Probability (18/07/2013)

The experience in dealing with practical data during the last session and the analytical use it could be put to for making managerial decisions left us wanting for more and the 5^th and 6^th session did not disappoint.

The class was introduced to the concept of ‘Probability’. Most of us have been exposed to this concept during our school days and to be honest I was a little confused about how a topic discussed purely as an arithmetic concept during our school days, could make sense for managers. However, as the class progressed I realized how wrong I was. The professor illustrated how Probability was much more than mere estimation of outcome of events that had no practical implications.

To begin with the professor introduced us to three methods of Estimation which are as follows -:

1             1)     Ratio Change ( varies from 0 to 1  eg.1/10)

2             2)     Percentage Change ( varies from  0 to 100  e.g. 50% )

3             3)     Probability ( varies from 0 to 1 e.g. 0.1)

RATIO CHANCE - A ratio represents, for every amount of one thing, how much there is of another thing.

Example 3/7

PERCENTAGE CHANGE -  In percentage, it’s all about "x" out of 100. Percentage is reported as a ratio but the sum adds up to 100.

Example 40% (40 out of 100)

PROBABILITY – Probability is the chance that something will happen - how likely it is that some event will happen. It is the measure of uncertainty of an event.

1 >=  P(E) >=  0

Sometimes you can measure a probability with a number: "10% chance of rain", or you can use words such as impossible, unlikely, possible, even chance, likely and certain.

Example: "It is unlikely to rain tomorrow".

3 Approaches to Calculating Probability –

1             1)      A Priori ( Prior Information )

2             2)      Empirical ( Information Collected )

3             3)      Subjective ( Judgemental )

A PRIORI – A priori approach to estimating probability is based on the idea that the probability of certain events/business decisions comes from reason alone and not from experience. In other words no prior information is required in such case.

An example for this approach in business context could be that a company creates three budgets, which it designates as worst case, most likely, and best case. An a priori conclusion would be that the business is equally likely to experience all three scenarios, which means that you would assign a 33% probability to each one. 

EMPIRICAL - Empirical probability estimates probabilities from experience and observation. It involves estimation on the basis of information collected to create a possible range within which the value of probability may lie.

An example for empirical method would be the study of capital asset pricing model which is often used to estimate a companies weighted average cost of capital.

SUBJECTIVE - Subjective probability estimates the outcomes of an event on the basis of a calculated guess . They are not based on any prior information or calculations. They only reflect a person’s opinion based on his past performances.

An example for this kind of probability would be its use in many business situations like estimating rates and/or dollar returns on investment decisions.

SAMPLE SPACE –  A sample space is defined as a universal set of all possible outcomes from a given experiment.

è Two events A & B are part of a sample space S. This sample space is represented as a set as in the diagram below :

Example: A firm puts in tenders for 3 projects A, B & C. What is the sample space with respect to projects for which tenders are accepted.

Sample Space, S = {A, B, C, AB, AC, BC, ABC, Null}

VENN DIAGRAM – A schematic diagram used in logic theory to depict collection of sets and represents their relationships, It is an illustration of the relationships between and among sets, group of objects that share something in common. Generally, they are used to depict set intersections and unions. The diagram uses circles and rectangles where circles represents sets with the position and overlap of the circles indicating the relationships between the sets.

3 Event Properties -:

1          1)      Mutually Exclusive Events

2          2)      Independent Events

3          3)      Dependent Events

MUTUALLY EXCLUSIVE EVENTS –  Two  events that have NO outcomes in common are called Mutually Exclusive. These are the events that cannot occur at the same time.

For Eg: Recording two separate roles of one die are mutually exclusive events. Whatever number the dice displays on its first role will have no impact on what number is rolled the second time.

INDEPENDENT EVENTS –  When two events are said to be independent of each other, this means that the probability that one event occurs in no way affects the probability of the other event occurring.

For Eg : You roll a die and flipped a coin. The probability of getting any number face on the die in no way influences the probability of getting a head or a tail on the coin.

DEPENDENT EVENTS –  When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event.

For Eg : If you were to draw two cards from a deck of 52 cards. If in your first draw you had an ace and you put that aside, the probability of drawing an ace on the second draw is changed completely because you drew an ace the first time.

** We have already defined dependent and independent events and saw how probability of one event relates to the other event. Keeping that concept in mind lets now look at Conditional Probability. Conditional Probability is the one which deals with defining dependence of events by looking at probability of an event given that some other event occurs first. The probability that an event occurs given that the other event has already occurred.

3 Laws of Probability -:

1           1)      Addition Law

2           2)      Conditional Law

3           3)      Multiplicative Law

Additional Law

 If two events A & B are mutually exclusive then,

   P(AUB) = P(A)+P(B)

                                                                                                                                                

This is the simplified version of the additional law. However, when A & B are not                                                                       mutually exclusive, A∩B = ∅, it can be shown that a more general law applies:

                P(AUB) = P(A) +P(B) – P(A∩B)

Conditional Law

When the probability of occurrence of an event, say B, depends on the whether or not the other event, say A, has occurred, it is called conditional probability. The probability of B occurring is conditional on the occurrence or otherwise of A.

The conditional probability of B occurring given that event A has occurred is written P(B/A).

              P(B/A) = P(A∩B)/P(A)

Multiplication Law

If A & B are independent events then,

P(A∩B) = P(A)P(B)

In words, the probability of independent events A & B occurring is the product of the probabilities of the events occurring separately.

BAYES THEOREM –  Bayes Theorem is a simple mathematical formula used for calculating conditional probabilities. It figures prominently in subjectivist or Bayesian approaches to epistemology, statistics and inductive logic.

• ·         An Extension of the Conditional Probability Formula
• ·         Usually used to find the cause of a given experimental result
• ·         P(Ai|B)=P(B|Ai)P(Ai)/∑_j=i^k(B|Aj)P(Aj).

This concluded the day’s sessions which helped me get rid of some myths related to probability and at the same time developed a greater sense of understanding of the topic.

 Written By - Nilay Kohaley

Group Members - 1) Pragya Singh

                               2) Pawan Agarwal

                               3) Priyanka Doshi

                               4) Poulami Sarkar