Do the 12 lines of a bingo card have equal chance of winning?

Solution 1:

I must admit, when I read the title of the question, I thought "Of course they all have an equal chance of winning. What a silly question." But then, I read the body, and was delighted to see that you had a valid point, and I was the silly one. I don't know how to solve for $5\times5$ bingo in any reasonable time, but for $3\times3$ bingo we have only $9!$ permutations of the numbers to consider. I wrote a python script to exhaustively test this.

from itertools import permutations
from fractions import Fraction

score = 9*[0]
card = 8*[0]

for p in permutations(range(9)):
    card[0] = {0,1,2}
    card[1] = {3,4,5}
    card[2] = {6,7,8}
    card[3] = {0,3,6}
    card[4] = {1,4,7}
    card[5] = {2,5,8}
    card[6] = {0,4,8}
    card[7] = {2,4,6}
    for number in p:
        for line in card:
            line -= {number}
        winners = len([line for line in card if not line])
        if winners:
            for idx, line in enumerate(card):
                if not line: score[idx] += Fraction(1, winners)
            break

for idx in range(8):
    print(idx, score[idx])

This produced the output

0 47520
1 45600
2 47520
3 47520
4 45600
5 47520
6 40800
7 40800    

This means that the diagonals are worst, and the lines along the edges of the card are best, with the horizontal and vertical lines through the center in the middle.

I should mention that in the case of ties, I awarded an equal fraction of the game to each winner, so that the sum of the scores does work out to $9!.$

Solution 2:

I simulated a standard bingo game, one million games with ten random cards for each game. I recorded the number of times each line triggered bingo, including bingo on multiple cards and multiple lines winning simultaneously on the same card.

The results:

• Horizontal through center wins $16.2\%$ of games
• Vertical through center wins $16.1\%$ of games
• Diagonals each win $15.9\%$ of games
• Other horizontal lines each win $5.8\%$ of games
• Other vertical lines each win $5.6\%$ of games

Note that this totals over $100\%$, accounting for multiple wins in a single game (there were nearly $1.1$ million wins in one million games)

After ensuring that these results are indeed accurate and giving some thought to why the diagonals win less than the central row and column, it is clear that this is due to intersecting lines. Each of the squares on the diagonals, when drawn, has the potential to trigger a win on either of two intersecting lines, while each square in the central row or column can only trigger a win on a single intersecting line.

I have also run the simulation without the free center square:

• Rows win $9.42\%$
• Diagonals win $9.17\%$
• Columns win $9.05\%$