Bayes' Theorem


  • Bayes' Theorem is used to revise previously calculated probabilities based on new information
  • Bayes' Theorem is extension of conditional probability(Dependent Events)

EXAMPLE
A desk lamp produced was found to be defective (D). There are three factories (A, B, C) where such desk lamps are manufactured. A Quality Control Manager (QCM) is responsible for investigating the source of this defect. Information available with QCM:

  • Of 100 units of the desk lamp, 35 come from both A and B while 30 come from C i.e. P(A) = P(B) = 0.35, P(C) = 0.30
  • If D is the probability of defective lamp then 
    • Probability of defect from A  = P(D | A) = 0.015
    • Probability of defect from B  = P(D | B) = 0.010
    • Probability of defect from C  = P(D | C) = 0.020

The QCM would like to answer the following question: If a randomly selected lamp is defective, what is the probability that the lamp was manufactured in factory C? i.e. FIND P(C | D)

 SOLUTION
Using Multiplication Rule for Dependent Events:

  1. P(C AND D) = P(D) * P(C | D), using Conditional Probability
  2. P(C | D) = P( C AND D) / P(D), using Conditional Probability 
  3. P(C | D) = [P(C) * P(D | C)]/P(D), using Multiplication Rule
    • Now the defective lamp can come from factory A or B or C. Therefore, defective probability P(D) = P(A AND D) OR P(B AND D) OR P(C AND D)
    • Using Mutually Exclusive and Multiplication Rule, we get
      • P(A AND D) = P(A) * P(D | A) = 0.35 * 0.015 = 0.00525
      • P(B AND D) = P(B) * P(D | B) = 0.35 * 0.010 = 0.0035
      • P(C AND D) = P(C) * P(D | C) = 0.30 * 0.020 = 0.006
    • P(D) = 0.00525 + 0.0035 + 0.006 = 0.01475
  4. P(C | D) = (0.30 * 0.020)/0.01475 = 0.4067 = 40.67%
The probability that the lamp was manufactured in factory C = 40.67%

Similarly, we can calculate:
The probability that the lamp was manufactured in factory A = 0.00525/0.01475 = 35.59%
The probability that the lamp was manufactured in factory B = 0.0035/0.01475 = 23.72%

 REFERENCES
https://www.greatlearning.in/great-lakes-pgpba/
https://onlinecourses.science.psu.edu/stat414/node/43


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