Ehrenfest chain transition probability

[shahawn11]

Homework Statement

I must show that un+1 = 1 + (1-2/N)un and use that to prove another equation to be true. Lastly I must use the result from problem 2 to solve problem 3.

Relevant Equations

Relevant equations are posted in image below

Here is the Ehrenfest Chain that the question is talking about:

I was able to solve parts 1 and 2 as shown in the image below. But I'm not really sure how I'd prove part three. Any help would be appreciated, thanks!

 

[lippyka]

Part 3 can be proved by induction. We will assume that the statement is true for some number n and show that it is true for n+1. Let E(n) be the statement that the Ehrenfest chain has 2^n black and 2^n white beads after n steps. The base case (n=1) is true since the Ehrenfest chain starts with 1 black and 1 white bead. Now assume that E(n) is true for some number n. To prove that E(n+1) is true, we need to show that the Ehrenfest chain has 2^(n+1) black and 2^(n+1) white beads after n+1 steps. By the inductive hypothesis, we know that the Ehrenfest chain has 2^n black and 2^n white beads after n steps. Furthermore, each step of the Ehrenfest chain increases the number of black beads by 1 and decreases the number of white beads by 1. Thus, after n+1 steps, the Ehrenfest chain has 2^n + 1 black beads and 2^n - 1 white beads. But since 2^n + 1 = 2^(n+1) and 2^n - 1 = 2^(n+1)-2, this means that the Ehrenfest chain has 2^(n+1) black and 2^(n+1) white beads after n+1 steps, which is what we wanted to show. Thus, by induction, we have shown that the Ehrenfest chain has 2^n black and 2^n white beads after n steps, for all n ≥ 1.