Need help understanding 2 different but similar probability formulas

[zzzsilver]

Can anyone give me some basis for these formulas or how you remember them? Thanks!
P(A \(B ∪ C)= P(A) − P(A ∩ B) − P(A ∩ C) + P(A ∩ B ∩ C)
For the first one, I thought that it was a (|) sign which means given but it really is (\) ( I just edited the problem where it says: P(A \(B ∪ C)... Can someone tell me if that means anything different than the (\) given sign? I think that is part of my problem... When I work it out I get the probability of A but not B U C....

P(A∪ B |C) = P(A|C) + P (B|C) - P(A∩ B |C)

[Al Bundy]

you don't need to , draw a venn diagram...
i did it for the first one just now...
shade in what is given.. all of circles b and c
and you want the whole circle A as P(A)
then if you subtract the a and b oval, the b and c oval and add back in the part in the middle (A and b and c) because it was included twice in those last subtractions...you have all of A remaining

[zzzsilver]

Ok got the first one still haven't figured out the second one

I was not familiar with the set notation "\". Based on going through problems using the said notation, I figured that it means " not including" so for the first problem that I was trying to figure out P(A \(B ∪ C) what that means is the probability of A not including B U C- and not P (A given B U C ) as I had first assumed. If anyone is familiar with this notation, please let me know if I am correct in my assumption..
PS the first one is the "|" sign. If anyone gets this equation P(A∪ B |C) = P(A|C) + P (B|C) - P(A∩ B |C) please let me know! Thanks!

[NoMoreExams]

I was not familiar with the set notation "\". Based on going through problems using the said notation, I figured that it means " not including" so for the first problem that I was trying to figure out P(A \(B ∪ C) what that means is the probability of A not including B U C- and not P (A given B U C ) as I had first assumed. If anyone is familiar with this notation, please let me know if I am correct in my assumption..
PS the first one is the "|" sign. If anyone gets this equation P(A∪ B |C) = P(A|C) + P (B|C) - P(A∩ B |C) please let me know! Thanks!

Not sure what your question is, A \ B means everything in A that's not also in B or A n B^{-1} or some equivalent notation. A|B means "A given B" or (A n B) / B

[zzzsilver]

Ok that's what I figured. Thank you for your clarification for equation 1... I was also wondering how to interpret this equation P(A∪ B |C) = P(A|C) + P (B|C) - P(A∩ B |C) but I think I get it now- it's basically saying that we are looking within the subset C and trying to find A U B and to do such we should find the prob of A that is in C and the prob of B that is within C and subtract the intersection- so essentially it is finding the P (A) + P (B) - PA ∩ B) but all considering that C is the full sample space....

[NoMoreExams]

Ok that's what I figured. Thank you for your clarification for equation 1... I was also wondering how to interpret this equation P(A∪ B |C) = P(A|C) + P (B|C) - P(A∩ B |C) but I think I get it now- it's basically saying that we are looking within the subset C and trying to find A U B and to do such we should find the prob of A that is in C and the prob of B that is within C and subtract the intersection- so essentially it is finding the P (A) + P (B) - PA ∩ B) but all considering that C is the full sample space....

You can see it from purely a mathematical standpoint of writing out the def.

P(A U B | C) = P([A U B] n C] / P[C] = P[(A + B - A n B) n C] / P[C] = P[A n C] / P[C] + P[B n C] / P[C] - P[A n B n C] / P[C] = P(A|C) + P(B|C) - P(A n B | C)

[zzzsilver]

Ok i see now... The key to solving that mathematically was seeing that the numerator P[(A + B - A n B) n C] could be distributed into several different probability statements... Thank you for that clarification!