Cn2 Probability

“Probability”

Arun Kumar, Ravindra Gokhale, and NagarajanKrishnamurthy

Quantitative Techniques-I, Term I, 2012Indian Institute of Management Indore

Describing Shape of a Bar Graph

Proportion of observations in a particular category.

Describing Shape of a Histogram

Proportion of observations in a particular class interval.

Probability

Proportion → sample

Probability → population

Example

Workforce distribution in the United States.

Industry ProbabilityAgriculture 0.130Construction 0.147Finance, Insurance, Real Estate 0.059Manufacturing 0.042Mining 0.002Services 0.419Trade 0.159Transportation, Public Utilities 0.042

Sample Space

Def: Set of all possible outcomes.

Ex.: Ω=Agriculture, Construction, . . . , Services, Trade,Transportation and Public Utilities

Simple Events

Simple event: An event in the finest partition of the samplespace.

Example: ω1=Agriculture, ω2=Construction.

Event

Def: Any subset of the sample space

Ex: Agriculture, Construction

Exercise

A bowl contains three red and two yellow balls. Two balls arerandomly selected and their colors recorded. Use a treediagram to list the 20 simple events in the experiment, keepingin mind the order in which the balls are drawn.

Other Approaches for Calculating Probabilities

Classical Approach: Assuming all outcomes to be equallylikely, the probability of an event is the number of favourableoutcomes divided by the total number of outcomes.Ex. Rolling a dice

Subjective Approach: Assigning probability to an event basedon one’s experience.

Example

Workforce distribution in the United States.

Industry ProbabilityAgriculture 0.130Construction 0.147Finance, Insurance, Real Estate 0.059Manufacturing 0.042Mining 0.002Services 0.419Trade 0.159Transportation, Public Utilities 0.042

Probability

P(Agriculture)

= 0.13

P(Either Agriculture or Construction or both) →P(Agriculture ∪ Construction) = 0.13+0.147=0.277.

P(Agriculture and Construction) →P(Agriculture ∩ Construction) =0.

P(Not in Agriculture) → P(Agriculturec) = 1-0.13=0.87.

Probability

P(Agriculture) = 0.13

P(Either Agriculture or Construction or both) →P(Agriculture ∪ Construction)

= 0.13+0.147=0.277.

P(Agriculture and Construction) →P(Agriculture ∩ Construction) =0.

P(Not in Agriculture) → P(Agriculturec) = 1-0.13=0.87.

Probability

P(Agriculture) = 0.13

P(Either Agriculture or Construction or both) →P(Agriculture ∪ Construction) = 0.13+0.147=0.277.

P(Agriculture and Construction) →P(Agriculture ∩ Construction)

=0.

P(Not in Agriculture) → P(Agriculturec) = 1-0.13=0.87.

Probability

P(Agriculture) = 0.13

P(Either Agriculture or Construction or both) →P(Agriculture ∪ Construction) = 0.13+0.147=0.277.

P(Agriculture and Construction) →P(Agriculture ∩ Construction) =0.

P(Not in Agriculture) → P(Agriculturec)

= 1-0.13=0.87.

Probability

P(Agriculture) = 0.13

P(Either Agriculture or Construction or both) →P(Agriculture ∪ Construction) = 0.13+0.147=0.277.

P(Agriculture and Construction) →P(Agriculture ∩ Construction) =0.

P(Not in Agriculture) → P(Agriculturec) = 1-0.13=0.87.

Compound Events

If A and B are two events then

Union event is A ∪ B

Intersection event is A ∩ B

Complement event is Ac

Venn Diagram Representation

8

A B

S

Disjoint events ‘A’ and ‘B’ A B

A

S

B

U

A U B

A

S

B

C

BS

Mutually exclusive and exhaustiveevents: A, B, C, and D

A

D

Probability Rules

1 P(A ∪ B) = P(A) + P(B)− P(A ∩ B)

2 P(Ac) = 1− P(A)

Mutually Exclusive

Def: Two events are mutually exclusive if they do not haveany common outcome.

Ex: Agriculture and Construction are mutually exclusiveevents.

Mutually Exclusive

A and B are mutually exclusive if P(A ∩ B) = 0.

This implies that for mutually exclusive events A and B,P(A ∪ B) = P(A)+P(B).

Pizza Venn Diagram

What is the sample space?

Sample space=Tomato only, Fish Only, Mushroom-Tomato,Mushroom-Tomato-Fish, Mushroom-Fish, No toppings.

What is the sample space?

Sample space=Tomato only, Fish Only, Mushroom-Tomato,Mushroom-Tomato-Fish, Mushroom-Fish, No toppings.

Probability of the events in the sample space

P(Tomato only)

=2/8; P(Fish only)=1/8.

P(Mushroom-Tomato) =2/8=1/4;P(Mushroom-Tomato-Fish)=1/8.

P(Mushroom-Fish) =1/8; P(No toppings)=1/8.

Probability of the events in the sample space

P(Tomato only) =2/8; P(Fish only)

=1/8.

P(Mushroom-Tomato) =2/8=1/4;P(Mushroom-Tomato-Fish)=1/8.

P(Mushroom-Fish) =1/8; P(No toppings)=1/8.

Probability of the events in the sample space

P(Tomato only) =2/8; P(Fish only)=1/8.

P(Mushroom-Tomato)

=2/8=1/4;P(Mushroom-Tomato-Fish)=1/8.

