In article <4714D24D.firstname.lastname@example.org>, Andersen <email@example.com> writes: > firstname.lastname@example.org wrote: > >>> Solutions to a) is p, and b) is 2p/(1-p). >> >> That can't be right. Try 2p/(1 + p). > > Sorry about that. You are absolutely right, 2p/(1+p) it is. > >> A strangely worded question. I assume that "we make an observation of >> a cow or a crow, with probability q and 1-q" 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". > > I don't find that strange. That just means that we want to verify our > hypothesis, and when we go out to the jungle, we observe cows 100q > percent of the time, and 100(1-q) percent of the time. Lets say we will > do 30 experiments, then this model is better, than maybe having to stick > around forever to wait for a crow, if 1-q is very small. > >> But anyway, observing the colour of a pre-determined non-cow 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 non-white >> 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 non-white cow if >> one existed. Since the number of observable non-white objects is >> extremely large (though presumably finite), the effect is tiny. > > I am not sure I follow. But could I rephrase it as follows. If we had no > information saying "all crows are black", then the two probabilities in > a) and b) should coincide?
One of the points I see being made is that if the population of objects is fixed and finite then by the process of elimination, sampling a crow leaves one less object that could be a non-white cow.
Suppose that I have a bag. In the bag are four statuettes. I tell you that the statuettes are of cows or crows and that they are painted black or white.
I further tell you that I populated this bag by flipping a coin eight times total to determine the color and type of each of the four statuettes.
You draw an item from the bag at random and use it as evidence for or against the assertion:
"All the statuettes of cows in the bag are painted white".
Now we have some symmetry to work with. Now drawing a black crow is supporting evidence that is just as strong as drawing a white cow. And drawing a white crow is also equally strong evidence. Each of the three leaves one less item that might serve as a counter-example to the stated assertion.