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Bayes' Theorem
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Bayes' theorem for reversing conditional probabilities, using prior and posterior probabilities with two events or three mutually exclusive events, e.g. P(B∣A)=P(A∣B)P(B)+P(A∣B′)P(B′)P(A∣B)P(B) and its three-event form.
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Bayes' Theorem with 2 Events
AHL 4.13
Bayes' theorem allows us to reverse conditional probabilities, determining the probability of an event based on prior knowledge of another event. If we have events A and B, Bayes' theorem states:
P(B∣A)=P(B)P(A∣B)+P(B′)P(A∣B′)P(A∣B)P(B)📖
In other words, Bayes' theorem lets us update our beliefs or predictions after observing new evidence. It's particularly useful when dealing with sequential information or adjusting probabilities based on new data.
Bayes' Theorem with 3 Events
AHL 4.13
In some cases, instead of complementary events B and B′, we have complementary events B1, B2 and B3. In this case Bayes' theorem can be generalized to
P(B1∣A)=P(B1)P(A∣B1)+P(B2)P(A∣B2)+P(B3)P(A∣B3)P(B1)P(A∣B1)
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