DaveG44131
DaveG44131
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September 11th, 2014 at 10:04:40 AM permalink
Hi I'm hoping a math wiz will be able to help me here. I found a bingo style table game online and want to know how good/bad a bet it is. It goes like this. The player makes a bet and is given a 1-75 bingo card with no free space. Rules/mechanics as follows

1) Coverall
2) 5 numbers called per round, 1 each B I N G & O
3) numbers are replaced after each round
4) Game lasts at most 22 rounds, if the player Bingos on or before the 22nd round they are paid, otherwise they lose their initial bet.

I also saw a miniature version using a 4x4 1-40 card (1-10,11-20,21-30,31-40) with 4 numbers called per round and lasting 12 rounds.

What are my chances of getting Bingo in these games?
Dieter
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September 11th, 2014 at 12:15:23 PM permalink
Quote: DaveG44131


4) numbers are replaced after each round



That one sounds like a killer.
May the cards fall in your favor.
beachbumbabs
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September 11th, 2014 at 8:16:40 PM permalink
I know you asked about the math, which I am not skilled enough to compute.

But, where in regular BINGO, you get closer and closer to your goal, in this game, you get further and further away because of the replacement.

The past performance now has no effect on future draws.

So, to get your first of any column, is 5 of 15
Then 4 of 15
Then 3 of 15
Then 2 of 15
Then 1 of 15

Your odds of a useful ball get worse and worse as rounds go on. And you have to do it x5 columns, all or nothing.

I would guess a house advantage of 20-25% at a SWAG.
If the House lost every hand, they wouldn't deal the game.
24Bingo
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September 11th, 2014 at 9:06:21 PM permalink
Quote: DaveG44131

1) Coverall
2) no free space

5) Game lasts at most 22 rounds, if the player Bingos on or before the 22nd round they are paid, otherwise they lose their initial bet.



Um... if I'm reading this right... do not play this game.
The trick to poker is learning not to beat yourself up for your mistakes too much, and certainly not too little, but just the right amount.
JB
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September 11th, 2014 at 9:08:40 PM permalink
Quote: DaveG44131

The player ... is given one standard 1-75 bingo card


Quote: DaveG44131

2) no free space


A standard bingo card has a free space. If the card has no free space, it is not a standard bingo card. Does the card have 4 or 5 numbers in the N column?
DaveG44131
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September 11th, 2014 at 9:40:53 PM permalink
5 numbers in the N column. Edited original post.
BTLWI
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September 13th, 2014 at 8:34:57 AM permalink
What does #3 mean?

The bingo balls that are drawn are put back in? So N-44 could be called 5 rounds in a game?

Or the not daubed numbers on your card change every round? For example your B-column is 4, 8, 10, 11, 12 the first round. They call B-12 so you get to daub it. Next round your B-column changes to 1, 2, 7, 10, daub?
beachbumbabs
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September 13th, 2014 at 9:32:37 AM permalink
Quote: BTLWI

What does #3 mean?

The bingo balls that are drawn are put back in? So N-44 could be called 5 rounds in a game?

Or the not daubed numbers on your card change every round? For example your B-column is 4, 8, 10, 11, 12 the first round. They call B-12 so you get to daub it. Next round your B-column changes to 1, 2, 7, 10, daub?



I'm not the OP, but yes, they could call N44 5 times in a game, because the numbers are replaced after each round, if what he said is correct. That's what's so deadly about the game. I don't know about the numbers changing, but I don't think they do, again referring to what he said.

The thing that's different about it from traditional BINGO, as well, is that each round guarantees one pick from each column. That might make up for some of what you lose in the replacement, mathematically. Still, not nearly enough to make it a good game, again guessing.
If the House lost every hand, they wouldn't deal the game.
DaveG44131
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September 13th, 2014 at 9:52:48 AM permalink
Quote: beachbumbabs

I'm not the OP, but yes, they could call N44 5 times in a game, because the numbers are replaced after each round, if what he said is correct. That's what's so deadly about the game. I don't know about the numbers changing, but I don't think they do, again referring to what he said.

The thing that's different about it from traditional BINGO, as well, is that each round guarantees one pick from each column. That might make up for some of what you lose in the replacement, mathematically. Still, not nearly enough to make it a good game, again guessing.



