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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 $P(A)=0$?

Teddy38
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TheHopefulActuary
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  • Hey there Kyle. You have asked 6 questions now, but you haven't accepted any of the answers given. Please review the answers to your other questions and accept some of the answers. – Thomas Oct 09 '12 at 17:31
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    According to your title, you're trying to show that $A$ and $B$ are independent. According to your first sentence, you're assuming that $A$ and $B$ are independent. This should be an easy proof then. – Chris Eagle Oct 09 '12 at 17:31
  • I just realized that Thomas, thanks for the heads up, I'm going to start doing that now. I'm pretty new to using this. – TheHopefulActuary Oct 09 '12 at 17:37

1 Answers1

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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.

GeoffDS
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  • This is not the test of independence. The test of independence is that $P(A)$ is the same regardless of whether $B$ happens. It implies that $P(AB)=P(A)P(B)$, but that is not what you asked. – Ross Millikan Oct 09 '12 at 17:37
  • My appologies everyone, I misunderstood the problem a little and then asked for help the completely wrong way. Thanks for your tries. – TheHopefulActuary Oct 09 '12 at 17:45
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    @Ross: Wikipedia 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. – hmakholm left over Monica Oct 09 '12 at 18:57