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P(A|B) may or may not be equal to P(A), i.e., the unconditional probability or absolute probability of A. If P(A|B) = P(A), then events A and B are said to be independent: in such a case, knowledge about either event does not alter the likelihood of each other. P(A|B) (the conditional probability of A given B) typically differs from P(B|A).
The additional input may be (a) a symmetry (invariance group); (b) a sequence of events B n such that B n ↓ B, P ( B n) > 0; (c) a partition containing the given event. Measure-theoretic conditioning (below) investigates Case (c), discloses its relation to (b) in general and to (a) when applicable.
In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events , hence the name.
P(A) is the proportion of outcomes with property A (the prior) and P(B) is the proportion with property B. P(B | A) is the proportion of outcomes with property B out of outcomes with property A, and P(A | B) is the proportion of those with A out of those with B (the posterior). The role of Bayes' theorem is best visualized with tree diagrams.
For working with , it is important that it be regular, that is: . For almost all x, (,) is a probability measure; For all A, (,) is a measurable function; In other words is a Markov kernel.
The opposite or complement of an event A is the event [not A] (that is, the event of A not occurring), often denoted as ′,, ¯,,, or ; its probability is given by P(not A) = 1 − P(A). [31] As an example, the chance of not rolling a six on a six-sided die is 1 – (chance of rolling a six) = 1 − 1 / 6 = 5 / 6 .
In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
If just one card is drawn from the deck, either a red card (heart or diamond) or a black card (club or spade) will be drawn. When A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B). [3] To find the probability of drawing a red card or a club, for example, add together the probability of drawing a red card and the probability of drawing a ...