A bag contains five white balls and five black balls. Your goal is to draw two black balls.
You draw two balls at random. Once you have drawn two balls, you put back any white balls, and redraw so that you again have two drawn balls. What is the probability that you now have two black balls? (Include the probability that you chose two black balls on the first draw.)
There are a few cases.
We draw two black balls. The probability that this occurs is 2/9.
We draw a black ball and a white ball, and then we put back the white ball, and draw another black ball. The probability that this occurs is 4/9*3/9 = 12/81 = 4/27.
We draw a two white balls, and then we put back the white balls, and draw another two black balls. The probability that this occurs is 4/9*5/9 = 20/81.
So the probability that we get two black balls in the end is 2/9 + 4/27 + 20/81 = 50/81.