Probabilidade Condicional

Conditional probability

Conditional probability tells you how likely event A is once you already know B happened: P(A|B) = P(A∩B) / P(B), which only makes sense when P(B) > 0. When P(A|B) = P(A), the two events are independent and learning about B tells you nothing new about A. Watch out for the inverse conditional fallacy, the habit of treating P(A|B) and P(B|A) as if they were interchangeable. Forensics has a famous version of this: "probability of innocence given a DNA match" is not the same thing as "probability of a match given innocence". To see it concretely, take a deck of 52 cards. P(king|face card) = P(king ∩ face) / P(face) = (4/52)/(12/52) = 1/3.

Applications

It turns up everywhere you reason from partial information. Medicine reads it as the sensitivity and specificity of a diagnostic test. Bayes' theorem uses it to update priors. Machine learning relies on it for Naive Bayes classifiers and for conditional independence in graphical models. Forensics needs it to interpret DNA matches without falling into the prosecutor's fallacy, and insurers use it to price risk. Weather forecasting leans on it too.

FAQ

Why is P(A|B) different from P(B|A)? The denominators differ, P(B) for one and P(A) for the other, so the two only line up when P(A) = P(B). Mixing them up is exactly what the prosecutor's fallacy does.

What if P(B) = 0? Then the conditional probability has no value at all. There is nothing to condition on if B never happens.

How does it relate to independence? A and B are independent precisely when P(A|B) = P(A), which is the same as saying P(A∩B) = P(A)·P(B).