prove that two events are independent if P(A|B) = P(A| not B)

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$P(A|B) = P(A|\overline{B})$ by definition is $$ \frac{P(A \cap B)}{P(B)} = \frac{P(A\cap \overline{B})}{P(\overline{B})} $$ Now using the fact $ P(A \cap B) +P(A \cap \overline{B}) = P(A) $ ,we get :$$ P(A \cap B) (1- P(B)) = (P(A) - P(A \cap B)).P(B) \implies P( A \cap B) - P( A \cap B)P(B) = P(A).P(B) - P(A \cap B).P(B) \implies P(A \cap B) = P(A).P(B) $$ Thus they are independent by definition.

Also intuitively , if they are independent, then the conditional probability of $A$ w.r.t $B$ and $\overline{B}$ has to be same as $A$ doesn't depend whether $B$ has occurred or not.

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Iqbal Hafizi
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Iqbal Hafizi

Updated on August 01, 2022

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  • Iqbal Hafizi
    Iqbal Hafizi over 1 year

    Prove that the two events $A$ and $B$ are independent if : $$ P( A|B) = P(A| \overline{B}) $$

  • Iqbal Hafizi
    Iqbal Hafizi almost 6 years
    thanks for your answer. can you tell me that how P(A and B) = P(A) * P(B) do you assume that they are independent ?
  • Chirantan Chowdhury
    Chirantan Chowdhury almost 6 years
    its just comes from the previous equation just cancelling the term $P(A \cap B).P(B)$
  • Iqbal Hafizi
    Iqbal Hafizi almost 6 years
    first you did a cross multiplication to get P(A and B).(1-P(B)) = [P(A)-P(A and B)].P(B) this I understand. but at next step I don't know how did you write P(A and B) - P(A and B)P(B) ? is it because you expanded the brackets from previous step, if so then it means P(B) is multiplied inside the bracket ?
  • Chirantan Chowdhury
    Chirantan Chowdhury almost 6 years
    @IqbalHafizi yes i just wrote it by just expanding.