Probability Concept and Bayes Theorem
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This document defines key probability concepts like experiment, outcome, event, joint probability, and conditional probability. It then uses Bayes' Theorem to calculate the posterior probability that a defective chip was manufactured by one of three suppliers (Schuller Sales) given the prior probabilities and conditional probabilities of defects from each supplier. The probability is calculated to be 25.64% that the defective chip came from Schuller Sales.
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Probability Concept and Bayes Theorem
- 2. • Probability= Likelihood= Chance • Three Term (1)Experiment A process that leads to the occurrence of one(and only one) of several possible observation. (eg- tossing a coin, microscope=>2 or more possible result) Experiment Outcome (2)Outcome A particular result of the experiment. (eg – tossing a coin => head & tail) (3)Event A collection of one of more outcomes of an experiment. 1st time=> 200 people tossing a coin 2nd time => 500 people tossing a coin
- 3. Independent Case Dependent Case (Joint Probability) P(A) Addition P(A or B)= P(A)+P(B) P(A) Multiplication P(A and B)= P(A) * P(B) Addition P(A or B)= P(A)+P(B)-P(A and B) Multiplication P(A and B)= P(A) * P(B|A) P(B) P(B)
- 4. Joint Probability • Joint Probability is the likelihood that two or more events will happen at the same time. Conditional Probability • A conditional probability is the likelihood that an event will happen given that another event has already happened.
- 5. In the 18th Century , Reverend Thomas Bayes, an English Presbyterian minister, ponder this question: “Does God really exist?” Being interested in the mathematics, he attempt to develop a formula to arrive at the probability that God does exist based on the evidence that was available to him on earth. Later, Laplace refined Bayes’ work and gave it the name “Bayes’ Theorem”.
- 6. Bayes’ Theorem is a method of revising a probability, given that additional information is obtained. For two event: P(A1|B)=P(A1) P(B|A1) P(A1)P(B|A1)+P(A2)P(B|A2) Prior Probability The initial Probability based on the present level of information. Posterior Probability A revised Prabability based on additional information.
- 7. • The manufacturer of VCR purchase LS-24 chip from the three suppliers called Hall Electronic, Schuller Sales and Crawford component. • 30 % of LS-24 chip are purchased from Hall Electronic. • 20% of LS-24 chip are purchased from Schuller Sales . • 50% of LS-24 chip are purchased from Crawford Component. • The manufacturer has the extensive history with the three suppliers and know that 3 % of LS-24 chip from Hall Electronic are defective .
- 8. • 5 % of LS-24 chip from Schuller Sales are defective. • 4 % of LS-24 chip from Crawford Component are defective. • When the LS-24 chips are arrive at the manufacturer, they are placed directly in a bin and not inspected. • A worker selects a chip for installation and finds it defective. • What is a probability that it was manufactured by Schuller Sale?
- 9. There are three event, that is , three supplier A1= The LS-24 was purchased from Hall Electronics A2= The LS-24 was purchased from Schuller Electronics A3= The LS-24 was purchased from Crawford Component Prior Probabilities P(A1)= .30 => The probability the LS-24 was manufactured by Hall Electronic P(A2)= .20 => The probability the LS-24 was manufactured by Schuller Electronic P(A3)=.50 => The probability the LS-24 was manufactured by Crawford Component The additional information is that the LS-24 chip is defective. B1 => The LS-24 is defective. B2 => The LS-24 is not defective
- 10. The following conditional probabilities are given P(B1| A1)=.03 The probability that an LS-24 chip produced by Hall Electronic is defective. P(B1| A2)=.05 The probability that an LS-24 chip produced by Schuller Sales is defective. P(B1| A3)=.04 The probability that an LS-24 chip produced by Crawford Component is defective.
- 11. Conditional Probability Prior Probability Probability P(A1)=.30 P(B1|A1)=.03 P(B2|A1)=.97 P(A2)=.20 P(B1|A2)=.05 P(B2|A2)=.95 P(A3)=.50 P(B1|A3)=.04 P(B2|A3)=.96 Joint Probability A1= Hall Electronic A2= Schuller Electronics A3= Crawford Component B1= Defective B2=Good P(A1 and B1) =P(A1)P(B1|A1) =.30x.03=.009 P(A1 and B2) =P(A1)P(B2|A1) =.30x.97=.291 P(A2 and B1) =P(A2)P(B1|A2) =.20x.05=.010 P(A2 and B2) =P(A2)P(B2|A2) =.20x.95=.190 P(A3and B1) =P(A3)P(B1|A3) =.50x.04=.020 P(A3 and B2) =P(A3)P(B2|A3) =.20x.95=.480
- 12. Event (Ai) Prior Probability (P(Ai)) Conditional Probability (P(Bi| Ai) Joint Probability P(Ai and Bi) Posterior Probability P(Ai|Bi) Hall .30 .03 .009 .009/.039=.2308 Schuller .40 .05 .010 .010/.039=.2564 Crawford .50 .04 .020 .020/.039=.5128 P(Bi)= .039 1.0000 Defective chip probability was manufactured by Schuller Sale=.2564(26%)