Bayes Theorem If the events A1,A2,.......Ak form a partition of sample space S, that is the events Ai are mutually exclusive and their union is S, and if B is any other event of S such that it can occur only if one of the Ai occurs then for any i. Bayes Theorem Q: # In a bolt factory, machines A, B and C manufacture 25, 35 and 40 percent of the total output respectively. Of their outputs, 5, 4, 2 percent, respectively are defective bolts. A bolt is selected at random and found to be defective. What is the probability that the bolt came from machine A? B? C?. Q:# A manufacturing firm employs three analytical plans for the design and development of a particular product. For cost reasons all three are used at varying times . In fact plans 1,2 and 3 are used for 30%, 20% and 50% of the products respectively. The “defect rate “ is different for the three procedures as follows P(D|p1)=0.01 P(D|P2)=0.03 P(D|P3)=0.02 if a random product was observed and found to be defective , which plan was most likely used and thus responsible? Q: # in a college 4 % of men and 1% of women are taller than 6 feet. Further more 60% of the students are women. Now if a student is selected at random and is taller than 6 feet , then what is the probability that the student is a women?