= Joint Probability = A '''joint probability''' is the likelihood of multiple events occurring, either simultaneously or sequentially. The math notation is ''P(A ∩ B)'', or sometimes more informally ''P(AB)''. <> ---- == Decomposition == The intersection of two events ''A'' and ''B'' is the same as event ''B'' times the [[Statistics/ConditionalProbability|conditional probability]] of ''A'' given ''B''. ''P(A∩B) = P(A|B) P(B)'' This can be expanded out to any number of events. ''P(A,,1,, ∩ A,,2,, ∩ ... ∩ A,,n-1,, ∩ A,,n,,) = P(A,,1,,|A,,2,, ∩ ... ∩ A,,n,,) ... P(A,,n-1,,|A,,n,,) P(A,,n,,)'' ---- == Independence == If two events are [[Statistics/Independence|independent]], then the joint probability is simply the product of their individual probabilities. ''P(A∩B) = P(A) P(B)'' This comes trivially from the above decomposition and the definition of independence (i.e., ''P(A|B) = (A)''). ---- == Dependence == If two events are instead [[Statistics/Independence|dependent]], the joint probability is not directly calculable. ---- CategoryRicottone