Question 1117717
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As I read your question, the actual given information is<br>
P(A|B)=0.30; P(B|A)=0.60; P(A and B)=0.18.<br>
I personally find it easiest to see what is going on in this kind of problem using a Venn diagram.<br>
Whether with a Venn diagram or with symbols, P(A|B)=0.30 means that P(A and B) is only 0.30 times P(B):<br>
0.18 = 0.30*P(B)  -->  P(B) = 0.18/0.30 = 0.60<br>
Then since P(B) is 0.60 and P(A and B) is 0.18, P(B and A') is 0.60-0.18 = 0.42.<br>
Similarly, P(A and B)=0.18 and P(B|A)=0.60 means<br>
0.18 = 0.60*P(A)  -->  P(A) = 0.18/0.60 = 0.30<br>
and that makes P(A and B') = 0.30-0.18 = 0.12.<br>
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The preceding is a formal way of solving your problem.<br>
Now here is the actual method I used for getting the answer of P(B and A')=0.42.<br>
(Look at a Venn diagram to follow the logic of this method.)<br>
Since P(A|B)=0.30, P(A'|B)=0.70.<br>
Then the ratio of P(B and A') to P(B and A) is 0.70:0.30.<br>
And so P(B and A') is {{{((0.70)/(0.30))*0.18 = 0.42}}}