Bingo -- estimating number of players

I've been spending a lot of time analyzing bingo again, specifically as played at the Suncoast. As with poker, in bingo the enemy is not the casino, but the other players. Critical to calculating the value of playing is knowing the number of cards you are playing against.

To ask a practical question, suppose you are playing a coverall game, and all you know is the first player to win did so on the 55th ball. There was only one winner. What is the most likely number of cards at play?

Here is some information from my probabilities in bingo page, which may be relavent, or may be red herrings:

Probability of a bingo in exactly 55 calls = 0.0000421251.
Probability of a bingo in 55 calls or less = 0.0000965367.

Discuss.

An infinite crowd of mathematicians enters a bar.
The first one orders a pint, the second one a half pint, the third one a quarter pint...
"I understand", says the bartender - and pours two pints.

Theorem. A cat has nine tails.

Proof. No cat has eight tails. Since one cat has one more tail than no cat, it must have nine tails.

I suppose you put that joke up to get people to re-read the post.

Although I occasionally play, I've never given much thought to the math other than the obvious: The greater the ratio of your cards to total cards, the greater your chances of winning - up to a point. (The "point" is where you have so many cards that you can't keep up.)

That said, I looked at your bingo page.

First off, I submit that the number of samples used in the last chart is woefully too few. I base that on the first couple lines where there are hits, followed by zeros. These just stick out like a sore thumb as being "wrong". Ditto for the unevenness of the bottom lines. There's probably similar issues in the middle that merely remain hidden.



Next I looked at the coverall lines of the other charts. There's hardly any difference between the results for 2,000 cards vs 10,000 cards.

Given that, what's the point of trying to figure out an answer to the original question?

Quote: DJTeddyBear

Given that, what's the point of trying to figure out an answer to the original question?

Heck, I'm still trying to figure out what the original question is. (As well as how to spell relevant).

The number of outstanding cards or how many outstanding cards are "loaded" with a particular number?

Assuming all cards are fair and its an honest game, I would assume that all players are experienced enough to know how to handle the vast array of cards that some play per game.

Surely there is some "average" that will allow someone to merely see the occupancy of the room and have a good idea of how many cards are in play.

My guess:


Now practically, how does that help for the next session? Could you count the number of individuals, then estimate a "cards per player" average? Does that information translate into an +EV opportunity to buy more cards? Most bingo rooms limit the number of coveralls you can purchase to prevent someone from completely stacking the deck. I suppose a team could have a chance, but wouldn't they always buy the max number anyway?

I think it's a neat problem. There IS a most likely number for these types of questions, but finding it can be tedious. You could actually construct the conditional probability distribution if you wanted to.

Given that someone got a coverall with 55 cards, what is the conditional probability that X number of cards were in play? The mode of the distribution is the answer.

My official guess is:



Side comment/observation:

Ten thousand cards in play?????????????

Lets see now.... 100 Blueies each playing ten cards would be a thousand cards

You've got some rooms at the 750 occupancy level and some at 600 and some at 500, but many at the 350 capacity level too.

Not all games are equally attended and many rooms are half empty.

I'd say that estimating the number of Bluies there early in the morning and even allowing 10 cards per little old lady, ten thousand cards in play is way too high an estimate.

JB, I see we agree with your first answer, and my second, which does come to 1/p. I also had a very complicated method to get at the same answer, which I'm surprised it did.

However, I can't help but feeling that it is biassed. It seems like one is cherry picking the time to do the analysis. Maybe this is a bad example, but suppose the true split of Obama/Romney voters was 50/50. Then suppose I do a survey until Obama is two standard deviations ahead, assuming a mean of 0.5, which would eventually happen. Then I release my survey to the media. It would unfairly make Obama look like he was ahead.

One source of bias in this analysis is the assumption that any numbers of cards in play is likely (or even possible). I think you have assumed that all numbers of cards are equally likely, which is an unrealistic assumption. It is necessary to do the analysis without further info, but unrealistic.

For instance, if the room only holds 1000 people, it is possible, but unlikely, that the room has 10000 cards in play. It should follow that given a coverall was hit in 55, it is most likely that there are less than 10000 cards in play.

Quote: FleaStiff

Ten thousand cards in play?????????????

Lets see now.... 100 Blueies each playing ten cards would be a thousand cards...

Quote: dwheatley

For instance, if the room only holds 1000 people, it is possible, but unlikely, that the room has 10000 cards in play. It should follow that given a coverall was hit in 55, it is most likely that there are less than 10000 cards in play.

10,000 cards is not unreasonable. Particularly for larger commercial bingo halls. I'm not talking about the basement of the church down the road here...


Paper bingo cards are sold in 3 basic formats: A single card, a strip of three, or a page of 9. Most bingo halls primarily use the 9-up page for their basic games, and the 3-up and singles for the specials.

