Chapter 4

Probability

• Jacob Bernouli (1654-1705), Abraham de Moivre (1667-1754), the Reverend Thomas Bayes (1702-1761) and Joseph Laagrange (1736-1813) developed probability formulas and techniques.

• Probability the was successfully applied at the gambling tables and, more relevant to social and economic problems.

• The mathematical theory of probability is the basis for statistical applications in both social and decision making research.

Probability:

Basic Terminology in Probability

• Two broad categories of decision making problems:• Deterministic Model• Probabilistic (Random) Models.

• Probability is the chance something will happen.• Probabilities are expressed as fractions/decimals between

zero and one.• Assigning a probability of zero means that something can

never happen, a probability of one indicates that something will always happen.

• In probability theory, an event is one or more of the possible outcomes of doing something.

Basic Terminology in Probability

Basic Terminology in Probability

• The activity that produces such an event is referred to in probability theory as an experiment.

• Events are said to be mutually exclusive if one and only one of them can take place at a time.

• When a list of the possible events that can result from an experiment includes every possible outcome, the list said to be collectively exhaustive.

Basic Terminology in Probability

Three Types of Probability

Three Types of Probability

• There are three basic ways of classifying probability. These three represent rather different conceptual approaches to the study of probability theory:

Three Types of Probability

• Classical Approach:• Defines ‘Probability’ as ratio of favorable outcomes to the

total outcomes.• Also known as priori probability.• It assumes a number of assumptions so most restrictive

approach and it is least useful in real life situations.

• Relative Frequency Approach:• Defines ‘Probability’ as observed relative frequency of an

event in a very large number of trials.• It assumes less assumptions but requires the event to be

capable of being repeated a large number of times.

Three Types of Probability

Three Types of Probability

• Subjective Probability:• Deals with specific or unique situations typical of the

business or management world.• Based upon some belief or educated guess of the

decision maker. • Subjective assessments of probability permit the widest

flexibility of the three concepts, also known as Personal Probability.

Three Types of Probability

Probability Rules

• Most managers who use probabilities are concerned with two conditions:1. The case where one event or another will occur.2. The situation where two or more events will both occur.

• A single probability means that only one event can take place, it is called marginal or unconditional probability.

• Addition Rule for Probability:P ( A or B) = P (A) + P(B) – P (A and B)For Mutually Exclusive Events:

P ( A or B) = P (A) + P(B)

Probability Rules

Probabilities Under Conditions of Statistical Independence

• Occurrence of one event has no effect on the probability of the occurrence of any other event.

• There are three types of probabilities under statistical independence

1. Marginal: – Probability of the occurrence of an event

2. Joint :– P (AB) = P(A) X P(B)

3. Conditional:– P(A/B) = P(A) and P(B/A) = P(B)

• In statistical independence, assumption is that events are not related.

Probabilities Under Conditions of Statistical Independence

Probabilities Under Conditions of Statistical Dependence

• When the probability of some event is dependent on or affected by the occurrence of some other event.

• As Independent case there are three types of probabilities under statistical independence

1. Conditional:– P (A/B) = P(AB)/ P(B) and P(B/A) = P(AB)/P(A)

2. Joint:– P(AB) = P(A/B) X P(B) and P(AB) = P(B/A) X P(A)

3. Marginal:– Marginal probabilities under statistical dependence

are computed by summing up the probabilities of all the joint events in which the simple event occurs.

Probabilities Under Conditions of Statistical Dependence

Revising Prior Estimates of probabilities: Bayes’ Theorem

• Bayes’ Theorem expresses how a subjective degree of belief should rationally change to account for evidence.

• Probabilities can be revised as more (additional) information is gained. New probability is known as ‘Posterior Probability’.

• In the Bayesian interpretation, Bayes' theorem is fundamental to Bayesian statistics, and has application in fields including science, engineering, medicine and law.

• In the Bayesian (or epistemological) interpretation, probability measures a degree of belief. Bayes' theorem then links the degree of belief in a proposition before and after accounting for evidence.

Revising Prior Estimates of probabilities: Bayes’ Theorem