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Bayes combined the decomposition with [[Statistics/JointProbability|joint probability identities]] to arrive at this more solvable theorem.

Conditional Probability

A conditional probability is the likelihood of an event happening given that another event happens. The math notation is P(A|B), as in the probability of A given B.

Contents

  1. Conditional Probability
    1. Decomposition
      1. Bayes Theorem
    2. Independence
    3. Conditional Independence

Decomposition

decomposition.svg

Bayes Theorem

Bayes combined the decomposition with joint probability identities to arrive at this more solvable theorem.

bayes.svg

Independence

If two events are independent, then probabilities of one do not change from being conditioned on the other.

Put simply, if the conditioning probability is not 0, then:

P(A|B) = P(A)

P(B|A) = P(B)

A conditioning probability of 0 will cause the conditional probability to be undefined.

Conditional Independence

If events A and B are conditionally independent, then:

P(A|B,C) = P(A|C)

P(A,B|C) = P(A|C) P(B|C)

This interrelation is sometimes notated as ((A⫫B)|C).

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Statistics/ConditionalProbability (last edited 2024-03-19 19:42:37 by DominicRicottone)