Probability Theory: An Introduction (2012)

This post is about probability theory. It will serve as an introduction to the theory of chances.

Probability Theory

Uncertainty is everywhere i.e. nothing in this world is perfect or 100% certain except the Almighty Allah the Creator of the Universe. For example, if someone bought 10 lottery tickets out of 500 and each of the 500 tickets is as likely as any other to be selected or drawn for the first prize then it means that you have 10 chances out of 500 tickets or 2% chances to win the first prize.

Similarly, a decision maker seldom has complete information to make a decision.
So, probability is a measure of the likelihood that something will happen, however, probability cannot predict the number of times that something will occur in the future, so all the known risks involved must be scientifically evaluated. The decisions that affect our daily life, are based upon the likelihood (probability or chance) but not on absolute certainty. The use of probability theory allows the decision maker with only limited information to analyze the risks and minimize the gamble inherently. For example in marketing a new product or accepting an incoming shipment possibly containing defective parts.

Probability can be considered as the quantification of uncertainty or likelihood. Probabilities are usually expressed as fractions such as {1/6, 1/2, 8/9} or as decimals such as {0.167, 0.5, 0.889} and can also be presented as percentages such as {16.7%, 50%, 88.9%}.

Types of Probability

Suppose we want to compute the chances (Note that we are not predicting here, just measuring the chances) that something will occur in the future. For this purpose, we have three types of probability

1) Classical Approach or Prior Approach

In a classical probability approach, two assumptions are used

• Outcomes are mutually exclusive
• Outcomes are equally likely

Classical probability is defined as “The number of outcomes favorable to the occurrence of an event divided by the total number of all possible outcomes”.
OR
An experiment resulting in $n$ equally likely mutually exclusive and collectively exhaustive outcomes and “$m$” of which are favorable to the occurrence of an event A, then the probability of event A is the ratio of $\ \frac {m}{n}$. (D.S. Laplace (1749-1927).

Symbolically we can write $$P(A) = \frac{m}{n} = \frac{number\,\, of\,\, favorable\,\, outcomes}{Total\,\, number\,\, of\,\, outcomes}$$

Some shortcomings of the classical approach

• This approach to probability is useful only when one deals with card games, dice games, or coin tosses. i.e. Events are equally likely but not suitable for serious problems such as decisions in management.
• This approach assumes a world that does not exist, as some assumptions are imposed as described above.
• This approach assumes symmetry about the world but there may be some disorder in a system.

2) Relative Frequency or Empirical Probability or A Posterior Approach

The proportion of times that an event occurs in the long run when conditions are stable. Relative frequency becomes stable as the number of trials becomes large under uniform conditions.
To calculate the relative frequency an experiment is repeated a large number of times say “n” under uniform/stable conditions. So if an event A occurs m times, then the probability of the occurrence of the event A is defined by
$$P(A)=\lim_{x\to\infty}\frac{m}{n}$$

if we say that the probability of a number n child will be a boy is $\frac{1}{2}$, then it means that over a large number of children born 50% of all will be boys.

Some Critics

• It is difficult to ensure that the experiment is repeated under stable/uniform conditions.
• The experiment can be repeated only a finite number of times in the real world, not an infinite number of times.

3) Subjective Probability Approach

This is the probability based on the beliefs of the persons making the probability assessment.
Subjective probability assessments are often found when events occur only once or at most a very few times.
This approach is applicable in business, marketing, and economics for quick decisions without performing any mathematical calculations.
The Disadvantage of subjective probability is that two or more persons facing the same evidence/problem may arrive at different probabilities i.e. for the same problem there may be different decisions.

Real-Life Example of Subjective Probability

• A firm must decide whether or not to market a new type of product. The decision will be based on prior information that the product will have high market acceptance.
• The Sales Manager considers that there is a 40% chance of obtaining the order for which the firm has just quoted. This value (40% chance) cannot be tested by repeated trials.
• Estimating the probability that you will be married before the age of 30 years.
• Estimating the likelihood (probability, chances) that Pakistan’s budget deficit will be reduced by half in the next 5 years.

Note that subjective probability theory is not a repeatable experiment, the relative frequency approach to probability is not applicable, nor can equally likely probabilities be assigned.

Important Terminologies of Probability Theory

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