PROBABILITY - Bayes’ Theorem

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PROBABILITY - Bayes’ Theorem. Business Statistics Dr. Gunjan Malhotra Assistant Professor Institute of Management Technology, Ghaziabad gmalhotra@imt.edu mailforgunjan@gmail.com. Introduction – Bayes’ Theorem. Developed by Reverend Thomas Bayes (1702-1761).

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### PROBABILITY -Bayes’ Theorem

Dr. Gunjan Malhotra

Assistant Professor

Institute of Management Technology,

gmalhotra@imt.edu

mailforgunjan@gmail.com

Introduction – Bayes’ Theorem
• Developed by Reverend Thomas Bayes (1702-1761).
• Bayesian decision theory’s purpose is to develop the solution of problems which involves decision making under uncertainty.
• It is an extended use of the concept of conditional probability given by
• It allows revision of the original probability with new information.
• In other words, it involves estimating unknown probability and making decisions on the basis of new (sample) information.
Bayes’ Theorem

Application

of Bayes’

Theorem

Computation

of

Prior

Probabilities

Obtaining

New

Information

Computation

of

Posterior

Probabilities

Bayes Theorem
• So say we know how to compute P(A|B). What if we want to figure out P(B|A)? We can re-arrange the formula using Bayes Theorem:
Bayes’ Theorem

Bayes’ theorem is used to revise previously calculated probabilities after new information is obtained.

• where:

Bi = ith event of k mutually exclusive and collectively exhaustive events

A = new event that might impact P(Bi)

Bayes’ Theorem – Example

Example: L. S. Clothiers

Aproposed shopping center

will provide strong competition for

downtown businesses like L. S. Clothiers.

If the shopping center is built, the owner of

L. S. Clothiers feels it would be best to

relocate to the center.

The shopping center cannot be built unless a

zoning change is approved by the town council. The

planning board must first make a recommendation, for

or against the zoning change, to the council.

Bayes’ Theorem

• Prior Probabilities

Let:

A1 = town council approves the zoning change

A2 = town council disapproves the change

Using subjective judgment:

P(A1) = .7, P(A2) = .3

Bayes’ Theorem
• New Information

The planning board has recommended against the zoning change. Let B denote the event of a negative recommendation by the planning board.

Given that B has occurred, should L. S. Clothiers revise the probabilities that the town council will approve or disapprove the zoning change?

Bayes’ Theorem

Conditional Probabilities

Past history with the planning board and the town council indicates the following:

P(B|A1) = .2

P(B|A2) = .9

P(BC|A1) = .8

P(BC|A2) = .1

Hence:

Bayes’ Theorem
• To find the posterior probability that event Aiwill
• occur given that event B has occurred, we apply
• Bayes’ theorem.
• Bayes’ theorem is applicable when the events for
• which we want to compute posterior probabilities
• are mutually exclusive and their union is the entire
• sample space.
Bayes’ Theorem
• Posterior Probabilities

Given the planning board’s recommendation not to approve the zoning change, we revise the prior probabilities as follows:

= .34

Bayes’ Theorem

• Conclusion

The planning board’s recommendation is good news for L. S. Clothiers. The posterior probability of the town council approving the zoning change is .34 compared to a prior probability of .70.

Bayes’ Theorem Example 1- (Ques.)
• A drilling company has estimated a 40% chance of striking oil for their new well.
• A detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have had detailed tests.
• Given that this well has been scheduled for a detailed test, what is the probability

that the well will be successful?

Bayes’ Theorem Example 1 (Ans.)

• Let S = successful well

U = unsuccessful well

• P(S) = 0.4 , P(U) = 0.6 (prior probabilities)
• Define the detailed test event as D
• Conditional probabilities:

P(D|S) = 0.6 P(D|U) = 0.2

• Goal is to find P(S|D)

Bayes’ Theorem Example

So the revised probability of success, given that this well has been scheduled for a detailed test, is 0.667

Apply Bayes’ Theorem:

Bayes’ Theorem Example

• Given the detailed test, the revised probability of a successful well has risen to 0.667 from the original estimate of 0.4

Sum = 0.36

Practice Questions
• Levine – pg – 144 – Q 4.30
• pg 144 – Q 4.33
• Pg 145 – Q 4.35,