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?
find key problem
0
$\begingroup$
probability
-
0Is that $1/s$ or $1/3$? – 2017-02-27
-
0should be 1/3 in the kitchen – 2017-02-27
-
0Study Bayes theorem. There are three possibilities, they are in the kitch and you did find them (this didn't happen) has a probability of .8 times 1/3 or 4/15. That they are in the kitchen and you didn't find them (this might have happened) .2 times 1/3= 1/15, or they are in the other room 2/3 or 10/15. The proability that they are in the kitchen and you missed them (1/15) divided by all the possible probabilities (10/15+ 1/15 =11/15) is the probability they are in the kitchen: (1/15)/(11/15) is 1/11. – 2017-02-27
1 Answers
0
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: $~\P(K\mid S_k^\complement)$
-
0what does Sk mean ? – 2017-02-27
-
0Sorry, I thought it was self commenting. Maybe I should have used $F_K$, "found in the kitchen"? – 2017-02-28