Suppose a basket contains 2 white balls and 1 black ball. We are going to pick balls as follows: (1) Pick out a ball equally likely at random from all balls in the basket and put it in your box. (2) Pick out a ball equally likely at random from the basket. If its color is the same as the last ball chosen then put in the box and repeat from step (2). If its color is not the same, put it back in the basket and repeat from step (1).
My Idea: There are 2 outcomes on the first pick. If it's a black ball (P=1/3), the second and last pick will be white. If it's a white ball (P=2/3), I'll see what the 2nd pick gives me. If the 2nd pick is W (P=1/2), the last one left is Black, which isn't the result we want. So, if the 2nd pick is B (P=1/2), I'll have to put it back in the basket and pick again. If I pick B (P=1/2), the last one is white.
Thus, Prob. = 1/3 + (2/3 * 1/2 * 1/2) = 1/2. Correct?