At my local bingo hall, there are several ways to get bingo, including the traditional methods. I will use this card as an example for the patterns:
Along with the column (5 possible drawn from 5 balls, 1 possible from 4), diagonal (2 possible with 4 balls drawn), and row (5 possible from 5 balls, 1 possible from 4), there are a few more.
Postage stamp: 4 numbers in the corner, e.g. B2, B15, I21, I30 - this can be done in all corners.
4 corners: All 4 numbers in the corners - B2, B10, O66, O63.
4 edges: All 4 numbers on the middle of each outer column/row - B5, N32, N45, O74.
Small 4 corners: All numbers on the corners of the inner ring surrounding the Free Space - I30, I16, G52, G58
Small 4 edges: All numbers on the middle of each column/row surrounding the Free Space - I29, N44, N33, G48
How would you go about calculating all the probabilities of all these after $n$ amount of balls are drawn?
I assume that a ball can only be drawn once. Let's say the game is played on a $N\times N$ board. Let one kind of bingo (say, a column-bingo) be given as having $k$ balls in a specific position.
First draw $n$ balls and place them randomly. The question now is "what is the probability that $k$ of them have ended up in the $k$ specified positions?" There are $\binom{N^{2}-1}{n}$ possible configurations in total, of which $\binom{N^{2}-1}{n-k}$ results in bingo (you place the $k$ balls that gets you the bingo first and then count how many ways there are of distributing the rest of the $n-k$ balls), so the answer is $$\frac{\binom{N^{2}-1}{n-k}}{\binom{N^{2}-1}{n}}.$$
If you want the total probability for bingo multiply result this by the total number of different ways of achieving bingo.