Exercise 13.29

In our analysis of the wumpus world, we used the fact that each square contains a pit with probability 0.2, independently of the contents of the other squares. Suppose instead that exactly $N/5$ pits are scattered at random among the $N$ squares other than [1,1]. Are the variables $P_{i,j}$ and $P_{k,l}$ still independent? What is the joint distribution ${\textbf{P}}(P_{1,1},\ldots,P_{4,4})$ now? Redo the calculation for the probabilities of pits in [1,3] and [2,2].