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# Bayes Theorem

Bayes Theorem1 is a logical extension of the conditional probability arguments we looked at in the Venn diagram section. We saw that:

$P(A \mid B)=\frac{P (A \cap B)}{P(B)}$and $P(B \mid A)=\frac{P (B \cap A)}{P(A)}$

Since $P(A \cap B)=P(B \cap A)$   :

Hence:  $P(B|A)=\frac{P(B)P(A|B)}{P(A)}$which is Bayes Theorem

and, in general,

The following example illustrates the use of this equation. Many more are given in the section on Bayesian inference.

### Example

Three machines A, B and C produce 20%, 45% and 35% respectively of a factory's wheel nuts output. 2%, 1% and 3% respectively of these machines outputs are defective.

a) What is the probability that any wheel nut randomly selected from the factory's stock will be defective? Let X be the event that the wheel nut is defective and A, B, and C be the events that the selected wheel nut came from machines A, B and C respectively:

P(X)     = P(A). P(X/A)+P(B). P(X/B)+P(C). P(X/C)

= (0.2).(0.02)+(0.45.)(0.01)+(0.35).(0.03)

= 0.019

b) What is the probability that a randomly selected wheel nut comes from machine A if it is defective?

From Bayes Theorem,

In other words, in Bayes Theorem we divide the probability of the required path (probability that it came from machine A and was defective) by the probability of all possible paths (probability that it came from any machine and was defective).

1Rev. Thomas Bayes (1702-1761) - English philosopher. A short biography and a reprint of his original paper describing Bayes Theorem appear in Press (1989).

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