Using Bayes formula

Question

suppose the cave system contains 100 caves, of which 90 caves are without a hidden treasure and 10 caves contain a buried gold object. In 70% of the caves with a hidden treasure, the Wumpus has left behind his usual stench from digging about and bumping into walls. In the remaining caves with hidden treasures in them, the Wumpus has left no trace, i.e. they are non-smelly. Furthermore, all the caves without a hidden treasure are free from smell, since the Wumpus has fled the cave system. The agent is now located in a random cave and can perceive that it is not smelly. What is the probability that this cave contains a hidden treasure?

how can I solve this using Bayes formula? prob. of cave buried treasure = 10/100. prob. of cave Hidden treasure = 0.7 *HT- stands for Hidden treasure

p(HT/Cave) = (1/100*10/100)/(1/100*70/100*29/100*1/100)

i have found the solution but i am not sure it is right or not ? can anyone help me?

Answer 1

Let denote the event that the caves without a hidden treasure.

Let denote the event that there is a treasure.

Now apply Bayes rule on (|)=()(|)/()=(()(|))/(()(|)+()(|)).

Answer 2

You want to know what's the probability of finding a hidden treasure if the cave is not smelly. So you want to calculate P(HT|notsmelly). Using Bayes' theorem that would be:

According to your data,

P(notsmelly|HT) = 0.3 and P(HT) = 0.1

P(notsmelly) is given by the 90 empty caves plus the 30% of the 10 treasured caves. So that would be 0.93.

So your probability should be (0.3 x 0.1)/0.93 = 0.0322 which is about 3%.