P(Mushroom-Fish) =1/8; P(No toppings)=1/8.

Probability of the events in the sample space

P(Tomato only) =2/8; P(Fish only)=1/8.

P(Mushroom-Tomato) =2/8=1/4;P(Mushroom-Tomato-Fish)

=1/8.

P(Mushroom-Fish) =1/8; P(No toppings)=1/8.

Probability of the events in the sample space

P(Tomato only) =2/8; P(Fish only)=1/8.

P(Mushroom-Tomato) =2/8=1/4;P(Mushroom-Tomato-Fish)=1/8.

P(Mushroom-Fish)

=1/8; P(No toppings)=1/8.

Probability of the events in the sample space

P(Tomato only) =2/8; P(Fish only)=1/8.

P(Mushroom-Tomato) =2/8=1/4;P(Mushroom-Tomato-Fish)=1/8.

P(Mushroom-Fish) =1/8; P(No toppings)

=1/8.

Probability of the events in the sample space

P(Tomato only) =2/8; P(Fish only)=1/8.

P(Mushroom-Tomato) =2/8=1/4;P(Mushroom-Tomato-Fish)=1/8.

P(Mushroom-Fish) =1/8; P(No toppings)=1/8.

Union Rule

What is the probability that your slice will have tomato ormushroom?

Ans. 6/8=3/4

Union Rule

What is the probability that your slice will have tomato ormushroom?

Ans. 6/8=3/4

Intersection Rule

What is the probability that your slice will have tomato andmushroom?

Ans. 3/8

Intersection Rule

What is the probability that your slice will have tomato andmushroom?

Ans. 3/8

Complement Rule

What is the probability that your slice will not have tomato?

Ans. 3/8

Complement Rule

What is the probability that your slice will not have tomato?

Ans. 3/8

Conditional Probability

You have pulled out a slice of pizza that has tomato on it.What is the probability that your slice will have mushrooms?

Ans. 3/5.

Conditional Probability

Def: Probability of event A in event B.

Notation: A|B

Multiplication rule

P(A ∩ B) = P(A)P(B |A)

P(A ∩ B) = P(B)P(A|B)

Independent Venn Pizza

Statistical Independence

Two events are said to be independent if occurrence of onehas no effect on the chances for the occurrence of the other.

Statistical Independence

Using the Statistically Independent Pizza, are eventsmushroom and tomato independent?

Statistical Independence

Two events A and B are considered independent whenP(A|B)=P(A).

Independence

Exercise 1

Is Gender related to whether someone voted in the lastmayoral election? Answer the question using the jointprobabilities given in the table below.

Table: Is gender related to whether someone voted in the lastmayoral election

GenderVoted in the last mayoral election Female MaleYes 0.25 0.18No 0.33 0.24

Statistical Independence

If two events A and B are independent then

1 P(A ∩ B) = P(A)P(B)

Law of Total Probability

Given a set of events S1, S2, . . . , Sk that are mutually exclusiveand exhaustive, and an event A, the probability of the event Acan be expressed as

P(A) = P(S1).P(A|S1) + P(S2).P(A|S2)

+P(S3).P(A|S3) + . . . + P(Sk).P(A|Sk)

Exercise 2

A business group own three five-star hotels (say, A, B, and C)in India. By studying the past behavior of the revenueobtained from the three hotels month by month, it has beenobserved that the probability of increase in revenue of either Bor C or both of them is 0.5. If A’s revenue increases in a givenmonth, the probability of increase in B’s revenue is 0.7, theprobability of increase in C’s revenue is 0.6, and the probabilityof increase in both B and C’s revenue is 0.5. However if A’srevenue does not increase in a given month, the probability ofincrease in B’s revenue is 0.2, the probability of increase in C’srevenue is 0.3, and the probability of increase in both B andC’s revenue is 0.1. What is the probability that the revenue ofall the three hotels, A, B, and C increases in a given month?

Exercise 3

You are a physician. You think it is quite likely that one of your patients has strep

throat, but you are not sure. You take some swabs from the throat and send them to

a lab for testing. The test is (like nearly all lab tests) not perfect. If the patient has

strep throat, then 70% of the time the lab says YES but 30% of the time it says NO.

If the patient does not have strep throat, then 90% of the time the lab says NO but

10% of the time it says YES. You send five succesive swabs to the lab, from the same

patient. You get back these results, in order; YNYNY. What do you conclude?

These results are worthless.

It is likely that the patient does not have the strep throat.

It is slightly more likely than not, that patient does have the strep throat.

It is very much more likely than not, that patient does have the strep throat.

Bayes’ Rule

Let S1, S2, . . . , Sk represents k mutually exclusive andexhaustive sub-populations with prior probabilitiesP(S1),P(S2), . . . ,P(S2). If an event A occurs, the posteriorprobability of Si given A is the conditional probability

P(Si |A) =P(Si).P(A|Si)∑kj=1 P(Sj).P(A|Sj)

Exercise

Strep Throat Exercise

Bibliography

An Introduction to Probability and Inductive Logic, by IanHacking

Introduction to Probability and Statistics, by WilliamMendenhall, Robert J. Beaver, and Barbara M. Beaver

Practice of Business Statistics, by David S. Moore, GeorgeP. McCabe, William M. Duckworth, and Stanley L. Sclove

Bradley A. Warner, David Pendergrift, and TimothyWebb,“That was Venn, This is now”, Journal ofStatistical Education, Volume 6, Number 1, 1998