The blue-hairs will take the 3-up and singles and tape them together into a 9-up format.

It's typical for each player to play two 9-ups per game, side by side. Arm reach as well as table size prevents playing them one above the other.

As a result, you can generally estimate that each player is playing 18 cards, although you also see players with three 9-ups. Yep. 27 cards.

Some bingo halls also have electronic games available. Players typically get the device, purchase as many games as their budget will allow, put it on auto-pilot, and then play 18 paper cards at the same time. Either that, or they go into the video room where they play slots rather than paper bingo...

Auto-pilot. Yep. The device is linked to the caller's computer system, so that the display doesn't have to show the cards. Players typically set them to just show summaries. I.E. X cards need 1 number, etc. When there is a winner, the card pops up on the display, but it is the player's responsibility to notice it and yell "Bingo." Many bingo halls have a rule that you must call bingo on the winning number.


I don't know where we were with a fair estimate of the number of players per card but here's a statistic for you: The Foxwoods Bingo hall (admitedly, one of the largest), has a capacity of 3,600. They frequently approach or hit capacity when they run the bigger games. 10,000 cards at Foxwoods is a slow day.

Quote: Wizard

However, I can't help but feeling that it is biassed. It seems like one is cherry picking the time to do the analysis.

I understand your point. My simple mind wonders if doing a similar calculation for all of the various games in the session and averaging the results would produce a better estimate?

The 75 numbers are not evenly distributed. Because there is a free square in the n column, there is only 4 numbers of 15 that need to be collected to fill that column. Would that have any bearing on the distribution and the results?

Quote: boymimbo

The 75 numbers are not evenly distributed. Because there is a free square in the n column, there is only 4 numbers of 15 that need to be collected to fill that column. Would that have any bearing on the distribution and the results?

Not for a full card - it's like you have a keno ticket with 24 numbers on it, and the house drew 55 numbers out of 75 instead of 20 out of 80. Even for patterns with multiple ways to win (like a straight line) it really doesn't matter; the calculations are more complex, but not because of the smaller N column.

The problem is that the ticket doesn't have an even distribution:

You choose 5 numbers from 15 to fill the B, 5 from 15 for I, 4 from 15 to fill the N, 5 from 15 to fill the G, and 5 from 15 to fill the O. But the distribution of the numbers drawn are even.

So, there are 3,003 combinations of B columns, 3,003 combinations of I columns, 1,365 combinations of N columns, 3,003 combinations of G's and 3,003 combinations of O's.

Multiply these all together and you get 1.11008 E+17 combinations of bingo cards, which is not the same as combin (75,24).


If you simply say combin 75,24, you get 2.57787 E+19 combinations of bingo cards. Because the total number of combinations is lower, won't that not change the answer?

Quote: boymimbo

The problem is that the ticket doesn't have an even distribution

But it doesn't matter. Let's simplify it to a 5-spot keno ticket. If I pick 1-2-3-4-5 on my keno ticket, am I less likely to win than someone who picked 1-16-31-46-61?

I kind of understand. The fact is that there are 2.57787E+19 combinations of numbers to be picked. The number of tickets are limited to the restrictions on what numbers you can pick. Therefore, the odds of winning after 24 numbers, etc are still valid?

Quote: boymimbo

I kind of understand. The fact is that there are 2.57787E+19 combinations of numbers to be picked. The number of tickets are limited to the restrictions on what numbers you can pick. Therefore, the odds of winning after 24 numbers, etc are still valid?

I'm not sure if I can explain it very well. I will say that I was quite surprised when I first learned that the partitioning of numbers into B's/I's/N's/G's/O's, and the smaller N column, have no impact on the math. It was the Wizard who first told me that (I wish I could say that I discovered it on my own, but I didn't). Perhaps he can explain it better than I can the next time he checks this thread.

First, it is not unusual for there to be over 10,000 cards in play in a Vegas bingo session. The 9 pm sessions at the Suncoast and Red Rock have large prize pools, and draw well over 10,000 cards even Mon-Thur. I'm working on estimating the average number of cards purchased per player, but I think it is a little over 100. For those who haven't played bingo in Vegas, most cards are electronic, played on machines, which can keep track of up to 4096.

Second, the probability of a bingo in 55 numbers or less is easily calculated as combin(75-24,55-24)/combin(75,55) = 0.000096537. Yes, the odds change according to the distribution by column. This figure is an overall average.

The way I look at is there are 24 numbers on the player's card. Without considering the letter, each ball drawn has the same 24/75 chance of being on the card. The coverall probability is simply the number of combinations of the balls drawn not on the card divided by the number of all combinations.

Please have a look at my new Bingo Calculator. The purpose of it is to estimate the number of cards in play, according to the distribution of balls called and number of winners.

