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Jan 27, 2017 · The conditional is the intersection of A and B divided by the whole area of B. But why is $P (A\cap B|C)/P (B|C) = P (A|B \cap C)$? Can you give some intuition? Shouldn't it be: $P (A\cap B \cap C)/P (B,C) = P (A|B \cap C)$? probability. conditional-probability. Share. Cite. Improve this question. Follow. edited Jan 27, 2017 at 9:21.
The conditional probability of A given B, denoted P(A ∣ B), is the probability that event A has occurred in a trial of a random experiment for which it is known that event B has definitely occurred. It may be computed by means of the following formula: P(A ∣ B) = P(A ∩ B) P(B)
If $A$ and $B$ are two events in a sample space $S$, then the conditional probability of $A$ given $B$ is defined as $$P (A|B)=\frac {P (A \cap B)} {P (B)}, \textrm { when } P (B)>0.$$. Here is the intuition behind the formula. When we know that $B$ has occurred, every outcome that is outside $B$ should be discarded.
Definition. Two events A and B are conditionally independent given an event C with P(C) > 0 if P(A ∩ B | C) = P(A | C)P(B | C) (1.8) Recall that from the definition of conditional probability, P(A | B) = P(A ∩ B) P(B), if P(B) > 0. By conditioning on C, we obtain P(A | B, C) = P(A ∩ B | C) P(B | C) if P(B | C), P(C) ≠ 0.
If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B ", or "the probability of A under the condition B ", is usually written as P (A|B) [2] or occasionally PB(A).
When A and B are independent, P (A and B) = P (A) * P (B); but when A and B are dependent, things get a little complicated, and the formula (also known as Bayes Rule) is P (A and B) = P (A | B) * P (B).
