Bingo probabilities

What is the probability of a specific ball being the bingo ball in a single/double/multiple game with 600 cards?

1 in 75.

Quote: DJTeddyBear

1 in 75.

The numbers in the N row Numbers don't get their fair share because of the free spade in the center.

The free space simply means that 4 of the 12 traditional bingo patterns have a slight advantage.

But the odds of any specific ball being the last ball called giving someone bingo, remains 1 in 75.

Quote: DJTeddyBear

The free space simply means that 4 of the 12 traditional bingo patterns have a slight advantage.

But the odds of any specific ball being the last ball called giving someone bingo, remains 1 in 75.

Bingo cards come in what's called a "perm set" which is a pre-designed set of cards that meets many conditions. For example, there can be no straights, the rows can't be all even or all odd, etc. What it comes down to is that companies who make cards or market bingo video games produce perm sets that do not have equal probability of each number for each column. For example, to avoid straights in the first column, cards are rich in B-1 & B-2 & B-14 & B-15 and poor in B-6 ... B-10. So, it is much more likely that 1,2,14,15 would be the ball that creates a bingo than 6,7,8,9,10.

There is much more to it than this, but this gives a quick reason why 1-in-75 is definitely not correct.

-- Ms. D.

The odds in Bingo are simple: if you are playing for some church/school/charity/service-club front, you are being cheated. If you are playing in Vegas, its legit and is probably 1.0 percent advantage/disadvantage. Any other statistic seems irrelevant other than perhaps the temperature/precipitation/parking data which can affect turnout.

Interesting.

While I'm well aware that there are a limited number of unique bingo cards being reproduced, and that this number is far lower than the actual number of different combinations possible, I would have assumed that the distribution of the numbers is equal.

Maybe there is psychology involved, same as in slot machines: The illusion you "almost" had the winning card/spin.

Quote: DJTeddyBear

Interesting.

While I'm well aware that there are a limited number of unique bingo cards being reproduced, and that this number is far lower than the actual number of different combinations possible, I would have assumed that the distribution of the numbers is equal.

Quite the opposite -- the distribution of cards in a perm is designed to minimize the number of simultaneous bingos that occur.

M.E. -

I'm not at all sure what you mean by "perm".


Here's what I know about bingo cards:

There's a limited number of different layouts printed. Less than 100,000. Perhaps far less, but at least 10,000.

How do I know that? Anyone who has played bingo in a hall that uses the card company's verification software knows how it goes. You yell bingo, then the floorperson comes over and announces the card number printed between the words "Free" and "Space". Moments later, the monitors show the layout of that card number, and highlights the bingo. In most bingo halls the floorperson also announces the serial number. Serial numbers are unique to each shipping pack of cards, but the same on every card in the package. These are used to verify that a player's card is not a counterfeit.

With that knowledge, it's easy to connect the dots that there are a limited number of different cards. Since I've never seen a card number with more than 5 digits, assume less than 100,000, but more than 10,000, different cards. Obviously, these cards are used over and over, but with somewhat random distribution in a bingo hall, they seem unique to the gambler.


Now, back to the topic. In those however many different cards, I assume that, for example, B1 appears as many times as any of the other B numbers, and that there's an even distribution among the five positions where it can appear. Ditto for the other 74 numbers.

It seems to me that anything else would not be a fair system.

Are you saying that's not the case? That the cards are not entirely fair?

I assume that the 'Ns' appear only on 4/15 of all cards, while all others appear on 5/15. So I would say any 'N' would be less than 1/75, and all other numbers greater than 1/75.

Quote: DJTeddyBear

There's a limited number of different layouts printed. Less than 100,000. Perhaps far less, but at least 10,000.

I would disagree, based on two points. 1) Additional numbers on bingo cards that are not on the free space, and 2) internal controls.

Regarding #1. You can usually find an additional set of numbers printed on bingo cards. It's usually on the back, or in a format where the number is printed through the entire pack of bingo cards (like holes in a way, not sure how to describe it.) Even if it's not, it seems much more likely that while those free space numbers are limited in number of digits, those free space numbers are being used to uniquely identify bingo cards in a separate manner. Bingo cards are purchased in bulk, and shipped to the casino. The casino probably receives the bingo cards with software or a disk of some kind that is fed into the computer system. I've sometimes noticed bingo cashiers "opening" a packet of bingo cards. I would argue that this packet of cards has some identifying number. Then, each bingo face within the packet has a serial number, that which is printed on the free space. By combining the package identifying number with the card face identifying number, the exact bingo card can then be replicated on screen.

I would guess that the data file arriving with the packet indicates what the bingo cards are for each packet and face identifying number combination. There is software you can purchase that prints bingo cards for you, along with an identifying number. In this case, it starts with the number '00001' and goes up to the number of faces that you print. The cards are printed randomly, and it's only stored within the computer system that number '00001' has a certain combination of numbers, and number '00002' has a different combination of numbers, etc. Thus one can then go back into the computer system, type in the number '00001' and see the exact same bingo face as what was printed out.

