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The World of Statistics and Probability (1)
By Dr. Magdi Abadir, PhD
Article 23: An introduction to Probabilities (9)
9. Bayes’ theorem
Consider the following situation:
A company produces machine components which pass through automatic testing. 5% of the components entering the testing machine are defective. However, the machine is not entirely reliable. If a component is defective there is 4% probability that it will not be rejected. If a component is not defective there is 7% probability that it will be rejected.
a) What fraction of all the components are rejected?
b) What fraction of all the components are not rejected?
c) What fraction of the components rejected are not defective?
d) What fraction of those not rejected are defective?
This type of problem was first tackled by Thomas Bayes and published in 1763. Though its significance was slow to catch on, it became crucial during Worls War II and exploded with computing power, becoming foundational in AI, medicine, and data science.
Now, let’s designate by D a defective component and by G a good one. Alos, let A be an accepted component and R a rejected one. Following the heading of the problem, the following diagram can be drawn:
a) The probability of all outcomes = 0.048+0.002+0.0665+0.8835=1 Rejected outcomes come from two sources only so that P(R)=P(R∩D)+P(R∩G) Therefore, the fraction to be rejected = 0.048+0.0665=0.1145
b) The probability of accepting a component = 1-0.1145=0.8855
c) The probability of non-defective components which were rejected = P(R∩G)=0.0665
d) The probability of defective components which were not rejected = P(A∩D)= 0.002.
The logic applied in that situation is the basis of applying Bayes’ theorem which is a direct application of the law of conditional probability.
Reference
1) W.J. DeCoursey (2003) “Statistics and Probability for Engineering Applications” Ed. Newnes, pages 34 – 36.
| Dr. Magdi Fouad Abadir, Ph. D.: Dr. M. F. Abadir is currently a professor with the Chemical Engineering Department at the Faculty of Engineering, University of Cairo, Egypt. His major interests are in the fields of high temperature science and technology. During his career, he has supervised more than 110 MSc and PhD theses and published more than a hundred papers mostly in international peer review journals. He currently teaches courses in High Temperature Technology and Industrial Statistics. He is also a consultant for several industrial businesses. |
