Probability Theory

Probability has gained a lot of importance and the mathematical theory of probability has become the basis for statistical applications in the areas of management, space technology and the like.  Various business decisions in real life are made under situations when a decision maker is very uncertain as to what will happen after the decisions are made.  The theory of probability is helpful in all such areas.  In particular it enables a person to make 'educated guesses' on matters where either full facts are not known or there is uncertainty about the outcome. 

 

The probability formulae and techniques were developed by Jacob Bernoulli, De Moiure, Thomas Bayes and Joseph Lagrange.  Later on, Pierre Simon and Laplace unified all these early ideas and compiled the first general 'theory of probability'.

 

Probability Concepts

 

Þ      Experiment - Any operation / process that results in two or more outcomes.

Þ      Random experiment – Any well-defined process of observing a given chance phenomena through a series of trials that are finite or infinite and each of which leads to a single outcome is known as a random experiment.

Þ      Possible outcome – The result of a random experiment.

Þ      Event – One or more possible outcomes of an experiment or a result of a trial or an observation.

Þ      Elementary Event – a single possible outcome of an experiment.

Þ      Compound Event – When two or more events occur in connect with each other.

Þ      Favourable event – the number of outcomes that result in the happening of a particular event.

Þ      Mutually exclusive events – Two events are said to be mutually exclusive or incompatible if the happening of any one of them precludes the happening of all others.

Þ      Dependent events or independent events – Two are more events are said to be independent if the happening of an event is not affected by the supplementary knowledge concerning the occurrence of any number of the remaining events.  The question of dependence or independence of events is relevant when experiments are consecutive and not simultaneous.

Þ      Exhaustive events – the total number of possible outcomes in any trial.

Þ      Equally likely events – Events are said to be equally likely, if taking into consideration all the relevant evidence, there is no reason to expect one in preference to the others.

Þ      Complementary events  - the number of unfavourable cases in an experiment.

 

 

 

THEORIES (OR TYPES) OF PROBABILITY

 

There are four basic ways of classifying probability based on the conceptual approaches to the study of probability theory.  They are:

 

1. Classical approach
2. Relative frequency of occurrence approach
3. Axiomatic approach
4. Subjective approach

 

Classical approach

 

The classical approach is based on the assumption that each event is equally likely to occur.  This is an apriori assumption (the term apriori refers to something that is known by reason alone) and the probability based on this assumption is known as apriori probability.  This approach employs abstract mathematical logic and hence is also called the 'abstract' or 'mathematical' probability.  This is the reason for considerable use of familiar objects like cards, coins, dice, etc., where the answer can be stated in advance before picking a card, tossing a coin or throwing a die respectively. 

 

If a random experiment results in 'N' exhaustive, mutually exclusive, and equally likely outcomes, out of which 'f' are favourable to the happening of an event 'E', then the probability of occurrence of 'E', usually denoted by P(E) and is given by

 

          P = P(E) = f/N =       Number of favourable outcomes

                                      ----------------------------------

                                         Total number of outcomes

 

The classical approach is highly hypothetical in its assumptions.  It assumes situations that are very unlikely but could conceivably happen. The limitation is that it is applicable only when the trials are equally likely or equally probable.  For instance, the probability that a candidate, attending an interview, will succeed is not 50% since the two possible outcomes, viz. success and failure are not equally likely.  Another limitation is that it is applicable only when the exhaustive number of cases in a trial is finite and only when the events are mutually exclusive.  Thus it is useful in card games, dice games, tossing coins, etc., but has serious problems when it is applied to less orderly decision problems that are encountered in the area of management. 

 

Relative Frequency of Occurrence Approach

 

The relative frequency of occurrence approach defines the probabilities as either:

 

v      The proportion of times that an event occurs in the long run when the conditions are stable, or

v      The observed relative frequency of an event in a very large number of trials.

 

In this approach, the probability of happening of an event is calculated knowing how often the event has happened in the past.  In other words, this method uses the relative frequencies of past occurrences as probabilities.  Hence, it is also called an 'empirical' approach to probability theory. 

 

For example, if an organization knows from past data that around 25 out of 300 employees entering each year leave the organization due to better opportunities elsewhere, then the organization can predict the probability of employee turnover as 25/300, i.e. 0.083 or 8.3%.

 

Another characteristic of probabilities established by the relative frequency of occurrence approach can be illustrated by tossing a fair coin 1000 times.  In this case, it is found that the proportion of getting either a head or tail is more initially but as the number of tosses increase, both a head or tail become equally likely and the probability of the event showing a head is 0.5 or the event showing a tail is 0.5.  Thus, accuracy is gained as the experiment is repeated and the number of observations is more.  But the limitation of this approach is the consumption of time and cost for such large repetitions and additional observations.  Moreover, predicting probability using this approach becomes a blunder if the prediction is not based on sufficient data.

 

Axiomatic Approach

 

There is no precise definition available for this approach, but this concept is considered as a set of functions based on the following axioms:-

 

v      The probability of an event ranges from 0 to 1.  That is, an event surely not to happen has probability 0 and another event sure to happen has probability 1.

v      The probability of an entire sample space is 1.  Mathematically P(S)=1.

v      If A and B are mutually exclusive events, then the probability of occurrence of either A or B denoted by P(AUB) is given by P(AUB) = P(A) + P(B).

 

Subjective Approach

 

Subjective probabilities are those assigned to events by the researcher based on past experiences or occurrences or on the basis of evidences available.  It may be an educated guess or intuition.  At higher levels of managerial decision making, where it becomes extremely important, specific and is demanded to be unique, subjective probability would be used.  It is therefore based on the personal beliefs of the person making the decision based on his past information, experiences, observed trends and estimation of futuristic situations.