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).
See also
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
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