I have a 5x5 grid filled with the numbers 1-25. 13 of the numbers are drawn randomly and marked. I can draw D of the remaining 12 numbers randomly and mark them off on my grid. I win if I eventually get a "bingo", i.e. 5 marks in any row, column, or either diagonal. My question is, given a value of D, what are my odds P(D) of winning? I should note that in the initial phase marking 13 numbers off, if a number would otherwise create a bingo, it is redrawn. This prevents me from winning without drawing, so P(0) = 0.

The smallest value where I'm guaranteed to win (P(D) = 1) is D=8: that leaves 25-13-8 = 4 unmarked "holes", which isn't enough to put a hole in all 5 columns. The following arrangement illustrates that P(7) < 1, since even after 7 draws there is no bingo:
Code:
_ = unmarked
X = initially marked
d = one of the D=7 drawn numbers

_dddd
d_ddX
XX_XX
XXX_X
XXXX_
This question is modeled after the game show Lingo's final round. The D value is determined by the contestants' performance during that round. Most people get D >= 6, and I've always suspected that the tension generated by wondering whether or not they'll draw luckily enough to win is mostly fabricated since they basically always win. There are very few final arrangements like the above for D=7 which result in a loss. I could write a simulation program, but I'd find it tedious. Perhaps someone else would enjoy the exercise, though, which is why I ask here. There's also a chance someone will find an exact answer, but I imagine it would be greatly complicated by the constraint implying P(0) = 0.