Combinations in Baccarat

Hi wizards! I'm studying baccarat now and I am lost. Base on my findings in the net the possible combinations of a 1-deck baccarat of the banker winning is 6737232640, the player is 654867443 and a tie is 1372227328. But how did they come up with these numbers? I'm sure they came up with a program to count this but do you have any idea how to mathematically solve this? or do you have a program that can do this? Because I have no background in programming, if u do can you share it with me?

As always thank you to whoever responds to this query.

Quote: MathCurious

Hi wizards! I'm studying baccarat now and I am lost. Base on my findings in the net the possible combinations of a 1-deck baccarat of the banker winning is 6737232640, the player is 654867443 and a tie is 1372227328. But how did they come up with these numbers? I'm sure they came up with a program to count this but do you have any idea how to mathematically solve this? or do you have a program that can do this? Because I have no background in programming, if u do can you share it with me?

The sum of the numbers - and note the actual player number is 6,548,674,432 - is 14,658,134,400. This is 52 x 51 x 50 x 49 x 48 x 47 - in other words, this is how many possible ways the first six cards from a single deck can be dealt.

It's a little hard to do it strictly mathematically, as you have to take into account the number of cards of each value after each step.
Here is how I do it with a computer:

First, let P1 = 0, 1, 2, ..., 9
For each P1, let B1 = 0, 1, 2, ..., 9
For each B1, let P2 = 0, 1, 2, ..., 9
For each P2, let B2 = 0, 1, 2, ..., 9
For each B2, let P3 = 0, 1, 2, ..., 9
For each P3, let B3 = 0, 1, 2, ..., 9
Set an array Shoe[ ] with 10 values: Shoe[0] = 16, and Shoe[1], Shoe[2], ..., Shoe[9] each = 4
(For more than one deck, set Shoe[0] = 16 x the number of decks, and the others = 4 x the number of decks)
To determine the number of "combinations" for a particular set of six cards, do this:
Start with Combos = Shoe[P1], then reduce Shoe[P1] by 1 (for example, if the first card is an Ace, there are initially four Aces in the deck, so Combos starts at 4, and Shoe[1] is reduced to 3 as there are now only 3 Aces remaining)
Multiply Combos by Shoe[B1], then reduce Shoe[B1] by 1
Multiply Combos by Shoe[P2], then reduce Shoe[P2] by 1
Multiply Combos by Shoe[B2], then reduce Shoe[B2] by 1
Multiply Combos by Shoe[P3], then reduce Shoe[P3] by 1
Multiply Combos by Shoe[B3], then reduce Shoe[B3] by 1
Note: never mind that you may not use cards P3 or B3; since it's possible that you might use 6 cards, you have to count all possible 6-card deals.
Now, determine the result of the hand; P1 and P2 are the player's first two cards, B1 and B2 are the banker's first two cards, P3 is the player's possible third card, and B3 is the banker's possible third card.
Another note: never mind that, in reality, if the player does not take a third card but the banker does, the banker would actually get card P3 instead of B3; the results are the same.
Add up the total number of combinations (the Combos values) of the deals where the player wins, then add up the number where the banker wins, then add up the number where there is a tie.

Quote: MathCurious

Hi wizards! I'm studying baccarat now and I am lost. Base on my findings in the net the possible combinations of a 1-deck baccarat of the banker winning is 6737232640, the player is 654867443 and a tie is 1372227328. But how did they come up with these numbers? I'm sure they came up with a program to count this but do you have any idea how to mathematically solve this? or do you have a program that can do this? Because I have no background in programming, if u do can you share it with me?

As you guessed, I wrote a program to simply loop through all the way the cards can come out. It would be very tedious to do it by hand. Baccarat is a pretty easy game to program. You should be able to do it with one introductory programming class.

Quote: Wizard

As you guessed, I wrote a program to simply loop through all the way the cards can come out. It would be very tedious to do it by hand. Baccarat is a pretty easy game to program. You should be able to do it with one introductory programming class.

This is 1 of those questions with lots of combination values that currently can't be done by hand unless your using my theorem. I hate to say it again guys but I know I can do this question by hand (or rather a spreadsheet but the same differences) but I think I should just wait till I publish my research. At least that's what I've been told I should do :/

But everyone should know I can do this by hand because I'm the only person who can solve the probabilities of the Royal Deck, which has far more combinations,& yet I can solve it by hand when no computer programmer can solve them!

Quote: USpapergames

But everyone should know I can do this by hand because I'm the only person who can solve the probabilities of the Royal Deck game, which has far more combinations,& yet I can solve it by hand when no computer programmer can solve them!

This is incorrect. Many of us looked at the Royal Deck you invented and said that we could calculate the probabilities "by hand" as you put it.

