**Introduction to Probability Theory**

**Introduction to Probability Theory**

Uncertainty is every where 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 500 tickets is as likely as any other to be selected or drawn for first prize then it means that you have 10 chances out of 500 tickets or 2% chances to win a first prize.

Similarly, a decision maker seldom have the complete information to make a decision.

So probability is a measure of likelihood that something will happen, however probability cannot predict the number of times that something will occur in future, so it is important that all the known risks involved 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 inherent. 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 the likelihood. Probabilities are usually expressed as fraction 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**

**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 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 “*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 ration of 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 shortcoming of classical approach

- This approach to probability is useful only when one deals with cards games, dice games or coin tosses. i.e. Events are equally likely but not suitable for serious problems such as decisions in managements.
- This approach assumes a world that does not exists, as some assumptions are imposed described above.
- This approach assumes a symmetry about 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 the 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 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 the stable/uniform conditions.
- Experiment can be repeated only a finite number of times in real world, not an infinite number of times.

**3) Subjective 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, 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 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 prior information that the product will have high market acceptance.
- The Sales Manager considers that there is 40% chances of obtaining the order for which the firm has just quoted. This value (40% chances) 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 budget deficit will be reduced by half in the next 5 years.

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

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