You have a bingo board with dimensions H (height) and W (width),
leaving of course (HW) total squares.

There are some number (N), (greater than (HW)), different bingo balls that can be chosen.

To "win" the numbers called out must create a certain pattern on your board.

We'll use a hook pattern as the only solution for now (identical to a
knight's movement in chess) See below:
(the x's are selected spaces)
x x

This is tricky, every time a space is chosen, the total number of spaces that can be chosen and still lead to a "win" is reduced. (inseparable events)

Also, any number of squares on the given board may be either free spaces, or unusable. This means that the total number of possible solutions is reduced, and some squares may no longer be able to provide a solution.

With all that said, how would you find an equation for the odd's that someone will create this pattern in the given board when a certain number, X, different random spaces called? Help?? Anyone??