Simple Math Help

[Poochymama]

For some reason I'm having trouble wrapping my brain around what seems to be a simple math concept.

I try to say this in the most general way possible

If circumstance P is true, then there is a 80% chance of x being true and a a 20% chance of y being true.

If circumstance W is true, then there is an 86% chance of x being true and a 14% chance of y being true.

Now lets say both P and W are true, what is the chance that x is true?

[TheNoobie]

No offense but why exactly are you asking this on C++ forums?

[Disch]

Judging from how the question is worded:

86% chance of x being true
20% chance of y being true

[m4ster r0shi]

I'd say the chance that x is true is 68.8%

If circumstance P is true, then there is a 80% chance of x being true and a a 20% chance of y being true.

This means that Prob(x/P)=0.8 and Prob(y/P)=0.2.

If circumstance W is true, then there is an 86% chance of x being true and a 14% chance of y being true.

This means that Prob(x/W)=0.86 and Prob(y/W)=0.14.

What is asked here is Prob(x/PW).

By definition, it's Prob(A/B)=Prob(AB)/Prob(B).
Therefore, Prob(x/P)=Prob(xP)/Prob(P)=0.8 and Prob(x/W)=Prob(xW)/Prob(W)=0.86.
Also, assuming P and W are independent, Prob(x/PW)=Prob(xPW)/Prob(PW)=Prob(xPW)/[Prob(P)*Prob(W)]

Now, if P and W are independent, so are xP and xW,
which means that Prob(xPW)=Prob(xPxW)=Prob(xP)*Prob(xW).

It easily follows that Prob(x/PW)=Prob(x/P)*Prob(x/W)=0.8*0.86=0.688.

Useful link -> http://en.wikipedia.org/wiki/Conditional_probability

EDIT: Mmm... This looks good in theory, but suppose we had this:

If circumstance P is true, then there is a 80% chance of x being true.
If circumstance W is true, then there is an 100% chance of x being true.

Then, if P and W are true, common sense says
that there's a 100% chance of x being true...
I think I'm missing something here...
I'll just go back to watching my movie...

EDIT2:

m4ster r0shi wrote:

Now, if P and W are independent, so are xP and xW

This is wrong. I'll just have to assume that xP and xW are independent too.

[m4ster r0shi]

Ok, the answer here really depends on what Prob(PW) is.

If P and W are independent,
EDIT: and xP and xW are independent,
then Prob(PW)=Prob(P)*Prob(W) and the answer is 68.8%.

If P is a subset of W, then PW=P. Therefore, Prob(PW)=Prob(P),
and the answer is Prob(x/PW)=Prob(xPW)/Prob(PW)=Prob(xP)/Prob(P)=Prob(x/P)=80%.

If W is a subset of P...
Well, this can't happen, because if it did, Prob(x/W) should not be greater than Prob(x/P).

Finally, here's an example where Prob(x/P)=50%, Prob(x/W)=100% and
Prob(x/PW)=100% -> http://www.4shared.com/photo/kNSzD2gF/probabilities.html