You choose one of two identical looking bags at random. One bag has three black marbles and one white marble. The other has three white marbles and one black marble.

After choosing a bag you draw one marble out at random. You notice it is black. You then put it back and draw another marble out of the same bag at random.

What is the probability that the second marble drawn is black?

With apologies to DJ and Brian Smith and Shannon Jones and Jesus Christ......

As someone (was it Jesus ?) astutely pointed out, when you draw a black ball, it was probably from the bag with three black balls and one white ball. Let's say you draw a ball of either color from a random bag 1600 times. If it is truly random, there will be roughly 800 picks from the 3-white-1-black-marble bag and 800 from the 3-black-1-white-marble bag. From the group of 800 from the 3-white-1-black bag, about 200 of these initial picks will be a black marble, and the subsequent draw, after that black marble is returned, will yield another (the same) black marble about 50 times (out of 200). From the 800 initial picks from the 3-black-1-white-marble bag, about 600 will be black; subsequent picks after the black marble is returned will yield approximately 450 black marbles (out of 600) on the second try. So after 800 (200 + 600) initial black marbles are picked from the two bags, 500 (50 + 450) black marbles are subsequently picked. So the probability of picking a black ball, after initially picking a black ball, is 5/8. (This is the same result that fell out of DJ's strange tree).

That may be wrong, but that is the analysis given in the latest edition of Marble Comics.... :-)

Posted by Dan on 2003-10-14 03:02:45