Thus, it would be fairly simple for a packet of bingo cards to be opened. That packet identifying number is then fed into the computer system. Every card from that packet would then be able to be identified with it's free space number, and a winning bingo appropriately determined. It's possible that a new packet could be opened prior to every game, however, it's more likely that a strict set of controls (reason #2) is in place that identifies which packets are currently being used. This could result in two different packets being used at the same time, which would explain why sometimes you would also hear the floorperson announce the packet identifying number, as you mentioned in your post.

konceptum -


Regarding cards sold to bingo halls, those are interesting arguments, except for one thing.

I've been to bingo halls where floorperson does NOT call out the serial number. And I've seen it where the winning card is displayed immediately, while the floorperson is calling the serial number.

Therefore, the serial number is used ONLY for security / anti-counterfeiting purposes.

This is confirmed at Arrow International - one of the leading suppliers of Bingo paper and other supplies.



Software that prints cards on demand can very well be an entirely different story.

Well, the very last paragraph on the page you referenced does state:

Quote:

When the floor workers are verifying the bingo card, the caller should ask: the color to determine that the winning face was played on the correct sheet in the program; the serial number to determine that the face was sold on the floor for the session; and finally the card verification number, assuming the game has electronic verification, to ensure the proper game pattern is the winner.

Which implies to me that at least the combination of serial number and card verification number are required.

I still believe that the serial number will have already been placed into the computer system, and this is why it is not normally read out during a bingo game. With proper controls in place, only those serial numbers of cards being used would be in the computer system for verification. I can see two possibilities. 1) When the card verification number is read out, the checker types in that number into the computer system. It then could give an option as to which serial number to check against. In the case where the serial number isn't correct, the checker could then select an alternate serial number. 2) If the card verification number is only a match to one of the serial numbers in the system, then it automatically pops up that card.

I've seen a similar thing to what you have described. The floorperson calls out the card verification number. The checker types it into the system, but perhaps cross references and/or selects the wrong serial number. While the floorperson is calling out the actual serial number, the checker has already told the system to switch to the other serial number. Thus the checker has it on the screen before the floorperson is done calling out the serial number.

This would probably be easy to test, assuming we couldn't just ask a bingo manager how things are done. Basically, go to a bingo hall and purchase a bingo card. Then leave. Come back a couple of days later, and play bingo. You'd have to get a bingo. At this point, assuming that the serial number of the packet in which your bingo card came out of is no longer in play, your bingo should probably be invalidated, since it's not a valid card that was sold at the session you're currently playing in. Needless to say, when the floorperson calls out the verification number, it won't match any of the cards for the serial numbers currently in the computer system. When they double check the serial number, they'll find that out. Of course, you'll be much maligned and hated for having wasted their time.

A - Color: Manufacturers sell packettes of multiple pages, each page with different border color, used in different games. Color is verified so a player doesn't use a page intended for a cheap payout game on a better payout game.

Some manufacturers use the same serial number on every page of the packette, some don't.

B - Pre-loaded Serial number: Larger bingo halls (such as Foxwoods) will use multiple serial numbers on the same game. But the video monitors show the winning card before the serial number is read by the floorperson.

C - Your 'test': Not only would you need to be lucky enough to actually get a bingo, I'm fairly sure that your plan is illegal. Probably a felony. But I'm also fairly sure that if you ask a bingo manager, he'll give you an honest answer that confirms what I've been saying.

---

But back to the original question:

There are 12 ways to get a normal bingo. 8 that require 5 numbers, 4 that use the free space. Of the 12, 4 require one 'N' space, one requires all four 'N' spaces, 4 that require all five 'B', 'I', 'G', or 'O' and the two diagonals and one horizontal, making 3 that require four numbers - one each of 'B', 'I', 'G', and 'O'.

Now, using all that info, and assuming there is an equal distribution and equally random placement of the numbers, can anyone figure out what the possibility of any 'B', 'I', 'G', or 'O' being the number called for Bingo, as well as an 'N' being the Bingo number?


For what it's worth, until I see math to the contrary, I'll maintain that it's 1 in 75 for any number.

I just created a quick simulation of a Bingo game. Parameters are as follows:

• 1,000,000 trials
• Each game is played by one player
• The card is randomly generated each game
• All numbers are equally weighted in card generation
• The middle space is already marked, and does not have a number
• Balls are drawn one by one, all equally weighted
• After each draw, a bingo check is made
• A bingo win is considered 5 in a row, horizontal, vertical, or diagonal
• No special games, i.e. No coverall, no postage stamps, no double bingos
• As soon as a bingo is made, the winning number is noted
• A new game with a new card is started; All balls are put back for draw

The probability that a B, I, G, or O was the winning ball is ~20.5%.
The probability that an N was the winning ball is ~18.0%.

I can see where the drop in N wins comes from.
Since the most used space in a bingo would be the center, and it is already marked, 3 out of 4 winning lines that use the center rely on B, I, G, and O. The 4th requires all N numbers.

