(This question is inspired by: If I have an event where the outcomes aren't uniformly distributed, how would I make the "fairest" bingo card out of the events? )
You have a $5 \times 5$ bingo card filled with $25$ distinct numbers, one per square. There is also a pot containing each number once and you draw them out one by one without replacement.
A line is any of the $5$ rows, $5$ columns, or $2$ main diagonals. A line is completed when its $5$ numbers have been drawn. A line wins if it is the first line to be completed.
Question: Do the $12$ lines have equal chance of winning? If not, which lines have higher chance of winning?
Why I ask: Define $T_l$ to be the time to completion for line $l$. It is obvious that all $T_l$ are equi-distributed, and thus all $E[T_l]$ are equal. However, because the lines overlap, the different $T_l$'s are dependent, and it is not clear to me that they have equal chances of being first.
In particular, imagine that the $5$-subsets are not arranged in rows, columns and diagonals, but are clustered in some non-uniform way. Then a subset that shares a lot of elements with other subsets might have a lower chance to win than a subset that does not share a lot of elements with other subsets. (I can provide a simple example if there is interest.)
On the bingo card, the $5$-subsets are pretty uniform but not exactly uniform, due to the diagonals. So my suspicion is that the line win chances are almost equal but not exactly equal. And I am curious as to which lines have higher chances.
I imagine (but would be happy to be proven wrong) that calculating the exact win prob for a line might be difficult/tedious, so qualitative arguments based on e.g. symmetry are also welcome.
Clarifications: A drawn number can complete multiple lines, so that needs some special handling. However what I am interested in is the question of equality, so I will accept any reasonable way to handle such "shared" wins, i.e. if $N>1$ lines are completed at the same draw (and no line has been completed before this draw), then you can treat this as if...
they all win (in which case the sum of the $12$ win probabilities exceed $1$, but that doesn't matter since I am interested in which are higher/lower), or,
the whole experiment is repeated from the beginning (i.e. we condition on such shared wins not happening), or,
you flip an $N$-sided die to determine the winner (i.e. this effectively counts as $1/N$ win for each involved line), etc.