A boy gets lost in a park. The police tell the parents the following information:
There is a 50% chance he is in the picnic area and if he is in the picnic area there is a 20% chance they will be able to find him.
There is a 33.3% chance he is in the parking lot and if he is in the parking lot there is a 40% chance they will be able to find him.
Finally there is a 16.6% chance he is in the restaurant next to the park and if he is in the restaurant there is a 66.6% chance they will be able to find him.
The following questions are posed:
1. Given the boy was not found in the restaurant, what is the probability the boy is actually in the restaurant?
2.Given the boy was not found in the picnic area, what is the probability the boy is actually in the parking lot?
I think I approached the questions correctly but I want some clarification. P is the picnic area, Pa is parking lot, R is the restaurant and F is found while $F^c$ is not found.
For the first problem I did $P(R|F^c)=1-P(R|F)=.33$
For the second problem I did \begin{align} P(Pa|(M|F^c)) &= \frac{P(Pa)P(M|F^c)}{P(Pa)P(Pa|F)+P(P)P(P|F^c)+P(R)P(R|F^c)} \\ &= \frac{(.33)(.8)}{(.33)(.40)+(.5)(.8)+(.16)(.33)} \\ &=.45143639\end{align} Is this approach correct for 2 or am I looking at it wrong?
For Problem 1, the probability that the boy was not found in the restaurant is $\frac56+\frac16\cdot\frac13=\frac89$, because if he was not found in the restaurant then he is either not in the restaurant or he was but not found there. On the other hand, the probability that he is in the restaurant despite not being found there is $\frac16\cdot\frac13=\frac1{18}$. Therefore, the probability that the boy is in the restaurant given that he was not found there is $$\frac{1/18}{8/9}=\frac1{16}$$
The second problem is solved in the same way. Can you manage that one yourself?