Probabilities in Bingo

One of the little pleasures of our annual winter vacation is an evening Bingo party. After a day of sledding and cross-country skiing, it’s relaxing to indulge in a social game that requires minimal thought, affords young and old the same chance of winning, and has a strong element of suspense.

To play Bingo, each player has one or more cards divided into squares. Each card has five rows and five columns. The columns are labeled from left to right with the letters “B,” “I,” “N,” “G,” and “O.” The center square is designated “free.” The five squares in column B contain different numbers selected from the interval 1 to 15. Column I contains five numbers from the interval 16 to 30. Column N contains four numbers from the interval 31 to 45. Column G contains five numbers from the interval 46 to 60, and column O contains five numbers from the interval 61 to 75.

An announcer randomly selects numbers from 1 to 75, calling out each one while players mark the appropriate grid square if that number appears on their cards. In standard Bingo, the player’s goal is to be the first to mark an entire row, column, or diagonal. There are 12 winning configurations. Eight of these configurations do not involve the use of the center “free” space, whereas four (column N, row 3, and the diagonals) do.

What makes the game suspenseful is the tantalizing uncertainty about when someone will achieve a Bingo. It’s natural to wonder how long a typical game would last. More precisely, what is the average number of calls required to complete a game for a given number of players?

“The analysis turns out to be straightforward, but obtaining the numerical results would be tedious without a computer,” David B. Agard and Michael W. Shackleford note in the September College Mathematics Journal.

In presenting their data, Agard and Shackleford also remark that some care must be taken in doing the analysis. Several previous efforts had failed to take into account the fact that “the events representing completion of the 12 possible Bingos are neither independent nor mutually exclusive,” they point out.

One crucial tabulation involves determining the number of subsets of the 12 possible Bingos of a particular size that cover a particular number of squares. For example, four covered squares can produce one Bingo in four different ways, and five covered squares can produce one Bingo in eight different ways.

Overall, there is approximately a 50 percent chance that one card will require at most 41 calls to complete a Bingo, and approximately a 90 percent chance to complete a Bingo in at most 54 calls, the researchers conclude.

Probability distribution for the number of calls to achieve a Bingo

No. calls	5	10	15	20	25	30	35
Probability	.00002	.0008	.0059	.0229	.0640	.1435	.2719

No. calls	40	45	50	55	60	65
Probability	.4456	.6401	.8144	.9322	.9859	.9990

However, a typical game involves many more than one card. Because the assumption of independence does not hold for different Bingo cards, the analysis is much more complicated. Agard and Shackleford turned to computer simulation to find the probability distribution for the number of calls to achieve the first Bingo in a game played with m randomly generated cards.

“For a game involving 100 cards, there is virtually no chance it will take more than thirty numbers to produce a Bingo,” the researchers report. “For even a small game of ten cards it is a near certainty to have produced a Bingo by the fiftieth number called.”

Probability distribution for the number of calls to achieve the first Bingo in a game involving m cards

No. calls	4	8	12	16	20	24	28
m = 10	.00003	.0025	.0195	.0761	.2037	.4142	.6654
m = 50	.0002	.0127	.0921	.3179	.6627	.9207	.9943
m = 100	.0004	.0243	.1708	.5207	.8750	.9919	.9999

These odds apply only to the standard version of Bingo. Other variants also occur—X-out, four corners, postage stamp, blackout—and have their own probability distributions. Shackleford provides the cumulative probability distributions for the number of calls to complete these types of Bingos at his Web site: http://www.thewizardofodds.com/game/probbingo.html.

Indeed, with Bingo a common pastime in gambling casinos and elsewhere, it can be a serious business.