7.6 Bayes’ Theorem

SCBA D A  DB  DC  D In this Venn Diagram, S is the whole sample space (everything), and D overlaps the other three sets. We will be doing more with this type of Venn.

Bayes’ Theorem Let A 1, A 2, …, A n be a partition of a sample space S and let E be an event of the experiment such that P(E) is not zero. Then the posteriori probability P(A i |E) is given by Where This is just the same as before when we did conditional probability. It looks more complex, but is not.

Ex 1. The accompanying tree diagram represents a two- stage experiment. Use the diagram to find P(B|D)? A B C D D D DCDC DCDC DCDC 1/4 1/2 1/4 3/4 1/4 1/2 2/3 1/3

SCBA D Given the Venn Diagram do the following: a.Draw a tree diagram b.Find P(D) c.Find P(B|D) & P(D|B) d.Find P(D c ) A B C D D D DCDC DCDC DCDC b. P(D)= (.3)(.5) + (.2)(.25) + (.5)(.4) =.4 d.P(D c )= (.3)(.5) + (.2)(.75) + (.5)(.6) =.6 or we could just do 1 – P(D) = =.6

Homework #1 P , 9-12 all and 13 – 21 odd & worksheet 7.4 #2 P – 22 even, 23 – 27 odd #3 P – 15 odd #4 p – 37 odd p. 351 #34 #5 p all, 7 – 25 odd