
P(E) = ------- , 
            x+y
              y
P(E') = -------
            x+y
Therefore, P(E)+ P(E') = 1

Therefore, probability of happening + probability of non-happening = 1 and 0<P(E)<1, thus, maximum value of P(E) = 1 and the minimum value of P(E) = 0.
Probabilities are expressed either as ratio, fraction or percentage such as :
 1
---- or 0.5 or 50%.
 2 

They are presumed to be equally likely and the probability of getting a 6 on a single throw of a dice would be 1/6.

Limitations of Classical Approach :

  • The limitation of this definition is that, it is restricted only to the games of chance and the problems other than the game of chance cannot be explained.
  • This method is not applicable when it is not possible to calculate the total number of cases.
  • This method is not applicable in the case where outcomes of a random experiment are not equally likely.
  • The further subdivision of the possible outcome of experiments into mutually exclusive, exhaustive and equally likely is almost difficult.

Relative Frequency Approach/ Empirical Probability :

The scientific study of measuring uncertainty is known as 'probability'. Probabilities are empirically determined when their numerical values are based upon a sample or a census of data which gives a distribution of events.

According to Von Mises :
If an experiment is performed repeatedly under essentially homogeneous and identical conditions, then the limiting value of the ratio of the number of times the event occurs to the number of trials, as the number of trials becomes indefinitely large, is called the probability of happening of the event, it is being assumed that the limit is finite and unique.

Symbolically, if A is the name of an event, f is the frequency with which that event occurred, and n is the sample size, then :
             f
P(A) = -----
             n
'Priori probabilities' are not the basis of the relative frequency approach to the theory of probability. There are various situations where it is impossible to have equally likely events which form the basis of the classical theory of probability. Thus, the probability of the happening of an event is derived from the past experiences or from the relative frequencies of success in the past. f the output of a machine is 10% of unacceptable articles in the past then the relative frequency for those unacceptable articles would be 10% of the total items. Thus, the estimation of the relative frequency should be on the basis of extensive readings in the past.

The accuracy of the result is directly proportional to the large number of past readings. The probabilities that are calculated on the basis of past experience are called posterior probabilities, and have relative frequency approach. This is in contrast to the priori probabilities that are calculated through classical approach.

The priori probabilities are applied in the calculation of the games of chance whereas posterior probabilities are applied in the problems of economic and social phenomena where priori probabilities are not constant. The nature of the priori probabilities is deductible and is theory-based rather than being based on evidence experience and experimentation. Posterior probabilities which are also known as empirical probabilities, and are based on the past experience as well as on conducted experiments.

Thus the posterior probability or empirical probability (P) of an event is given by,
       No.of times an event is observed
P = ---------------------------------------------------------
      No.of trails experiment is conducted

If out of 1000 items produced by a machine in past, 60 were found to be defective, the probability of a detective article to be produced by this machine would be :
    60
----------- or 0.06.
  1000

Limitations of Relative Frequently Theory or Probability

  • In conducting large number of trials of the experiment there may not exist given homogeneous and identical conditions of the experiment.
  • No matter how large is the value of n, the relative frequency m/n, will not attain a unique value.
  • The defined probability (P) can never be obtained practically. An attempt can be made at a nearby estimate of P by adequately increasing n.

Subjective Approach to Probability :

The classical and empirical approach of probability is objective in description. Whereas the subjective access to probability of an event is considered to be the scope of one's certainty of a particular event to occur. Hence, if one considers an event that the student 'A' will pass the examination, then its probability cannot be estimated by either of the objective approaches that have been discussed above.

The passing and failing of a student are the events which are not equally likely cases and if these cases were equally likely, then by applying the classical approach of probability we could have derived the probability to be 1/2. Thus, in this case the experiment cannot be repeated under uniform conditions.

With the help of empirical approach it is not possible to comment upon the probability, of this event. Hence comparatively, subjective approach is useful in such cases because in this approach the probability of an event is the representation bf the degree of faith and belief of a rational person in the occurrence of that particular event that depends upon his judgment, personal point of view etc.

Considering this approach, the probability of an event is different for each person. Due to this it is known as subjective probability where the probability of an event is affected by the personal bias of its estimator. Therefore, it is also known as Personalised Theory of Probability on the presumption that any decision is the reflection of the personality of its decision-maker along with those important subjective elements that assign a probability to an event. Let us assume that we have data relating to the price of a share for the last 3 years and further supposing that out of 2000 quotations relating to this share price, there was price rise on 500 occasions, and then the empirical probability of a price rise in this share is :
 500
----------  or 0.25.
 2000

According to this data, the buying and selling of the shares should be done by some people separately due to the reason that the two approaches, i.e., relative frequency and subjective estimates are joined together, yielding out different ideas in their minds relating to price of these shares in future.
The final decisions are the reflection of the personality of the decision maker. Thus the decision taken under this theory is based on the available data in addition to the effects of those factors that are subjective in nature.

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