Conditional Probability

Once you know an event happened, every outcome outside it becomes impossible, so the chance of anything else has to be recalculated inside that smaller world. That is why learning a new fact changes the odds of something else that already happened: you are not changing the past, you are changing which outcomes are still in play. P(A given B) asks: out of all the times B happens, how often does A happen too? The formula P(A|B) = P(A and B) / P(B) just says to divide the overlap by the new, smaller sample space.

In a two-way table, conditioning means picking one row or one column and ignoring the rest of the table entirely. If B is 'tested positive,' you throw away everyone who tested negative, then ask what fraction of the remaining people actually have the disease. That is why P(A|B) and P(B|A) are usually different numbers: P(A|B) divides by the total in row B, while P(B|A) divides by the total in row A, and those row totals are rarely equal. Confusing the two is one of the most common errors in probability.

This idea is the engine behind Bayes' rule, which is really just conditional probability applied twice: once to find the overlap P(A and B), and once to find the total P(B) by adding up every way B could happen. It also connects forward to independence, since A and B are independent exactly when knowing B teaches you nothing, meaning P(A|B) equals P(A). Testing problems make the stakes clear: as a disease gets rarer, false positives from the enormous healthy population start to outnumber true positives, so a positive test means less and less.