Of course we are assuming that $A$ and $B$ are independent events. I know how to show that if $P(A)=1$ then $P(B)=P(AB)$, but how do we show that if $P(A)=0$?
How do you prove that no matter whether $P(A)=1$ or $0$, $A$ is independent from $B$.
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probability
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probability-distributions
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0I just realized that Thomas, thanks for the heads up, I'm going to start doing that now. I'm pretty new to using this. – 2012-10-09
1 Answers
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Kyle, from your title, it seems you are asking, if $P(A) = 0$, how can we prove that $A$ and $B$ are independent events? The condition that must hold for two events, $A$ and $B$, to be independent is
$P(AB) = P(A)P(B)$
So, if you want to prove $A$ and $B$ are independent, you need to show this. In this case, if $P(A) = 0$, what is the right hand side? And, since $P(AB)$ means the probability of $A$ and $B$ both happening, what do you think $P(AB)$ is when the probability of just $A$ happening is $0$?
Note, in your question, you actually ask something totally different. You assume $A$ and $B$ are independent and you want to prove $P(B) = P(AB)$. This is a vastly different question.
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1@Ross: [Wikipedia](http://en.wikipedia.org/wiki/Independence_%28probability_theory%29) uses $P(A\cap B)=P(A)P(B)$ as the definition. It seems to work better than $P(A)=P(A\mid B)$ especially when one of the events can have probability $0$: If $P(B)=0$, then $P(A\mid B)$ isn't even well defined, but an empty event is clearly independent of anything according to Wikipedia's definition. – 2012-10-09