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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. |
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| == Decomposition == | == Description == |
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| The intersection of two events ''A'' and ''B'' is the same as event ''B'' times the [[Statistics/ConditionalProbability|conditional probability]] of ''A'' given ''B''. | This is the probability of a set of two (or more) events occurring. This coincidence of events is an '''intersection''', which notated for e.g. two events as ''A∩B''. |
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| ''P(A∩B) = P(A|B) P(B)'' | In some cases, especially when the events feature binary outcomes, an intersection is informally notated like ''P(AB)''. |
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| 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,,)'' ---- |
These events can be sequential or simultaneous. To demonstrate the validity of the latter, note that the probability of ''A∩B'' is the same as the probability of ''B'' multiplied by the [[Statistics/ConditionalProbability|conditional probability]] of ''A'' given ''B'': ''P(A∩B) = P(A|B) P(B)''. |
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| == Independence == | === Independence === |
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| 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. |
'''Independence''' means the occurrence of one event has no impact on the probability of another event. For example, given two independent events ''A'' and ''B'', the joint probability is simply the product of their individual probabilities: ''P(A∩B) = P(A) P(B)''. |
Joint Probability
A joint probability is the likelihood of multiple events occurring.
Contents
Description
This is the probability of a set of two (or more) events occurring. This coincidence of events is an intersection, which notated for e.g. two events as A∩B.
In some cases, especially when the events feature binary outcomes, an intersection is informally notated like P(AB).
These events can be sequential or simultaneous. To demonstrate the validity of the latter, note that the probability of A∩B is the same as the probability of B multiplied by the conditional probability of A given B: P(A∩B) = P(A|B) P(B).
Independence
Independence means the occurrence of one event has no impact on the probability of another event. For example, given two independent events A and B, the joint probability is simply the product of their individual probabilities: P(A∩B) = P(A) P(B).