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| The intersection of two events ''A'' and ''B'' is the same as event ''A'' times the [[Statistics/ConditionalProbability|conditional probability]] of ''B'' given ''A''. | 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)'' |
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| ''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,,) | ''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)''). |
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).
Contents
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)).