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15.2: Conditional Probabilities and Bayes Rule

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    Let A and B be random events with probabilities P(A) and P(B). We can now say that the probability P(A ∩ B) that event A and B happen is given by

    \[P(A\cap B) = P(A)P(B|A) = P(B)P(A|B)\]

    Here, P(B|A) is the conditional probability that B happens, knowing that event A happens. Likewise, P(A|B) is the probability that event A happens given that B happens.

    Bayes’ Rule relates a conditional probability to its inverse. In other words, if we know the probability of event A to happen given that event B is happening, we can calculate the probability of B to occur given that A is happening. Bayes’ rule can be derived from the simple observation that the probability of A and B to happen together (P(A∩B)) is given by P(A)P(B|A) or the probability of A to happen and the probability of B to happen given that A happens (Equation C.4). From this, deriving Bayes’ rule is straightforward:

    \[P(A|B)=\frac{P(A)P(B|A)}{P(B)}\]

    In words, if we know the probability that B happens given that A happens, we can calculate that A happens given that B happens.

    This page titled 15.2: Conditional Probabilities and Bayes Rule is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Nikolaus Correll via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.