Conditional Probability

Algebra 2 · Axiom Academy

LESSON Conditional Probability How a probability changes once you know something has happened — the restricted sample space behind P(A given B). 1. Once B Happens, B Is the New Whole Conditional probability asks a sharper question than plain probability: given that event B has already happened , how likely is event A? We write it P(A given B) and read it "the probability of A given B." Let A = "roll a 6" and B = "roll an even number." With no information, P(A) = 1/6. But once you know the roll is even, only 2, 4, and 6 are still possible — three outcomes, one of them a 6 — so the probability jumps to 1/3. 2. The Formula: The Share of B That Is Also A To measure "how much of B is also A," divide the overlap by the size of the condition. That single ratio is the definition of conditional probability. Divide by P(B) because B — not the original sample space — is now the whole. Reading it off a two-way table 100 students, sorted by whether they play a sport (S) and whether they made the honor roll (H): Restrict to the sport column (50 students); 30 of them made honor roll: 3. Independence: When B Tells You Nothing Sometimes learning B does not change A at all. Then A and B are independent , and the conditional probability collapses right back to the plain one. P(King) = 4/52 = 1/13, and P(King given Red) = 2/26 = 1/13. Equal — the colour tells you nothing about the rank. P(King) = 1/13, but P(King given Face) = 4/12 = 1/3. Not equal — "face card" makes a King far more likely.

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