find key problem Suppose that my keys are in the kitchen with probability $1/3$ and in the living room with probability $2/3$. If I search a room that contains the keys, then I find them with probability $0.8$. Given that I have searched the kitchen once unsuccessfully, what is the probability of the keys being in the kitchen? 

Hint: this is simply an application of Bayes' Rule .
If $K$ is "the key is in the kitchen" and $S_k$ "a search in the kitchen finds the key", then you know $\def\P{\operatorname{\mathsf P}}~\P (K)=1/3~$ and $~\P(S_k\mid K)=0.8$ (and also $~\P(S\mid K^\complement)=0~$ obviously).
Find: $~\P(K\mid S_k^\complement)$
