Bayes' Theorem (Bayes Rule)
If 0 $<$ P(A), P(B) $<$ 1 then, $\mathsf {\large P(B|A)=\frac{P(A|B)\cdot P(B)}{{(A|B)\cdot P(B)+(A|B^c)\cdot P(B^c)}}}$
Example 1) Probability and Random Processes, Oxford, 3ED, p.11
Only two factories manufacture zoggles. 20% of the zoggles from factory A and 5% from factory B are defective. Factory A produces twice as many zoggles as factory B each week. 1) What is the probability that a zoggles, randomly chosen from a weeks production, is satisfactory? 2) If the chosen zoggle is defective, what is the probability that it came from factory A.
Example 2)
In Orange County 51% of adults are males. An adult is randomly selected for a survey concerning credit card usage.
a. What is the prior probability that the selected adult is female?
b. It is later learned, from the survey results, that the selected adult is a cigar smoker. Data from the Substance Abuse and Mental Health Services Administration indicates that in Orange County 9.5% of males and 1.7% of females smoke cigars. Given this additional information compute the probability that the selected adult is female.
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