Bayes Theorem and Total Probability
In this class, We discuss Bayes Theorem and Total Probability.
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The reader should have prior knowledge of total probability. Click Here.
We understand what is Bayes theorem equation if the total probability is involved.
Example:
In a bolt factory machines A, B, C manufactures 20%, 30%, 50% of bolts.
A, B, C output 6%, 3%, 2% of defective bolts.
A bolt is drawn at random and found that it is defective.
Find the probability that the bolt is from machine A. P(A|D) =?
Solution:
D = Defective bolt
P(A) = 20/100 = 0.2
P(B) = 30/100 = 0.3
P(C) = 50/100 = 0.5
P(D|A) = 6/100 = 0.06
P(D | B) = 3/100 = 0.03
P(D | C) = 2/100 = 0.02
From Bayes theorem P(A|D) = (P(D|A) P(A))/P(D)
Here P(D) is total probability.
Defective bolts come from three different machines.
P(A|D) = (P(D|A) P(A)) / (P(D|A)P(A) + P(D|B)P(B) + P(D|C)P(C))
P(A|D) = 12/31
Bayes Theorem: E1, E2, . . En, are mutually exclusive events, and P(Ei) is not equal to zero.
An arbitrary event D which is a subset of all the events Ui=1 to n Ei
P(Ei|D) = (P(Ei)P(D|Ei))/ (Σi=1 to n P(Ei)P(D|Ei))