Conditional Probability
Conditional probability is the chance of one event given that another has already happened — it's how you update a probability once you learn something new.
Definition
The conditional probability P(A | B) is the probability that event A occurs given that event B has occurred. Learning that B happened narrows the sample space down to just the outcomes inside B, and you ask what fraction of those also give A. It is the formal version of "now that I know this, how do the odds change?"
The formula & procedure
P(A | B) = P(A and B) / P(B), as long as P(B) > 0. Rearranged, this is the multiplication rule: P(A and B) = P(B)·P(A | B). Two events are independent exactly when P(A | B) = P(A) — knowing B tells you nothing about A. In a two-way table, P(A | B) is just the count in the A-and-B cell divided by the total for group B.
Worked example
A music app labels 500 saved tracks by mood and tempo. 180 tracks are "upbeat," and of those, 120 are also "fast." What is P(fast | upbeat)? Restrict attention to the 180 upbeat tracks; 120 of them are fast, so P(fast | upbeat) = 120/180 ≈ 0.667. Compare that with the overall fast rate: if 200 of all 500 tracks are fast, then P(fast) = 200/500 = 0.40. Because 0.667 ≠ 0.40, "fast" and "upbeat" are not independent — knowing a track is upbeat raises the chance it's fast.
The classic mistake
Swapping the condition — treating P(A | B) as if it equals P(B | A). These are usually different: P(fast | upbeat) is not the same as P(upbeat | fast). Two related slips are assuming events are independent without checking, and dividing by the whole grand total instead of the conditioning group's total.
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