The wumpus world is a toy problem in artificial intelligence. The setup goes like this; suppose you have a $n \times m$ grid. There is gold at one of the cordinates on the grid. However there is also pits and a monster called the wumpus which will kill you if you walk on top of them. Your goal is to find the gold. If a coordinate has a pit then all adjacent coordinates will contain a breeze. When you start out exploring the grid, you are cordinate $(1,1)$ and you know nothing besides whatever information is contained in $(1,1)$ (that is breeze, no breeze).

For my question we can ignore the wumpus and gold. I am interested in the prior and posterior probability of a particular cell containing a pit. For example, let

'-' $\leftarrow$ a safe cell

'x' $\leftarrow$ unexplored cell

'?' $\leftarrow$ cell possibly contains pit then here is a possible state.

? x x b ? x - b x - x x

Assume each coordinate excluding $(1,1)$ has probability $.20$ of containing a pit. let $P(i,j)$ denote the probability that col $i$ row $j$ contains a pit. Using the example above,

1. $P(1,4)$

2. $P(2,3)$

3. $P(1,4) \cap P(2,3)$

4. In general how do you compute $P(i,j)$ given all the information you know up to that point. That is, you may no there are breezes in some cells and not in other cells.

5. Just for fun. Probability at the beginning of every cell except $(1,1)$ having a pit.