On a table there are two bags. The first contains one white marble. The second bag contains one red and one white marble. Blindfolded, Billy takes one marble out of the second bag and puts it in the first bag. He then takes off his blindfold and takes a marble out of bag one. It is a white marble.
What is the probability that the other marble in the first bag is also a white marble?

Dutch National Science Quiz 12/24/2004

(In reply to re: more manually by Bruce)

I'm not convinced about it. I agree with Owl's precedent comment (which has been useful for me).

Really, Billy can't hold just "a white marble". When he picks up the marble, he picks a "concrete" white marble, w1 or w2. But while if he picks w2 he is sure that the marble remaining in the bag is w1, if he picks w1 he is in doubt about what marble rests in the bag: both w2 and r has the same probability (1/2) to be there in this case.

I think this is enough to see that, once Billy has picked up a white marble, the probability for the other white marble in the bag is higher than 1/2 (1/2 is just the probability in the case Billy has taken w1, but as he could also take w2, remaining w1 in the bag, probability increases). (To 2/3: not proven here).

As Owl points, this suppose that Billy is taking the marbles randomly. This fact has some importance because implies that the probability of the three possible events (Billy has w1 and in the bag is w2; he has w2 while w1 is in the bag; and he has w1 and r is in the bag) is the same (=1/3). If, instead, Billy deliberately watches in the bag to choose a white marble, the probability of the three events change. In this case the probability of Billy holding w1 while r is in the bag is =1/2 and is twice the probability of the each other two events (w1, w2 and w2, w1: 1/4 for each one).

Edited on April 19, 2005, 10:18 pm

Posted by armando on 2005-04-19 15:56:16