help with venn diagram probabilities please

[natenatmom]

I need to correct a problem and turn it in again. The question gives the following probabilities... P(B)=1/4, P(A/B)=1/2, P(A/Bbar)=1/3. It ask you to determine P(A) and P(Bbar/A). I got 1/4 for A and and 3 for Bbar/A. The teacher said I was working with the right formula and that the formula remains the same even if the values change. I got part of the problem correct. It asked for P(ABbar) and I got 1/3 for that which was right. Can someone help me correct the other two parts?

 

[pka]

Please review what you have written!
Is it that you are given P(B)=(1/4), P(A|B)=(1/2) and P(A|B<SUP>-</SUP>)=(1/3)?
Are we to assume that B<SUP>-</SUP> is the complement of B?
If so the P(B)=1−P(B<SUP>-</SUP>).

Moreover, \(\displaystyle P(A|B) = \frac{{P(AB)}}{{P(B)}}\quad \Rightarrow \quad P(AB) = (1/2)(1/4) = 1/8\).
Is that what you understand to be the case?

Also \(\displaystyle P(AB^ - ) = (1/3)(3/4) = 1/4\) meaning \(\displaystyle P(A) = P(AB) + P(AB^ - ) = (1/8) + (1/4) = (3/8)\).

Please advise if I have misunderstood what you meant?

 

[natenatmom]

Yes, that is what is given by the question. I don't know about assuming B- is the complement of B. It doesn't say anything about that. I guess so ???

 

[pka]

But does what I posted help you?
You have given some answers that are not consistant with what I did.

 

[natenatmom]

Well no, I's still confused because in your formula for P(A), you give 1/4 as the value for P(ABbar) but I had gotten 1/3 as my answer for that and the teacher said it was correct.

 

[pka]

Well no, I's still confused because in your formula for P(A), you give 1/4 as the value for P(ABbar) but I had gotten 1/3 as my answer for that and the teacher said it was correct.

That is exactly what I feared!
In your first posting you said: \(\displaystyle P(B) = \frac{1}{4}\quad \& \quad P(A|\overline B ) = \frac{1}{3}\). Are those numbers correct?

Assuming that \(\displaystyle \overline B\) means the complement of the event \(\displaystyle B\) then we have \(\displaystyle P(\overline B ) = 1 - P(B) = \frac{3}{4}\).

Therefore, \(\displaystyle P(A|\overline B ) = \frac{{P(A\overline B )}}{{P(\overline B )}}\quad \Rightarrow \quad P(A\overline B ) = \left( {\frac{1}{3}} \right)\left( {\frac{3}{4}} \right) = \left( {\frac{1}{4}} \right)\).

Because I have no desire to confuse you, maybe you can show us how you got 1/3. Perhaps, I have misunderstood the notation, particularly Bbar; or even how the problem is using P(A/B). Is P(A/B) the same as “the probability of A given B”?