I'm looking for feedback on the problem below from Jeffreys' probability primer. I think (a) is 0.0009 and (b) is 1 in 1000. Is this correct?
(a) In an urn with 1000 balls, one is green and the rest are red. A ball is drawn at random and seen by no one but a slightly colorblind witness, who reports that the ball was green. What is your probability that the witness was right on this occasion, if his reliability in distinguishing red from green is .9, i.e., if pr(He says it is X|It is X) = .9 when X = Red and when X = Green?
(b) Suppose that the balls were all numbered, from 1 to 1000, and the witness knows this fact. A ball is drawn, and he tells me that it was numbered 25, what is the probability that he is right?” In answering you are to “assume that, there being no apparent reason why he should choose one number rather than another, he will be likely to announce all the wrong ones equally often.”
Thanks!
