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Bayes ’ Theorem

Posterior Probabilities Given the planning board’s recommendation not to approve the zoning change, we revise the prior probabilities as follows:. Bayes ’ Theorem. = .34. Bayes’ Theorem. Conclusion

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Bayes ’ Theorem

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  1. Posterior Probabilities Given the planning board’s recommendation not to approve the zoning change, we revise the prior probabilities as follows: Bayes’ Theorem = .34

  2. 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.

  3. Tabular Approach Step 1 Prepare the following three columns: Column 1 - The mutually exclusive events for which posterior probabilities are desired. Column 2 - The prior probabilities for the events. Column 3 - The conditional probabilities of the new information given each event.

  4. Tabular Approach (1) (2) (3) (4) (5) Prior Probabilities P(Ai) Conditional Probabilities P(B|Ai) Events Ai .2 .9 A1 A2 .7 .3 1.0

  5. Tabular Approach Step 2 Column 4 Compute the joint probabilities for each event and the new information B by using the multiplication law. Multiply the prior probabilities in column 2 by the corresponding conditional probabilities in column 3. That is, P(Ai IB) = P(Ai) P(B|Ai).

  6. Tabular Approach (1) (2) (3) (4) (5) Joint Probabilities P(Ai I B) Prior Probabilities P(Ai) Conditional Probabilities P(B|Ai) Events Ai .14 .27 .2 .9 A1 A2 .7 .3 1.0 .7 x .2

  7. Tabular Approach • Step 2 (continued) We see that there is a .14 probability of the town council approving the zoning change and a negative recommendation by the planning board. There is a .27 probability of the town council disapproving the zoning change and a negative recommendation by the planning board.

  8. Tabular Approach Step 3 Column 4 Sum the joint probabilities. The sum is the probability of the new information, P(B). The sum .14 + .27 shows an overall probability of .41 of a negative recommendation by the planning board.

  9. Tabular Approach (1) (2) (3) (4) (5) Joint Probabilities P(Ai I B) Prior Probabilities P(Ai) Conditional Probabilities P(B|Ai) Events Ai .14 .27 .2 .9 A1 A2 .7 .3 1.0 P(B) = .41

  10. Step 4 Column 5 Compute the posterior probabilities using the basic relationship of conditional probability. The joint probabilities P(Ai IB) are in column 4 and the probability P(B) is the sum of column 4. Tabular Approach

  11. Tabular Approach (1) (2) (3) (4) (5) Posterior Probabilities P(Ai |B) Joint Probabilities P(Ai I B) Prior Probabilities P(Ai) Conditional Probabilities P(B|Ai) Events Ai .14 .27 .2 .9 A1 A2 .3415 .6585 1.0000 .7 .3 1.0 P(B) = .41 .14/.41

  12. Homework P142-2 P 143-6,9,10 P146-15,17 P152-23 P153-28 P158-30,33 P165-39,42

  13. End of Chapter 4

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