I have to answer the following question
A traveller arrives at dusk at a small village. She knows that 5% of the villagers are vampires, 10% are werewolves and the remainder are, for want of a better word, normal. She meets a stranger, and asks him if he is normal. She knows that normal people always answer this question truthfully. She also knows that vampires are basically honest, so that any particular vampire has only a 10% chance of lying to this question. Werewolves are less honest and have a 30% chance of lying.
- What is the probability that the stranger will claim to be normal?
My idea was the following:
$P$(claim to be normal) = $P$(one says the truth|the stranger is a normal person) + $P$(one doesn't say the truth|the stranger is a vampire) + $P$(one doesn't say the truth|the stranger is a werewolf). However, this clearly does not return the right solution since the probability of a normal person saying the truth is 1 already.
I feel like the problem is fairly simple, but am completely stuck. I also think that Bayes' theorem should be implemented at some point, but not too sure when and how.
Any help would be greatly appreciated.