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. 3839).
See also
Conditional Probability,
InclusionExclusion Principle,
Independent Statistics,
Total
Probability Theorem
Explore with WolframAlpha
References
Papoulis, A. "Bayes' Theorem in Statistics" and "Bayes' Theorem in Statistics (Reexamined)." §35 and 44 in Probability,
Random Variables, and Stochastic Processes, 2nd ed. New York: McGrawHill,
pp. 3839, 7881, and 112114, 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
MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/BayesTheorem.html
Subject classifications