You are correct on both counts. The same number can be called more than once and each round guarantees a pick from each column. Your card never changes as BTLWI suggests. Also, I would think being guaranteed up to 110 balls being called (22 rounds x 5 balls per round) would also make up some of the "with replacement" difference, since the average Coverall game takes approximately 73 balls called to win.
ThatDonGuy
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September 13th, 2014 at 6:13:41 PM permalink
If it's Coverall, I did a Monte Carlo with 50 million runs, and won 1/763 of the time.
miplet
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September 13th, 2014 at 9:07:59 PM permalink
Quote: ThatDonGuy

If it's Coverall, I did a Monte Carlo with 50 million runs, and won 1/763 of the time.


Spread sheet transitional matrix (or whatever they are called): 1 in 766.867413 google doc here
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Dieter
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September 14th, 2014 at 7:22:33 AM permalink
Are there payouts for patterns other than coverall?

Those might make it a playable game.
May the cards fall in your favor.
GWAE
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September 14th, 2014 at 7:35:50 AM permalink
Quote: miplet

Spread sheet transitional matrix (or whatever they are called): 1 in 766.867413 google doc here



I can not imagine how long it took people to do that before excel existed.
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ThatDonGuy
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September 14th, 2014 at 8:33:31 AM permalink
Quote: miplet

Spread sheet transitional matrix (or whatever they are called): 1 in 766.867413


I started out trying one of those, but didn't have the "House light bulb moment" to figure it out. It never dawned on me that it's really five identical independent problems ("what is the probability that, if you draw a number from {1, 2, ..., 15} 22 times with replacement, each of the numbers from 1 to 5 will be drawn at least once?").

For those of you playing at home:
Assume the card is B1-5, I16-20, N31-35, G46-50, and O61-65
Concentrate on just the Bs:
Let S(n,k) be the probability of k different numbers from 1-5 being drawn after n Bs have been drawn.
S(0,0) = 1
S(n,0) = S(n-1,0) x 10/15
S(n,k) = (S(n-1,k-1) x (6-k)/15) + (S(n-1,k) x (1 - (6-k)/15)), for 0 < k < 5
S(n,5) = (S(n-1,4) x 1/15) + S(n-1,5)
S(22,5) is the probability that you have drawn all of the numbers from 1 to 5 in 22 draws
The probabilities of drawing 16-20, 31-35, 46-50, and 61-65 are the same, so the overall probability of winning is S(22,5)5.
98Clubs
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September 14th, 2014 at 5:16:53 PM permalink
busted.
Some people need to reimagine their thinking.
DaveG44131
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September 16th, 2014 at 9:11:51 AM permalink
Quote: miplet

Spread sheet transitional matrix (or whatever they are called): 1 in 766.867413 google doc here



I tried to copy your Google doc to analyze the 4x4 variant described in my OP and got 1 in approximately 828. Is this right?
DaveG44131
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September 16th, 2014 at 9:13:27 AM permalink
Quote: Dieter

Are there payouts for patterns other than coverall?

Those might make it a playable game.



Some machines had payouts for patterns other than coverall, but then the payout for coverall was reduced.
miplet
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September 16th, 2014 at 10:02:53 AM permalink
Quote: DaveG44131

Quote: miplet

Spread sheet transitional matrix (or whatever they are called): 1 in 766.867413 google doc here



I tried to copy your Google doc to analyze the 4x4 variant described in my OP and got 1 in approximately 828. Is this right?


I'm getting 361.0339019 . I added another sheet.
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DaveG44131
DaveG44131
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September 16th, 2014 at 11:06:30 AM permalink
Quote: miplet

Quote: DaveG44131

Quote: miplet

Spread sheet transitional matrix (or whatever they are called): 1 in 766.867413 google doc here



I tried to copy your Google doc to analyze the 4x4 variant described in my OP and got 1 in approximately 828. Is this right?


I'm getting 361.0339019 . I added another sheet.



Thanks for checking. I had a feeling I miscopied something.
Dieter
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September 16th, 2014 at 11:56:24 AM permalink
Quote: DaveG44131

Some machines had payouts for patterns other than coverall, but then the payout for coverall was reduced.



Yeah, but an all or nothing payout eats into your playing money quickly.

Some little wins along the way mean you have a better chance of not going broke before the big one hits.
May the cards fall in your favor.
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