Uncertainty arises in the wumpus world because the agent's sensors give only partial, local information about the world. In this example the agent faces a situation in which each of the three reachable squares- , , and - in a grid world might contain a pit, because the squares and were breezy (no breeze was detected on square ). Pure logical inference can conclude nothing about which square is most likely to be safe, so a logical agent might be forced to choose randomly. A probabilistic agent can do much better than the logical agent.

The goal is to calculate the probability that each of the three squares contains a pit. (For the purposes of this example, we will ignore the wumpus and the gold.) The relevant properties of the wumpus world are that (1) a pit causes breezes in all neighboring squares, and (2) the square does not contain a pit. The given set of random variables are:

- We use one Boolean variable for each square, which is true iff square actually contains a pit.
- We also have Boolean variables that are true iff square, is breezy; we include these variables only for the observed squares-in this case, , ], and .

In our analysis of the wumpus world in the excercise hour, we used the fact that each square contains a pit with probability 0.2, independently of the contents of the other squares. Suppose here instead that exactly pits are scattered uniformly at random among the squares other than .

- Are the variables and still independent?
- What is the joint distribution now?
- Redo the calculation for the probabilities of pits in and .