Cn2 Probability
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Transcript of 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