SOLUTION: If A and B are events in the same space such that P(A)=0.7, P(B)=0.2, and P(A∩B)=0.1, compute P(A∩B)¯. [input a number, e.g. 0.123]

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Question 1089593: If A and B are events in the same space such that P(A)=0.7, P(B)=0.2, and P(A∩B)=0.1, compute P(A∩B)¯. [input a number, e.g. 0.123]
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!


Let w,x,y, and z be the probabilities of those regions in
the Venn diagram. 

Then 

P(A) = 0.7 = w + x
P(B) = 0.2 = x + y
P(A∩B) = 0.1 = x
P(AUBUCUD) = 1 = w + x + y + z

Substituting 0.1 for x from P(A∩B) = 0.1 = x

P(A) = 0.7 = w + 0.1
P(B) = 0.2 = 0.1 + y
P(AUBUCUD) = 1 = w + 0.1 + y + z

0.7 = w + 0.1        0.2 = 0.1 + y
0.6 = w              0.1 = y

Substituting those:

P(AUBUCUD) = 1 = 0.6 + 0.1 + 0.1 + z

  1 = 0.6 + 0.1 + 0.1 + z
  1 = 0.8 + z
0.2 = z

So the Venn diagram is now:



That's the Venn diagram.

I don't know what you mean by  P(A∩B)¯. I've never seen that
line up there before.

Is it the complement of P(A∩B)?  If so, all we needed to do was
subtract P(A∩B) from 1 or 1 - 0.1 = 0.9.  But surely they wouldn't 
have given you all that other un-needed information if that was all 
they wanted.  If you know what regions you wanted just add up the 
probabilities in those regions.

If you like, you can explain what you mean by P(A∩B)¯ in the 
thank-you note form below and I'll get back to you by email.

Edwin
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