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Program : MBA

Semester : I

Subject Code : MBF103

Subject Name : Statistics for Management

Book Code : B1129

Unit number : 2

Unit Title :Probability

Lecture Number : 9 & 10.

Lecture Title : Introduction to Probability

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Introduction to probability

Objectives:

Examine the use of probability theory in decision making    Recognise the approaches to probability    Apply the rules of probability for calculating different kinds of

probabilities     Apply the Baye's probability theorem by taking new information

into account     Apply the concept of random variables to real life situations

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Lecture Outline

• Introduction

• Definition of Probability

• Basic Terminology & Approaches to probabilities

• Rules of probability

• Conditional probability

• Solving problems basing on probability

• Baye’s probability

• Difference between Bayes & Conditional probability

• Introduction to Random varaible

• Summary

• Check Your Learning

• Activity

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Introduction

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Every human activity has an element of uncertainty. Uncertainty affects the decision making process. In your daily lives, you very often use the word ‘probably’, like, probably it may rain today; probably the share price may go up in the next week. Therefore, there is a need to handle uncertainty systematically and scientifically.

Mathematicians and statisticians developed a separate field of mathematics and named it as ‘Probability Theory’. The theory of probability helps us to make wiser decisions by reducing the degree of uncertainty.

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DEFINITION OF PROBABILITY

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Probability is a numerical measure which indicates the chance of occurrence of an event “A”. It is denoted by P(A).

It is the ratio between the favorable outcomes to an event “A” (m) to the total outcomes of the experiment (n). In other words, P(A)=m/nwhere, ‘m’ is the number of favourable outcomes of an event ‘A’and ‘n’ is the total number of outcomes of the experiment.

herdingcats.typepad.com/my_weblog/2008/12/pro...

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Basic Terminology in Probability

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•Experiment: An operation that results in a definite outcome is called an experiment.

•If a coin stands on its edge, it is not an experiment.

Random Experiment: When the outcome of an experiment cannot be predicted, then it is called random experiment or stochastic experiment

• Sample Space: Sample Space or total number of outcomes of an experiment is the set of all possible outcomes of a random experiment and is denoted by ‘S’.

• Event: Events may be a single outcome or combination of outcomes. Event is a subset of sample space.

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Basic Terminology in Probability

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• Equally likely Events (Equi-probable events): Two or more events are said to be equally likely if they have equal chance of occurrence.

• Mutually Exclusive Events: Two or more events are said to be mutually exclusive if the occurrence of one prevents the occurrence of other events.

• Exhaustive Set of Events: A set of events is exhaustive if one or other of the events in the set occurs whenever the experiment is conducted.

• Complementation of an Event: The complement of an event is given by P(Á)= 1-P(A)

• Independent Event: Two events are said to be independent of each other if the occurrence of one is not affected by the occurrence of other or does not affect the occurrence of the other.

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Approaches to Probability

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•Classical/ Mathematical/ Priori Approach: Under this approach the probability of an event is known before conducting the experiment. Getting a head in tossing a coin.

• Statistical/ Relative frequency/ Empirical/ Posteriori Approach:Under this approach the probability of an event is arrived at after conducting an experiment

• Subjective Approach: Under this approach the investigator or researcher assigns probability to the events either from his experience or from past records.

• Axiomatic Approach: This approach is based on set theory.

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Rules of Probability

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•Managers very often come across with situations where they have to take decisions about implementing either course of action A or course of action B or course of action C. Sometimes, they have to take decisions regarding the implementation of both A and B.

Addition Rule: The addition rule of probability has provided with 3 different steps: i) If ‘A’ and ‘B’ are any two events then P(AuB)= P(A)+P(B)-P(AΠB)ii) If ‘A’ and ‘B’ are two mutually exclusive events then P(AuB)= P(A) + P(B)iii) If A, B and C are any three events then P(AuBuC)= P(A)+P(B)+P(c)-P(AΠB)-P(BΠC)-P(CΠA)+P(AΠΠBC)

Multiplication Rule: Multiplication Rule is given when A & B are two independent events then P(AΠB)=p(A)P(B)

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Conditional Probability

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Sometimes we wish to know the probability that the price of a particular petroleum product will rise, given that the finance minister has increased the petrol price. Such probabilities are known as conditional probabilities.

If A and B are dependent events then the probability of occurrence of A and B is given by

P(A∩B)=P(A).P(B/A) =P(B).P(A/B)

It follows that P(A/B)=P(A∩B)/P(B)P(B/A)=P(A∩B)/P(A)

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Solving Problem on Probability

•Following are the steps which need to follow while solving problems on Probability:

Define the events Find total outcome of the experimentFind probability of each eventIf words ‘either - or’ is used, check whether the events are mutually exclusive or not, to apply addition rule If words ‘both or and’ are used, check whether the events are independent or dependent, to apply proper multiplication rules To find total outcome of the experiment use 2n or 6n in the case of coin or die respectively, where ‘n’ is the number of coins or dice thrown at a time or a coin or die thrown ‘n’ times. In all other cases use nCr.

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Baye’s Probability

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Baye’s Theorem states that if A1, A2,……. An are ‘n’ mutually exclusive and exhaustive events and B is a common event to all theorems then probability of occurrence of A1 given that B has already occurred is given by

n

iii ABPAP

ABPAPBAP

1

111

)/().(

)/().()/(

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Difference - Baye’s & Conditional Probabilities

Baye’s Probabilities Conditional Probabilities

Finds the probability of population value, given the sample value

Finds the probability of getting a sample value given the population value

Possible to incorporate latest information

Not possible to incorporate latest information

Possible to incorporate cost aspects

Not possible

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Random Variable

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If we can assign a real-valued function to every value of the variable in the sample space,Such thati.P(Xi)=P[X=Xi] for all Iii.P(Xi) ≥ 0 for all Iiii.ΣP(Xi) = 1 then it is called Random Variable

If Xi is a discrete random variable then P(X) is known as probability mass function of X.If Xi is a continuous random variable then P(X) is known as probability density function of X and is denoted by f(X).

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Summary

Probability plays an important role in decision making process.

When multiple events are involved in an experiment, the concerned

probabilities are calculated using addition and multiplication rules of

probability.

Baye‘s theorem deals with the probability of the occurrence of an

event to the occurrence or non-occurrence of an associated event.

This is an important theorem helpful for managers in business

decisions.

Random variable is a not a variable. It is a function. It can be discrete

or continuous.

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Check Your Learning

1.What is probability distribution of a random variable?

Ans: Probability of the number of heads forms a probability distribution. A

systematic representation of random variable with its value and probabilities

is called a probability distribution of that random variable.

2. In a pack of cards how many king cards are available

Ans: 4

3. Which Addition Rule we use, if both A & B are two mutually exclusive events?

Ans: P(AUB)= P(A)+P(B).

4. Define classical approach with one example?

Ans: Classical approach the probability of an event is known before conducting

the experiment. Drawing a king from well shuffled pack is one of its

example.

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Activity

1. Take a pack of cards & try to analyze how many kings, Queens &

Jacks are present. And also analyze how many:

A’s

Heart shape cards

Spades

Cloves

Diamond shape cards are present.

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Unit-5 Probability