Today, having plenty of better things to do, I decided to amuse myself by calculating the odds involved in the game of bingo. Specifically, I am considering the standard US version of the game, in which the numbers run from 1 to 75, the cards use a 5 x 5 grid with the central cell marked as “free,” and the winner is the first player to complete a horizontal, vertical or diagonal row of five marked cells.
What, I wondered, are the chances of two or more players getting bingo at the same time and having to share the prize? Well, in order to calculate this I first had to find the chances of a single player getting bingo as each number is called. This can happen after as few as 4, or as many as 71 numbers have been called, although these are both very remote possibilities: only about 1 chance in 300,000, and 1 chance in 700,000 respectively. Ninety-nine percent of the time, the bingo will come somewhere between calls 14 and 62, with the most likely time being on the 43rd call (about a 1-in-25 chance).
Armed with these figures I was able to calculate the chances of sharing a prize. This of course depends on the number of players in the game, but not as much as you would expect. (With more players, there is a better chance that one of them will go bingo fairly early on in the game, at a point when the chances of any other individual player winning are still quite small.)
No. of players |
Chance of 1 winner |
Chance of 2 winners |
Chance of >2 winners |
1 |
100.00% |
0.00% |
0.00% |
2 |
97.17% |
2.83% |
0.00% |
3 |
95.96% |
3.95% |
0.09% |
4 |
95.21% |
4.62% |
0.17% |
5 |
94.67% |
5.10% |
0.23% |
6 |
94.24% |
5.47% |
0.29% |
7 |
93.89% |
5.77% |
0.34% |
8 |
93.60% |
6.02% |
0.38% |
9 |
93.34% |
6.24% |
0.42% |
10 |
93.11% |
6.43% |
0.46% |
15 |
92.24% |
7.16% |
0.61% |
20 |
91.62% |
7.66% |
0.72% |
30 |
90.73% |
8.38% |
0.90% |
40 |
90.07% |
8.89% |
1.04% |
50 |
89.55% |
9.29% |
1.15% |
100 |
87.83% |
10.59% |
1.58% |
150 |
86.73% |
11.39% |
1.88% |
200 |
85.90% |
11.98% |
2.12% |
This table shows that even with up to 40 players, there’s less than a 1-in-10 chance that the prize will have to be shared (and an even smaller small chance that it will have to be split among 3 or more players). Bingo is obviously well suited to being played by groups of different sizes.
Another attraction of bingo is that most non-winners feel they came very close to winning. In order to calculate the figures above, I looked at all possible configurations of cell markings on a card (well, my computer did the actual looking, as there are a total of 2^24, or 16,777,216 possibilities). Of these, there were 10,624,010 configurations that a non-winning player might hold (those that did not include a row of five marked cells). And 8,950,584 (over 84%) of these non-winning combinations were “pregnant,” that is, they had one or more rows with exactly four marked cells. As a result, most players in a game will be only one number short of having a winning card, before one lucky player calls “bingo”. This not only increases the excitement of the game, it also encourages repeat play. No wonder bingo is so popular!