How to work out this probability proof?
If Π(A) denotes the odds on A and Π(A|B) the odds on A given B, deduce that
Π(A|B) / Π(A) = P(B|A) / P(B |Ā )
I really am struggling with this this question, even though it seems pretty straightforward. Some help will be hugely appreciated.
- SamwiseLv 71 decade agoFavorite Answer
Hey! I thought this WAS a mathematics forum!
If you'll excuse me, I'm going to write Ā as ~A, because I don't have an equivalent character for ~B and I'm going to need one.
Now, let's simplify the whole business by considering all the combined cases possible with respect to the occurrence of events A and B: let
P(A & B) = q/(q+r+s+t)
P(A & ~B) = r/(q+r+s+t)
P(~A & B) = s/(q+r+s+t)
P(~A & ~B) = t/(q+r+s+t)
We've just reduced the model to a typical Venn diagram consisting of two partially overlapping circles inside a larger circle, with the four spaces thus created marked q, r, s, and t. We can now express odds and probabilities using these four variables.
Π(A|B) = q/s
Π(A) = (q+r)/(s+t)
P(B|A) = q/(q+r)
P(B|~A) = s/(s+t)
The proof is now just a matter of algebraic manipulation:
Π(A|B) / Π(A)
= (q/s) / [(q+r)/(s+t)]
= (q/s) * [(s+t)/(q+r)]
= P(B|A) / P(B|~A)
It is pretty straightforward, but only after finding a way to express odds and probabilities, including conditional ones, in terms of a common set of variables.