# Bayes' formula

## Inhaltsverzeichnis

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### Merke

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{*LOSnr34*}
Calculate and interpret covariance given a joint probability function.

Bayes' formula is used for updating probabilities based on new information.

### Methode

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We can use Bayes' formula to find P(A|B) when only P(B|A) and P(A) are known:

P(A|B) = [P(B|A)∙P(A)]/P(B) Bayes' formula

= [P(B|A)∙P(A)]/[P(B|A)∙P(A) + P(B|AC)∙P(AC)],

using the total probability rule P(B) = P(B|A)∙P(A) + P(B|AC)∙P(AC) for the denominator.

### Beispiel

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In 2009, 40 percent of all Level I candidates passed their exams. Of those who passed, 70 percent had done a prep course. Of those who did not pass, 40 percent had done the same course. Find the probability that a randomly chosen candidate, who had done a prep course, passed his or her Level I exam.

To do so, we must find P(pass|course). Using Bayes's formula, we calculate like this:

P(pass|course) = P(course|pass)*P(pass)/P(course)

= P(course|pass)*P(pass)/

[P(course|pass)*P(pass) + P(course|passc)*P(passc)

= 0.7*0.4/[0.7*0.4 + 0.4*0.6]

= 0.28/0.52

= 0.5385

= 53.85 percent.