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| 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)''. | 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)''. |
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.