The conditional probability of an event assuming that has occurred, denoted , equals
(1)
which can be proven directly using a Venn diagram. Multiplying through, this becomes
(2)
which can be generalized to
(3)
Rearranging (1) gives
(4)
Solving (4) for and plugging in to (1) gives
(5)
See alsoBayes' Theorem,
Fermat's Principle of Conjunctive Probability,
Total Probability Theorem Explore this topic in the MathWorld classroom Explore with Wolfram|AlphaMore things to try:
ReferencesPapoulis, A. "Conditional Probabilities and Independent Sets." ยง2-3 in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 33-45, 1984. Referenced on Wolfram|AlphaConditional Probability Cite this as:Weisstein, Eric W. "Conditional Probability." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConditionalProbability.html
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