The Bets

http://www.docstoc.com/docs/26993860/Artificial-Reasoning 30/85 - The Bets
Bets on B (5/8) - you bet $5 and bookmarker bets $3
Bets on ~B (3/4) - you bet $6 and bookmarker bets $2

You suffer a net loss both ways. See Pay Off Matrix.

So it needs to be P(B) = P(~B)

or if a and b are mutually exclusive then P(avb) = P(a) + P(b)

so subjective bliefs (betting rates) should follow the basic rules of probability so that there's no sure-loss

http://en.wikipedia.org/wiki/Bayes%27_theorem
P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}.
P(A|B) = (P(B|A)*P(A))/P(B)