# Constructing an “Announcement Bingo” card with highest chance of win

There's an event coming up where \$FAVORITE_COMPANY is going to announce a bunch of upcoming products. A fan of that company has 25 anticipated/desired announcements, ranked by likelihood, and wants to arrange them into a Bingo card with the highest possible chance of winning. Optionally, house rules might count "four corners plus center square" as a valid Bingo.

My first thought was to rank squares by how many Bingoes each one is part of:

Then, picturing the center as a free space, I realized that Bingoes crossing the center square are more likely since you only need four spaces for a Bingo:

After this I quickly realized that I was in over my head:

• The center square isn't the only square that affects probability of Bingoes including it, especially if it's merely the most likely announcement, not a free space; the probabilities of every square in a Bingo must be taken into account when ranking each square within it.
• If we do count four corners + center as a Bingo, our first five placements can be the five most likely announcements without worrying about dumping the highest-chance items into one row/column/diagonal. Otherwise, or after first five placements in any case, we have to figure out whether to evenly distribute items or front-load the most likely items into a single Bingo.

So, three questions:

1. What order should I place items into a single card to get the highest probability of a win?

2. If I were to generate several cards with high probabilities of a win, so that consecutive groups of items are shuffled among similar-probability squares, what would an accurate diagram like my two images above look like?

3. Approximately how much does the spacing between announcement probabilities affect all this, if at all?

EDIT: For the purpose of this scenario, the order of announcements doesn’t matter. If there are multiple cards, they are all examined once at the end of the event, and everyone with at least one Bingo ties for the win with no tie-breakers.

A natural way to interpret the problem is that each announcement $$j$$ independently has a probability $$p_j$$ of being made. The number of announcements actually made can range from $$0$$ to $$25$$. Then a bingo card wins if any of its $$12$$ lines (optionally count "$$4$$ corners plus center" as a $$13$$th "line") is good. But in this setting, the order of announcement doesn't matter. In particular your card doesn't have to race against other cards. In this case, the prob of winning can be exactly calculated (via Inclusion-Exclusion), although the sum is tedious.
If you have to race against other cards, then the order of announcement matters, and it is unclear (at least to me) what is the best / most natural way to model that. Maybe randomly permute the $$25$$ announcements first and then for each one announce it with its probability? Or should the random permutation be biased so that a higher-prob announcement has a higher chance of coming first?