In this modified version of bingo game, there are total of $n^2$ bingo balls, half of the bingo balls will be randomly rolled out (in the case of odd number of balls, round up to nearest integer). What are the maximum possible of bingo lines (vertical, horizontal, and diagonal) can an $n \times n$ bingo card achieve?
Now, let’s sax the maximum number of bingo line for any given $n$ is $m$
And from that, what are the probability of getting $m$ lines for the given $n$.