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 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(A1 ∩ A2 ∩ ... ∩ An-1 ∩ An) = P(A1|A2 ∩ ... ∩ An) ... P(An-1|An) P(An)


Independence

If two events are 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 dependent, the joint probability is not directly calculable.


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Statistics/JointProbability (last edited 2024-03-19 15:03:29 by DominicRicottone)