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I'm working through some slides for a probability course and without having anyone to ask, I'm having a hard time checking whether my reasoning is correct.
The exercise is as follows:

Let A and B be two groups of events with probabilities $P(A)=0.3$; $P(B)=0.4$ and $P(A\cap B)=0.2$. Find $P(\overline{A} \cap B)$.

It's trivial that

$P(\overline{A}) = \Omega - P(A) = 0.7$

However, how do I find $P(B\setminus A)$?

I know that

$P(A\cap B) = P(B)P(A\mid B)$

Should I therefore go with the following somehow?

$P(\overline{A}\cap B) = P(B)P(\overline{A}\mid B)$

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Note that since $A\cap B$ and $\overline{A}\cap B$ are disjoint, with union $B$, we have $$\Pr(B)=\Pr(\overline{A}\cap B)+\Pr(A\cap B).$$

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  • $\begingroup$ Thanks a lot, as often happens, the right solution is the simpler one. $\endgroup$ – Nit Feb 24 '15 at 18:50
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    $\begingroup$ You are welcome. I drew a picture, which makes everything clear. Then translated what the picture said into probability language. $\endgroup$ – André Nicolas Feb 24 '15 at 18:52

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