As I wrote before, in bingo the enemy is not the casino, but the other players. How many cards the other players have is like ammunition, preventing you from winning. It is critical to know how many cards you are fighing against, to estimate your odds.

So, please have a look and let me know what you think.

That's an awesome tool, Wizard, been awhile since I've been at a good Bingo game. The Jewish Temple in a nearby town had a great Sunday Bingo game years and years ago, but I don't believe they are doing that anymore. Each round was $0.25/card and then the last round was $1.00/card. I'm unsure of how they came up with the prizes, but the last round would occasionally have a purse of over 1K.

It's a really cheap way to have a great time, in my opinion. That's especially true if you're willing to limit yourself to 1-2 cards/game. I always enjoyed it, sit around, chat people up between games, read the paper, have a snack.

In fact, I hit the 1K cover all on the last game of the day on one occasion, ended up getting chopped two ways, though.

Thanks for the kind words. I'm glad somebody had a look. The calculator was just announced on the Odds site. JB kindly cleaned up the appearnce a bit.

Quote: Mission146

It's a really cheap way to have a great time, in my opinion. That's especially true if you're willing to limit yourself to 1-2 cards/game. I always enjoyed it, sit around, chat people up between games, read the paper, have a snack.

Well call me crazy or a flea or anything you want,
but I have enjoyed strolling over to Ellis Island from the
Strip and playing their free bingo games in the
morning. (If you play $ 20 on machines the previous
day, you have qualified to play in their bingo game).

I do see the appeal of the game - it is suspenseful
when you are just 1 or 2 squares away from a
prize. We have never played at a bingo hall back
home, so there is of course also a novelty aspect to
this.

I played at a white folding table near the bar with
a group of locals. They were friendly enough and
shared some tips on shows they have seen and
enjoyed.

Just glanced at the "active members" box on the main page,
and right now it's just me. I don't think I have ever seen
that before ! Hello, hello, is anybody out there ? Major Tom
to Ground Control, Major Tom to Ground Control..............

Well, back to reading the Sunday paper and my cup of coffee.

My work in bingo continues in bingo. Lately I've been mainly playing at the Red Rock. These figures, especially average number of cards is still rather rough, so consider this a progress report.

This table shows for each session at the Red Rock the average number of cards in play, guaranteed prize pool for a level 1 card, and expected return assuming the player buys the number 2 Electronic Express special, which averages to a cost of 50 cents per level 1 card. The ER column is the expected return. This is based only on sessions with a fairly low Cash Ball jackpot of under $2,000. When it gets above some certain point it induces too much competition, and isn't worth playing.

Session	Average cards	Prize pool	ER
9:00 AM	2,501	750	60%
11:00 AM	3,385	1125	66%
1:00 PM	4,664	1750	75%
3:00 PM	3,696	1250	68%
5:00 PM	2,178	750	69%
7:00 PM	5,320	1750	66%
9:00 PM	5,190	3000	116%
11:00 PM	2,814	750	53%

The main thing to take away from this is the 9 PM session is a solid advantage play. I've known this for years. The problem is they limit the number of cards you can buy, I think because of me, so the expected profit for sitting there for an hour is about $19 only.

Quote: JohnnyQ

Just glanced at the "active members" box on the main page,
and right now it's just me. I don't think I have ever seen
that before ! Hello, hello, is anybody out there ? Major Tom
to Ground Control, Major Tom to Ground Control..............

Well, back to reading the Sunday paper and my cup of coffee.

Gee, just before you logged on there were 17 people on line? Are you related to EvenBob ?

Quote: Wizard

The main thing to take away from this is the 9 PM session is a solid advantage play. I've known this for years. The problem is they limit the number of cards you can buy, I think because of me, so the expected profit for sitting there for an hour is about $19 only.

I won't pretend to understand the higher level mathematics involved in the previous post, but this "the 9 PM session is a solid advantage play" is good to know. As with most Vegas "advantage plays" it is never really a matter that one would trek across town for but it is a matter that if one has happened to have already made that trek, the knowledge is of great value. Think perhaps of those who for whatever reasons just happen to be spending the evening at Red Rock, its 8:00pm and they all trek somewhere else ignorant of the slight, but real, advantage they could have had. Or think of that same group who, knowing of the slight advantage, spend an extra hour there and sling a few extra drinks waiting for the 9:00pm game.

Wizard, I'm confused. Why is the 9PM session a solid advantage play with more players? I thought the advantage was supposed to be when there are fewer cards in play... Also, how do you figure out the guaranteed prize pool?

I played a lot of bingo lately, some for the Wizard and some on my own. I took some friends to Fiesta bingo (paper only) and they loved it! It is fun to do paper only. Unfortunately, I haven't won anything since my win my first time playing at Palace Station... :(