
Re: interesting probability problem
Posted:
Oct 16, 2007 9:49 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).
That can't be right. Try 2p/(1 + p).
> 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?
A strangely worded question. I assume that "we make an observation of a cow or a crow, with probability q and 1q" means this: we toss a biased coin, or whatever, to decide whether to observe a cow or a crow. If we need to observe a crow then we pick one crow at random from the population of crows, and see whether it is white or black. Why bother, one might ask, since we are told to "assume all crows are black".
But anyway, observing the colour of a predetermined noncow object (such as a crow) can't possibly affect the probability that all cows are white. What *would* have an effect is if we pick a nonwhite object at random and find that it is not a cow (for example, it is a crow). The key thing is that we *could* have picked a nonwhite cow if one existed. Since the number of observable nonwhite objects is extremely large (though presumably finite), the effect is tiny.

