PROBABILITY - Bayes’ Theorem

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). • 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 • We start with conditional probability definition: • 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,

PROBABILITY - Bayes’ Theorem