Math 2.0

definition

Conditional probability

Definition

Let (Ω,F,P)(\Omega, \mathcal{F}, P) be a probability space and let A,B∈FA, B \in \mathcal{F} be events with P(B)>0P(B) > 0. The conditional probability of AA given BB is P(A∣B)=P(A∩B)P(B).P(A \mid B) = \frac{P(A \cap B)}{P(B)}.

Remarks

For fixed BB with P(B)>0P(B) > 0, the map A↦P(A∣B)A \mapsto P(A \mid B) is a probability measure on (Ω,F)(\Omega, \mathcal{F}). It gives all its mass to BB. For one roll of a fair die, P({6}∣even)=(1/6)/(1/2)=1/3P(\{6\} \mid \text{even}) = (1/6) / (1/2) = 1/3. In general P(A∣B)≠P(B∣A)P(A \mid B) \ne P(B \mid A). Bayes' theorem relates the two. The definition does not apply when P(B)=0P(B) = 0. Conditioning on such events needs the conditional expectation.

Explanations

L1 human

Once you know that BB happened, the outcomes outside BB are no longer possible. So you keep only the part of AA inside BB, and you rescale by P(B)P(B) so that the total is 11 again.

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  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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Sources

  1. Web https://en.wikipedia.org/wiki/Conditional_probability, Kolmogorov definition

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