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))