Probability
Ravindra S. Gokhale IIM Indore

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“It is a truth very certain that, when it is not in our power to determine what is true, we ought to follow what is most probable”
- Rene Descartes

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Probability Models
 Concerned with the study of random (or chance) phenomena  Random experiment: An experiment that can result in different outcomes, even though it is repeated in the same manner every

time
 Managerial Implication: Due to this random nature additional capacities for any setup are required

 Although random, certain statistical regularities are to be captured to form the model

 Model: An abstraction of real world problem

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Sample Space
 A set of all possible outcomes of a random experiment  Denoted by ‘S’  Examples:

•
• • • • • •

Tossing of a coin: S = {‘Head’, ‘Tail’}
Throwing of a dice: S = {1, 2, 3, 4, 5, 6} Marks of a student in a 20 mark paper with only integer negative marking possible: S = {-20, -19, …, -1, 0, 1, …, 19, 20} Number of people arriving at a bank in a day: S = {0, 1, 2, …} Inspection of parts till one defective part is found: S = {d, gd, ggd, gggd, …} Temperature of a place with a knowledge that it ranges between 10 degrees and 50 degrees: S = {any value between 10 to 50} Speed of a train at a given time, with no other additional information: S = {any value between 0 to infinity}
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Sample Space (cont…)
 Discrete sample space: One that contains either finite or countable infinite set of outcomes

•

Out of the previous examples, which ones are discrete sample spaces???

 Continuous sample space: One that contains an interval of real numbers. The interval can be either finite or infinite

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Events
 A collection of certain sample points  A subset of the sample space  Denoted by ‘E’

 Examples:

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Getting an odd number in dice throwing experiment S = {1, 2, 3, 4, 5, 6}; E = {1, 3, 5}

•
•

Getting a defective part within the first three inspected parts
S = {d, gd, ggd, gggd, ggggd, …}; E = {d, gd, ggd} Event of a component not failing before time t: S = [0, ∞); E = [t, ∞)

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Events (cont…)
 Complement of an event E1: E1’ = S – E1  Intersection of two events: E3 = E1  Union of two events: E4 = E1 U E2 U U E2

 Two events A and B are mutually exclusive if: A

B=Ø

 ‘Union’ and ‘Intersection’ can be extended to more than two events

 Two events A and B are mutually exclusive and exhaustive if:
A

U

B = Ø and A U B = S

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Graphical Representation of Events
Venn Diagram S A B A
A

S B U
B

Disjoint events ‘A’ and ‘B’

S A
AUB

B

A C

B D

S

Mutually exclusive and exhaustive events: A, B, C, and D

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Graphical Representation of Events (cont…)
Tree Diagram

Defective

NotDefective

Head

Tail

Defective

NotDefective

Head

Tail Head

Tail

Defective

NotDefective

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Probability
 Probability of an event represents the ‘relative likelihood’ that performance of the experiment will result in occurrence of that event

 P(A) = Probability of the event A in the sample space
 Examples:

• •

A = Getting an odd number in dice throwing experiment

P(A) = 0.5
B = Getting two heads in two successive tosses of a coin P(B) = 0.25

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Concept of ‘Sample’ and ‘Population’
 A population is the entire collection units in which we are interested

 A sample is a subset of a population

 The objective of ‘inferential statistics’ is to make an inference about a population of interest based on the information obtained from the sample from that population

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Probability in the Context of Sample and Population
 Proportion  Sample

 Probability



Population

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The Approaches for Determining Probability
 Classical Approach: probability is based on the idea that certain occurrences are equally likely (e.g. throwing of dice)

 Relative Frequency Approach: probability based on long term relative frequency --- requires a large amount of historical data

 Subjective Probability: probability based on one’s judgement (e.g. how may days will it take for a consignment to reach from Indore to Delhi?)

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The Three Axioms of Probability
 For any event A, 0