The book asks the question:
A bag contains $a$ white and $b$ black balls. Balls are chosen from the bag according to the following method:
A ball is chosen at random and is discarded.
A second ball is then chosen. If its color is different from that of the preceding ball, it is replaced in the bag and the process is repeated from the beginning. If its color is the same, it is discarded and we start from step 2.
In other words, balls are sampled and discarded until a change in color occurs, at which point the last ball is returned to the urn and the process starts anew. Let $P_{a,b}$ denote the probability that the last ball in the bag is white. Prove that $P_{a,b}=0.5$
Hint: Use induction on $k=a+b$.
I tried doing this and got :
Denote the probability that in step 2 the 2 balls were in the same color by $p$, then with probability $p$ we are returning to step 1 with $k-2$ balls and with probability $1-p$ we are returning to step 1 with $k-1$ . From the induction hypothises we get that:
$P_{a,b}=0.5p + 0.5(1-p)=0.5$.
Did I do this right ?