## Determining the colour of the last ball in a bag

There exists a bag with a white and b black balls. Balls are chosen by the following method:
i) A ball is chosen at random and discarded.
ii)A second ball is then chosen. If colour is different from that of the preceding ball, it is replaced in the bag and the process repeated from the beginning. If the colour is the same, it is discarded and we go back to step ii.

$P_a_,_b = \frac{1}{2}$

The hint is to use induction on $k = a + b$

Could someone show me how to start this, as I cannot see how induction would work. I am not looking for a full solution!