Probability of knowing the source of a sound after hearing it I am given this problem by my professor and I can't figure a reasonable answer, though I know it'll be easy.
Suppose to hear a sound on your roof at midnight. It could've been a thief or an animal, respectively with probability
$$ P(S \mid X=\text{thief})= 0.8, \quad P(S \mid X=\text{animal})=0.3 $$
We know that $P(X=\text{thief})=0.001$.  
We want to find the probability of the sound to have been caused by a thief after hearing that sound, so it should be $P(X=\text{thief}\mid S)$ (I think).
No other clue is given, I have tried to solve it via Bayes' Theorem, putting  
$$P(X=\text{thief}\mid S)= \frac{P(S\mid X=\text{thief})\cdot P(X=\text{thief})}{P(S)}$$ 
but this is assuming $P(S)=1$, because you did hear a sound. Moreover, this would mean that the result will be $0.8\cdot 0.001= 0.008$, which seems counter-intuitive. The text of the problem is unclear, I know, but it's all I've got.

Hint to compute P(S):
$$ P(S) =P(S,X=\text{thief})+P(S,X=\text{animal})$$
