Consider the following random process. Initially, there are n red balls in an urn. At each step, a ball chosen uniformly at random from the urn is removed, and then a blue ball is placed into the urn. (Note that after each step, there are n balls in the urn.) All random choices in this process are made independently. After n steps, what is the expected fraction of balls in the urn that are red? In other words, if Xt denotes the number of red balls in the urn after t steps, what is the value of E(Xn/n) as n -→ o?
Consider the following random process. Initially, there are n red balls in an urn. At each step, a ball chosen uniformly at random from the urn is removed, and then a blue ball is placed into the urn. (Note that after each step, there are n balls in the urn.) All random choices in this process are made independently. After n steps, what is the expected fraction of balls in the urn that are red? In other words, if Xt denotes the number of red balls in the urn after t steps, what is the value of E(Xn/n) as n -→ o?
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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Transcribed Image Text:Consider the following random process. Initially, there are n red balls in an urn. At each step, a ball chosen
uniformly at random from the urn is removed, and then a blue ball is placed into the urn. (Note that after
each step, there are n balls in the urn.) All random choices in this process are made independently.
After n steps, what is the expected fraction of balls in the urn that are red? In other words, if Xt denotes
the number of red balls in the urn after t steps, what is the value of E(Xn/n) as n -→ o?
(a) About 0.63
(b) About 0.50
(c) About 0.37
(d) About 0.33
(e) 0
(f) None of the above
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