 Solution (source code)

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= Solution
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A <stationary distribution> is a probability row vector $\pi$ satisfying $\pi P=\pi$. Its coordinates obey $\pi_-(1-p)=\pi_+q$ and $\pi_-+\pi_+=1$. Thus
$$
\boxed{\pi_-=\frac q{1-p+q},\qquad\pi_+=\frac{1-p}{1-p+q}.}
$$
Because both transitions between distinct states have positive probability, the finite <Markov chain> is irreducible; its positive self-transition probabilities also make it aperiodic.

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