# Question

A bag contains a white and b black balls. Balls are chosen from the bag according to the following method:

1. A ball is chosen at random and is discarded.

2. 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 Pa,b denote the probability that the last ball in the bag is white. Prove that

Pa,b = 1/2

Use induction on k ≡ a + b.

1. A ball is chosen at random and is discarded.

2. 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 Pa,b denote the probability that the last ball in the bag is white. Prove that

Pa,b = 1/2

Use induction on k ≡ a + b.

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