Assume "standard" bingo (75 numbers) with the columns ranging the following inclusive "semi-random" values B: 1 to 15, I: 16 to 30, N: 31 to 45, G: 46 to 60, O: 61 to 75. By semi-random I mean restricted to a small range (15 at a time). There is a free space in the middle of the 5x5 playing board. Numbers (from 1 to 75 inclusive) are randomly drawn one a time without replacement (without any repeats in each game) and with equal probability of being drawn. A win (Bingo) is defined as a completed line segment of 5 adjacent squares made only from the drawn numbers but which may include the free space (and must include it if it is beneficial). A bingo card has 25 of these board squares arranged in a 5x5 matrix.
So my question is if there are 20 players, each with a unique playing card (randomly generated by computer out of I think 552 septillion possible playing cards), what are the chances/probability that 2 or more players will get Bingo on the same drawn number?. For example, someone could win bingo with as few as 4 drawn numbers but likely it would take much more. So I am asking if balls are drawn until at least one person wins, what are the chances that at least 2 people will win at the same time? The game is considered finished / decided when there is at least 1 winner for that game. You can assume that all players are good enough not to make any mistakes (not true in real bingo but assume here).
I am not sure how to set this up mathematically and because there are so many possible bingo cards, computer simulation of all of them is not a good idea. Perhaps what can be done with simulation is to first simulate 20 legitimate bingo cards (out of 552 septillion), and then have the computer draw one random number at a time until we have at least 1 winner. Do this for maybe 1 million trials and count how many have simultaneous multiwinners. For example, after 11 balls drawn there are no winners yet for that game but on the 12 drawn ball, there are 2 or more winners. I would like to know how often the multiwinner situation occurs.
I could probably do the simulation with a fair amount of work but wanted to know if this problem can be done mathematically or if is too difficult to set up.
One concern I see is that if one card has for its first column (the B column), from top to bottom, 1, 4, 7, 10, and 15 and some other card has for column B (also from top to bottom), 15, 10, 7, 4, and 1 in that order. Problem there is even though the cards have different order for the B column numbers, if those 5 numbers are drawn, both players may win at the same time. So my point is it makes a difference if we say (or not) that multiple bingo cards cannot have the same exact 5 numbers in a row, column, or diagonal but just in a different order. That might be an interesting problem in itself to figure out how many fewer "legit" bingo cards there are with that constraint so someone could comment about it but the actual question is about the multiwinner probability. I think the answer to the no permutation bingo card restriction is about 111 quadrillion legit bingo cards.
