23 A bag contains a white and b black balls. Balls are chosen from the bag according to the following method: A. A ball is chosen at random and is discarded. B. 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.

23 A bag contains a white and b black balls. Balls are chosen from the bag according to the following method: A. A ball is chosen at random and is discarded. B. 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.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

23 A bag contains a white and b black balls. Balls are chosen from the bag
according to the following method:
A. A ball is chosen at random and is discarded.
B. 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
Hint: Use induction on k = a + b.
Pa,b
-
1
2

Expert Solution

Step 1: Given information

The provided information is as follows:

The $P subscript straight a comma straight b end subscript$ denotes the probability that the last ball in the bag is white.

The white ball is the a and the black ball is the b.

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Follow-up Question

I have a question about Step 4. I cannot figure out why

P(E a,b| WcBc) = P(Ei, j). My guess is that it is because of the inductive hypothesis. However, I don't know why: I don't know what P(Ei,j) means and I don't what P(E a,b| WcBc) represents.

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