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# Bayes' Theorem

Let and be sets. Conditional probability requires that

 (1)

where denotes intersection ("and"), and also that

 (2)

Therefore,

 (3)

Now, let

 (4)

so is an event in and for , then

 (5)
 (6)

But this can be written

 (7)

so

 (8)

(Papoulis 1984, pp. 38-39).

Conditional Probability, Inclusion-Exclusion Principle, Independent Statistics, Total Probability Theorem

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## References

Papoulis, A. "Bayes' Theorem in Statistics" and "Bayes' Theorem in Statistics (Reexamined)." §3-5 and 4-4 in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 38-39, 78-81, and 112-114, 1984.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 810, 1992.

## Cite this as:

Weisstein, Eric W. "Bayes' Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BayesTheorem.html