Trusting an unreliable neighbour about the cat

On any given day your cat escapes the house with probability $0.001$. When you phone your neighbour to ask, they answer yes or no, but lie with probability $0.1$, independently of what actually happened. The neighbour tells you the cat got out today. What is the probability that it really did?

About $0.0089$. By Bayes, the posterior is (true yes) over (all yeses): $\dfrac{9\times10^{-4}}{9\times10^{-4}+0.0999}=\dfrac{9\times10^{-4}}{0.1008}$. False reports from the 99.9% of quiet days swamp the rare true ones — a base-rate effect.