A box contains three white and three black balls. A randomly selected ball is removed from the box. If the selected ball is white, a black ball is added to the box. If the selected ball is black, a white ball is added to the box. (The selected ball is not put back into the box.) This process is repeated twenty more times.

(a) Let A be the event the first ball selected is white and B be the event the second ball selected is white. Show that A and B are not independent. You can show that A and B are not independent by showing any one of the following inequalities hold.

P(A, B) does not equal P(A)P(B)

P(A | B) does not equal P(A)

P(B | A) does not equal P(B)

P(A | B) does not equal P(A | B^c) ----C means complement

P(B | A) does not equal P(B | A^c) --- C means complement

There are other inequalities that would also show that A and B are independent, but the most common approach is to use one of the first three given above.

(b) What is the probability that the twenty-first randomly selected ball is white?