
Re: interesting probability problem
Posted:
Oct 16, 2007 8:47 AM


On Oct 16, 11:25 am, Andersen <andersen_...@hotmail.com> wrote: > Bertsekas has the following exercise in his probability book: > > " > Consider a statement whose truth is unknown. If we see many examples > that are compatible with it, we are tempted to view the statement as > more probable. Such reasoning is often referred to as inductive > inference (in a philosophical, rather than mathematical sense). Consider > now the statement that ?all cows are white.? An equivalent statement is > that ?everything that is not white is not a cow.? We then observe > several black crows. Our observations are clearly compatible with the > statement, but do they make the hypothesis ?all cows are white? more likely? > > To analyze such a situation, we consider a probabilistic model. Let us > assume that there are two possible states of the world, which we model > as complementary events: > > A : all cows are white, > C(A) : 50% of all cows are white. [C(A) is the complement event of A]. > > Let p be the prior probability P(A) that all cows are white. We make an > observation of a cow or a crow, with probability q and 1?q, > respectively, independently of whether event A occurs or not. Assume > that 0 < p < 1, 0 < q < 1, and that all crows are black. > > (a) Given the event B = {a black crow was observed}, what is P(AB)? > (b) Given the event C = {a white cow was observed}, what is P(AC)? > " > >  > Solutions to a) is p, and b) is 2p/(1p). From this he draws the > conclusion that a) does not affect the hypothesis A, while b) > strengthens it. Is this reasoning correct? I mean event B should have > the same effect as event C, as B supports A, since A is equivalent to > "everything that is not white is not a cow". What is wrong in my reasoning?
Two quick observations. First, the solution to b) must be wrong, because the expression is > 1 if p is > 1/3. Second, the states of nature A and C(A) are very strangely chosen, as there is a lot of middle ground. C(A) should be "There exists a cow that is not white". If you then try to apply Bayes' theorem, you find that P(Ccomp(A)) is no longer 1/2.

