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?

(In reply to re(2): Have you lost your marbles ? by Dan)

What you're forgetting this time is that, if you picked a black marble on your first try, either bag is not equally likely; there is a 75% chance that you have picked one of the three black marbles in the second bag, rather than the one black marble in the first bag.
Look at it this way: say you have two bags of four marbles, and all eight marbles are the same color. Any individual marble is just as likely as any other.
By the same reasoning, there is a 50/50 chance that you will pick a black marble or a white marble with a single trial. Three times out of four (not half the time), that one black marble will come out of the bag with three black marbles, and one time out of four the black marble will be the one with three white neighbors.

So, to give the same kind of analysis, out of 400 trials in which you picked the black marble, 300 of those times you will be holding the bag with three black marbles and one white marble, and you will pull a second black marble 225 times. The other 100 trials, you have the bag with one black and three white marbles, and get the black marble again 25 times on average.
All in all, that's an expected 250 out of 400, which does indeed reduce to 5/8, the same answer arrived at previously.

Posted by DJ on 2003-10-13 22:57:12