Yes, you're right. Further review of various materials, including the technical manuals from the website you indicated, show that only the bingo face number is required.

The equipment manuals state:

Quote:

the paper categories for the session being played will appear on the top of the category list and will not have parenthesis around them. If more than on category is programmed for a bingo session the E-max™ will perform an auto search of all the programmed categories in order to determine a valid winner.

If you are playing a different category of paper from the one that is programmed, you can choose a Category on the left side of the screen to specify the perm you wish to verify. Next enter the card number then touch Enter.

The Serial number window on the right hand side of the screen allows you to enter the serial numbers of the books sold for your session. This adds another dimension to your games security and integrity.

Since the serial number appears to be optional, clearly it is not used to verify the winning card.

Further, at a separate bingo supply company:

Quote:

Capitol Bingo Paper offers six unique Bingo Paper Series and 54,000 unique bingo faces with over 30 colors from which to choose.

Obviously, mathematically we know that there are well over 54,000 unique permutations for bingo faces. But it appears that one of the major printers of bingo cards only creates a limited number of them.

Various websites that I looked at which sell bingo calling equipment all mention that the software contains all known paper permutations from major printers. Thus, whatever the 54,000 unique bingo faces are that Capitol prints, they are obviously all stored in a database, which is included with bingo equipment.

Further, according to this:

Quote:

One of Arrow International's special areas of focus has been designing paper bingo permutations with features that are important to entertain and excite the bingo player during the actual bingo game. One strategy is to produce bingo cards that bring more players close to a bingo. “Many of the bingo series such as Arrow's 'Player Preferred UniMax' have built-in features like this,” Greg explained. “After years of developing bingo series we have learned that there are ways to design bingo card permutations to enhance certain playabilities. For example, some card series are designed to significantly increase the number of calls it takes on average to win a certain game, which in turn allows games to offer larger jackpots without increasing their average pay out."

So, it would appear that 1) All possible permutations are not printed. Rather, bingo hall owners and bingo product manufacturers have worked together to create sets of permutations. 2) These permutations are then available in some type of database format which is then included with calling equipment which allows an individual, knowing what card printer/manufacturer and what type of cards, to enter the face verification number to verify a bingo. 3) You're right that serial numbers are used only for internal controls.

-----

As to the question, if there was an equal distribution of numbers on a bingo card (and it appears that there isn't), then I would say the following:

Of the 12 possible normal bingos, 8 of them require a B. 8 of them require an I. 8 of them require a G. 8 of them require an O. But only 5 of them require an N.

I would argue that having to draw 5 numbers to create a bingo is less likely than drawing only 4 numbers to create a bingo. Thus the diagonals, the free-space row, and the N-column would seem most likely for winning bingo.

Taken a step further, drawing 4 N numbers would seem to be less likely than drawing 1 each of B, I, G, and O. Someone else would have to do the probability math on that for me. But I would also want to see the likelihood of drawing 1 each of B I G and O prior to drawing 4 N numbers, and vice versa.

Thus, I would find it more likely that a winning bingo would occur on either the B I G or O numbers.

Thanks for finding and reading that technical stuff.


Some bingo halls have a cover-all game that pays a certain amount if bingo is called before the 48th number, and pays a much lower amount if called after that.

Foxwoods takes this to an entirely different level. There are about 20 of the big bingo displays around the room (this is a BIG room), but half of them show the first 40 numbers of the last game of the day. (On stage, they have two complete caller's stations.) This allows players to pre-spot their cards for that game, and to purchase more cards if none of their cards are close.

And that plays right into this last item you quoted:

Quote:

For example, some card series are designed to significantly increase the number of calls it takes on average to win a certain game, which in turn allows games to offer larger jackpots without increasing their average pay out."

I'd love to know how they do that.

I'm inclined to think that whatever mechanism is used, would also mean that once in a while, there will be a winner with a far less than average number of calls, and/or that when there is a winner, there would occasionally be an inordinate number of multiple winners.

---

Back to the original topic...

Quote: konceptum

Taken a step further, drawing 4 N numbers would seem to be less likely than drawing 1 each of B, I, G, and O. Someone else would have to do the probability math on that for me. But I would also want to see the likelihood of drawing 1 each of B I G and O prior to drawing 4 N numbers, and vice versa.

I agree.

Is it safe to assume that there is an equal likelihood of any B, I, G, or O being the Bingo number, and a different equal likelihood that any N is the Bingo?

Quote: Dween

I just created a quick simulation of a Bingo game.

...

The probability that a B, I, G, or O was the winning ball is ~20.5%.
The probability that an N was the winning ball is ~18.0%.

Hmmm.... I'm assuming that the 20.5% is for EACH letter. Combined that's 82% which makes sense if N is 18%

The caller at our local bingo parlor had a growth appear on his neck. He just got the report from the doctor today. Luckily the growth was B9.

LOL that was cute