If wolves are currently migrating into the forest, there is a 90% chance that the next migration will also be by wolves, and a 10% chance that it will be by bears. However, if bears are the migrators, there is a 60% chance that the next migration will also be by bears, and a 40% chance that it will be by wolves. On average, how many migrations will occur before the next wolf migration occurs?

This is a Markov Chain problem, where $w_2 = 0$ (second time wolves) is our absorption state. Furthermore, $w1$ is the state that the current migration is by wolves, and $w_2$ is the next migration by wolves. Furthermore, $b$ is the bear migration state. $w_1 = 0.9 w_2 + 0.1 b + 1$ $b = 0.4 w_2 + 0.6 b + 1$ We can rewrite this as $0.4b = 0.4 w_2 + 1$ $b = w_2 + \frac{5}{2}$ $w_1 = 0.9 w_2 + 0.1 (w_2 + \frac{5}{2}) + 1$ $w_1 = 0.9 * 0 + 0.1 * (0 + \frac{5}{2}) + 1$ $w_1 = \frac{5}{20} + 1 = \frac{25}{20}$ So, starting at state $w_1$, we expect it to take 1.25 migrations until the next migration of wolves.

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