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In Probability,

AA and BB are independent when

P(B∣A)=P(B)P(B|A) = P(B)

Whether AA happens or not does not affect the probability of BB.

By definition of P(B∣A)P(B|A),

P(A∩B)=P(A)P(B){P(A \cap B)} = P(A)P(B)

In this case, P(A∩B){P(A \cap B)} is the joint and P(A)P(B)P(A)P(B) is the marginal distributions.

If AA is independent of BB, BB is also independent of AA.

Furthermore, if AA and BB are independent, the following three are also independent.

  1. A&BCA \& B^C
  2. AC&BA^C \& B
  3. AC&BCA^C \& B^C