Session 5 - Session 6

Session 5 - Session 6

Probability-

The class started with the basic approach of measuring numbers and quantities.

The first 3 points introduced were –

a  a)       Ratio Change (1 out of how many)

b  b)      Percentage Change(how much  out of 100 the quantity depicts)

    c)       Probability (varies from 0-1 , max. probability=1 , min probability=0)

       Now let us further get acquainted with the concept of probability.

      Probability is a measure or estimation of how likely it is that something will happen. It shows how true a statement is from a given set of statements. It therefore deals with the measure of uncertainty. Value of probability varies between 0 (0%   chance , hence it  will not happen) and 1 (100% chance , i.e. it will happen). The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater the number of times such event is expected to happen.

Probability deals with two basic fundamentals-

       A)     Sample Space- Sum total of all possible results of an experiment.

       B)      Outcome/Event- the result obtained from performing an experiment. 

       How does probability help us in Business?

       Probability theory is applied in everyday life in risk assessment and in trade on financial markets. Governments apply probabilistic methods in environmental regulation, where it is called pathway analysis. A good example is the effect of the perceived probability of any widespread Middle East conflict on oil prices—which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such group think on pricing, on policy, and on peace and conflict.

Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacture's decisions on a product's warranty.

3 ways of measuring probability-

a) A priori Method-

Here the outcomes of an experiment are known already from before hand and we work with those datas known.

If we have a situation (a "random process") in which there are n equally likely outcomes, and the event A consists of exactly m of these outcomes, we say that the probability of A is m/n. We may write this as "P(A) = m/n" for short where ,P(A) is the probability of A happening.

Example-

If   we consider tossing a fair die, there are six possible numbers that could come up and , since the die is fair, each one is equally likely to occur. So we say each of these outcomes has probability 1/6.

A business related example would be-

In a priori probability, a Company XYZ's stock can only do one of three things: Go up, go down or stay the same. Accordingly, there is only a 33% chance that the stock will go up next day for a given value of the stock closing the previous day.

b) Empirical Method-

Here the probability result is deduced from information collected beforehand.

The empirical approach to assigning probability is used when data is available about the past history of the experiment.  The probability of an outcome is the relative frequency of the outcome. 

Ex- if out of 100 sales calls, you had 37 sales, the probability of a sale would be 37/100 = .37

c) Subjective Method-

Here the probability obtained is purely judgmental .

Subjective probability is an individual person's measure of belief that an event will occur. It always gives result in  a range .With this view of probability, 

Ex- It makes perfectly good sense intuitively to talk about the probability that the Dow Jones average will go up tomorrow.

INTERSECTION of events-

The intersection of events A and B (A and B are general ideas) is the EVENT that occurs when BOTH  A and B occur.  The way we write the intersection is to have A and B or it may be written A ∩ B. The way we write the probability of the intersection is to have P(A and B) or it may be written P(A ∩ B).

Venn Diagrammatic representation

Occurence of events can be represented via venn diagram also.

    The rectangle here represents the sample space.  On one variable we have event S1 and that takes up the space represented by circle S1.  Ignoring circle B, all the rest of the rectangle is S1c (the complement of S1).  A similar interpretation holds for S2. The shaded region denotes the intersection of the 2 sets.

Types of events-

 a)Mutually exclusive Events- Where both events cannot occur simultaneously

 b) Independent Events-Where both events do not  affect outcome of one another.

 c) Dependent Events - Where both the events are dependent on one another.

a)Mutually Exclusive Events-

 When two events (call them "A" and "B") are Mutually Exclusive it is impossible for them to happen together:

P(A and B) = 0

In a Deck of 52 Cards:

The probability of a King is 1/13, so P(King)=1/13,the probability of an Ace is also 1/13, so P(Ace)=1/13.

 When we combine those two Events:

The probability of a card being a King and an Ace is 0 (Impossible).The probability of a card being a King or an Ace is (1/13) + (1/13) = 2/13,which is written like this:P(King and Ace) = 0.

b)Independent Events-

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

 An example of two independent events is as follows; say you rolled 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 and Ace) = 0

c)Dependent Events-

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

For example, if you were to draw a two cards from a deck of 52 cards. If on your first draw you had an ace and you put that aside, the probability of drawing an ace on the second draw is greatly changed because you drew an ace the first time. Let's calculate these different probabilities to see what's going on.

There are 4 Aces in a deck of 52 cards

P(ace)=No. of aces in the deck / No. of cards in the deck.

On your first draw, the probability of getting an ace is given by:

P(ace)=4/52=1/13.

If we don't return this card into the deck, the probability of drawing an ace on the second pick is given by

P(ace)= No. of aces remaining in the deck / No. of cards on the deck remaining

         =3/51.

3 laws of Probability.

 a)ADDITION LAW

 b)CONDITIONAL LAW- When a condition is given and we need to find the probability based on that condition.

 c)MULTIPLICATIVE LAW - 

Conditional Law-

As we have seen, P(A) refers to the probability that event A will occur.  A new idea is that P(A|B) refers to the probability that A will occur but with the understanding that B has already occurred and we know it.  So, we say the probability of A given B.  The given B part means that it is known that B has occurred.

By definition

P(A|B) = P(A and B)/P(B).

Similarly

P(B|A) = P(A and B)/P(A).

Note P(A and B) =  P(B and A)

MULTIPLICATION RULE-

 Consider events A and B.  According to the Multiplication Rule,

P(AB)= P(A).P(B).

 What The Rule Means:

Suppose we roll one die followed by another and want to find the probability of rolling a 4 on the first die and rolling an even number on the second die. Notice in this problem we are not dealing with the sum of both dice. We are only dealing with the probability of 4 on one die only and then, as a separate event, the probability of an even number on one die only.

 P(4) = 1/6 

P(even) = 3/6

So P(4.even) = (1/6)(3/6) = 3/36 = 1/12

ADDITION RULE-

In probability we refer to the addition operator (+) as or. 

Sometimes we want to ask a question about the probability of A or B, written P(A or B) = P(A ⋃ B).

By the general addition rule P(A or B) = P(A) + P(B) – P(A and B).

Bayes Theorem-

Bayes' Theorem is a theorem of probability theory originally stated by the Reverend Thomas Bayes. It can be seen as a way of understanding how the probability that a theory is true is affected by a new piece of evidence. It has been used in a wide variety of contexts, ranging from marine biology to the development of "Bayesian" spam blockers for email systems. In the philosophy of science, it has been used to try to clarify the relationship between theory and evidence. Many insights in the philosophy of science involving confirmation, falsification, the relation between science and pseudosience, and other topics can be made more precise, and sometimes extended or corrected, by using Bayes' Theorem. These pages will introduce the theorem and its use in the philosophy of science.

Begin by having a look at the theorem, displayed below. Then we'll look at the notation and terminology involved.

P(T l E) =  P (E l T) X P(T)  /  P(E l T) X P(T) + P(E-T) X P(-T)

In this formula, T stands for a theory or hypothesis that we are interested in testing, and E represents a new piece of evidence that seems to confirm or disconfirm the theory. For any proposition S, we will use P(S) to stand for our degree of belief, or "subjective probability," that S is true. In particular, P(T) represents our best estimate of the probability of the theory we are considering, prior to consideration of the new piece of evidence. It is known as the prior probability of T.

This was in short an overview on today's lesson which was indeed informative and I could easily learn the basics of probability and how it can be used in business matters.

Written By-

Parthajit Sar

Group Members-

Parthajit Sar

Raghav Bhatter

Nitesh Beriwal

Nihal Moidu

Prachee Kasera

Neha Gupta