Battle of the bands

For convenience, denote $p_{I\rightarrow O} := p$ and $p_{O\rightarrow I} := q = 1-p.$

Let $f(n)$ be the probability that the Outsiders "win" eventually given that they start with $1 \le n \le 9$ supporters. We're looking for $f(2).$

We can do what's called a "first-step analysis"; that is, we can condition on what happens in the first day. With probability $p,$ there will be $n+1$ supporters of $O$ after the first day, and with probability $q$ there will be $n-1$ supporters of $O.$ Thus, $f(n) = p\cdot f(n+1) + q \cdot f(n-1).$ The characteristic polynomial for this recurrence relation is $pr^2 - r + q,$ and since $p \ne \frac{1}{2},$ $f(n) = c_1 + c_2 \left(\frac{q}{p}\right)^n.$ Note that $f(0) = 0$ and $f(10) = 1,$ so $c_2 = -c_1 = \frac{1}{\left(\frac{q}{p}\right)^{10}-1}.$

Substituting in $\frac{q}{p} = \frac{1}{0.9}$ into $c_1$ and $c_2$ and evaluating $f(2)$ gives $\approx 0.1256.$

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Note that we've assumed that one group will completely take over by the end of high school. This is true with a probability that is very, very close to 1. How close? What's the distribution of how many days until one group has taken over completely?

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For a more computational approach, we can create a transition matrix and raise it to a large power, representing the number of days in high school.

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Can you think of any fun extensions or variants to this problem?