= Joint Probability = A '''joint probability''' is the likelihood of multiple events occurring. <> ---- == Description == This is the probability of a set of two (or more) events occurring. This coincidence of events is an '''intersection''', which notated for e.g. two events as ''A∩B''. In some cases, especially when the events feature binary outcomes, an intersection is informally notated like ''P(AB)''. These events can be sequential or simultaneous. To demonstrate the validity of the latter, note that the probability of ''A∩B'' is the same as the probability of ''B'' multiplied by the [[Statistics/ConditionalProbability|conditional probability]] of ''A'' given ''B'': ''P(A∩B) = P(A|B) P(B)''. === Expected Values === For discretely distributed X and Y, the [[Statistics/ExpectedValues|expected value]] is generally ''Σ,,y,,Σ,,x,, x y P(x,y)''. For continuously distributed X and Y, the expected value is generally ''∫∫,,Ω,, x y P(x,y) dxdy''. === Independence === '''Independence''' means the occurrence of one event has no impact on the probability of another event. For example, given two independent events ''A'' and ''B'', the joint probability is simply the product of their individual probabilities: ''P(A∩B) = P(A) P(B)''. ---- CategoryRicottone