Non-Coding Problem:
Given a bag of n balls each with a different color. Each time you pick up a pair of balls and paint both to one of their colors. What's the expected number of steps before all balls become the same color?
Analysis:
This is a problem from the "Green Cover Book". It's easy to see that we have the following transition probability.
Thus if we let \(\mu_i\) be the expected number of steps getting to one color when the bag has i distinct colors, then we have
\[
\mu_i = 1 + \frac{i(i-1)}{n(n-1)} \mu_{i-1}+\left(1-\frac{i(i-1)}{n(n-1)}\right) \mu_i.
\]
From this we can use induction to prove that
\[
\mu_i = \frac{i-1}{i} n(n-1).
\]
Therefore we have
\[
\mu_n = (n-1)^2.
\]
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