# Question: A bag contains a white and b black balls Balls

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.

## Relevant Questions

A round-robin tournament of n contestants is a tournament in which each of the pairs of contestants play each other exactly once, with the outcome of any play being that one of the contestants wins and the other loses. For a ...A ball is in any one of n boxes and is in the ith box with probability Pi. If the ball is in box i, a search of that box will uncover it with probability αi. Show that the conditional probability that the ball is in box j, ...Five distinct numbers are randomly distributed to players numbered 1 through 5. Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players 1 and 2 compare their ...A and B play the following game: A writes down either number 1 or number 2, and B must guess which one. If the number that A has written down is i and B has guessed correctly, B receives i units from A. If B makes a wrong ...If E[X] = 1 and Var(X) = 5, find (a) E[(2 + X)2]; (b) Var(4 + 3X).Post your question