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?
Ok, since I think we've cleared it up that the answer is 5/8, can I ask a further question: suppose that, as before, you select a bag at random and pull a black marble from it. You then replace it, and pull a second black marble, replacing that as well. What is the odds of picking a third black marble? What is the general formula of pulling a black marble on the nth try if the first (n-1) marbles pulled were all black? And what is the limit as n approaches infinite?
Extension to the problem
