Bingo Probabilities

A minimum of four matching calls are required for a bingo. If only one card is in play, the initial question is what the probability will be of having a bingo on the fourth, or fifth, or whatever, call. The cumulative probability of each of these numbers will be the probability of getting bingo in N calls or fewer. Two bingo authors (Durango Bill -- durangobill.com -- and the Wizard of Odds -- wizardofodds.com) have published detailed information on these calculations, which are understandably complex. The following is an extract of information from the calculations.

1. What is the average number of calls for a bingo? The average number of calls is around 8.6 with a field of 2000. The probability that any given card will get bingo in 8.6 calls or less is 0.0347% or so. With 1000 cards in play, the average number of calls to bingo is around 9.65.

2. What is the probability that a given card will generate bingo in 20 calls? Any card has a 2.29% chance of generating a bingo in 20 calls or less. The odds that 50 cards will generate a bingo in 20 calls or less are quite high, 68.56%. The odds that 2000 cards will generate a bingo in 20 calls are effectively 100%. In fact, the odds for 2000 cards in play will go to more than 99.5% at between 12 and 13 calls.

3. So, if one is playing 30 cards in a field of 2000, what are the chances of winning? The expected number of calls is between 8 and 9. With 30 cards, the chance of a bingo in 9 calls is about 1.4%. The chance that someone in the field of 2000 will have bingo in 9 calls is 61.2%.

More demanding patterns, obviously, will require a higher number of calls on average for a winner to emerge. For example, bingo the hardway takes an average of 11.41 calls. Double bingo takes over twice the number of calls for single (19.32 and 24.56 the hardway). A sixpack takes 9.51 calls on average, and a ninepack takes 21.79 calls. A coverall can be expected in 57.75 calls if 2000 cards are in play.

It should be clear that large numbers of cards in play increase the probability of having a winner in fewer calls, so that any given player’s card is less likely to be that winner. Equally obvious is that the chances of winning are fairly remote, even with 30 cards in play in a field of 2000.

Serious players should visit the websites explaining the mathematics of bingo probabilities. It is possible to set up spreadsheets to calculate cumulative probabilities, based on tables provided. Both Durango Bill and the Wizard of Odds will shed light on calculation issues, as well as probabilities for the other, somewhat harder patterns.