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A joint probability is the likelihood of multiple events occurring, either simultaneously or sequentially. The joint probability of two events is notated as ''P(A ∩ B)'', or sometimes more informally as ''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)''.
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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,,)''

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== 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)'').

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== Dependence ==

If two events are instead [[Statistics/Independence|dependent]], the joint probability is not directly calculable.

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

  1. Joint Probability
    1. Decomposition
    2. Independence
    3. Dependence

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)