We chose not to do it because there was no point -its not a game that exists anywhere, so who cares? - and because a few aspects of the game were not crisply defined. And, frankly, you acted as if the WOV forum mathematicians were circus animals that were supposed to spend time jumping through hoops simply because we were being challenged.

Somehow, you have confused a community-wide decision to ignore you with the idea that you are king of the world.

13^4 = 28,561 starting hands. From there you filter down the cases where player/banker take a third card

Pull out the handful of cases where the banker uses different rules than the player and that’s the house edge

This assumes infinite deck, which makes the calculations easier by several orders of magnitude yet gets you within a couple basis points of the exact answer for 8 decks

Quote: USpapergames

This is 1 of those questions with lots of combination values that currently can't be done by hand unless your using my theorem. I hate to say it again guys but I know I can do this question by hand (or rather a spreadsheet but the same differences) but I think I should just wait till I publish my research. At least that's what I've been told I should do :/

Be careful; you're beginning to sound like David Fabian.

There is no way to answer this question without at least a rudimentary understanding of computer programming. In that sense, it has been answered. Loop through all hands.

This spreadsheet (of mine) is in the public domain and has all the code in VBA:

https://researchers.one/doc/5f848e142c77e400043aadd8

Mike has written similar code, as have many others here.

Quote: gordonm888

This is incorrect. Many of us looked at the Royal Deck you invented and said that we could calculate the probabilities "by hand" as you put it.

We chose not to do it because there was no point -its not a game that Lexists anywhere, so who cares? - and because a few aspects of the game were not crisply defined. And, frankly, you acted as if the WOV forum mathematicians were circus animals that were supposed to spend time jumping through hoops simply because we were being challenged.

Somehow, you have confused a community-wide decision to ignore you with the idea that you are king of the world.

1) Never said I was king of the world???

2) You didn't need any game rules to solve the hand ranking probabilities!

3) The point was to prove that I was the only person who could solve those probabilities & I put a $3,000 reward for anyone who could! I believe you were the person who asked me to bump up the reward!

4) Never assuming the people on this form would just do anything for a challenge, but if you were up for the challenge of working on the most difficult game probabilities than my challenge should have been of interest to you.

5) Nobody said they could solve the probabilities at the end of the form and all that attempted failed & quiet early because they realized how naive they were thinking the question was easy.

Quote: ThatDonGuy

Be careful; you're beginning to sound like David Fabian.

I don't know who David Fabian is but I don't see the issue with stating something that is a fact. Do I need to bring back the $3,000 challenge to prove this or should I just show you that I have the correct probabilities with the correct hand rankings?

Quote: USpapergames

I don't know who David Fabian is but I don't see the issue with stating something that is a fact.

David Fabian, like you, claimed to have discovered some theorem - in his case, a "one-page proof of Fermat's Last Theorem using only methods already known in Fermat's day." He offered to reveal it for $1 million; needless to say, he got no offers, although later he withdrew the offer when he realized that something he thought was a theorem was only a conjecture. About nine years later, he returned with a claim that he really had solved it, although the proof was now 10 pages long, and he wanted to know what was the best way to make money from it. When the best answer he got was pretty much, "Reveal the proof, then, if it holds up to scrutiny, go on the lecture circuit," he and his proof pretty much disappeared.

I will gladly eat my skepticism if you have a theorem that pretty much revolutionizes probability/combinatorics. It's just that I for one have been burned too many times by promises. Ever hear the stories of the people who (a) claimed to find two new Mersenne primes, only to claim that they both ended in 3 (fairly elementary mathematics shows that every Mersenne prime > 3 ends in 1 or 7), or (b) claimed to have a "universal file compressor" that was guaranteed to compress 100% of all files of a particular size and then restore them (again, fairly elementary mathematics shows that the total number of files < N bytes is one less than the number of files of N bytes, so it is impossible to compress every file of N bytes uniquely, and if they're not unique, then they can't be restored as there is no way of knowing which of the duplicates was the original)?

Quote: USpapergames

I don't know who David Fabian is but I don't see the issue with stating something that is a fact. Do I need to bring back the $3,000 challenge to prove this or should I just show you that I have the correct probabilities with the correct hand rankings?

The Wizard has published on the internet the combination math for hundreds and hundreds of games.

Many of us have published combination math on these forums.

For a private client, I have developed the combination math for 8-card poker, where each player must form two 3-card hands (with pairs, flushes, straights, straight flushes, etc) and one 2-card hand. Try that, try 3-card flushes out of 8 cards with multiple players..

What have you done USpapergames? Worked out the combinations to a game you invented and no one has ever heard of or played?Sorry, that is the mathematical equivalent of jacking off into a Kleenex.

Quote: ThatDonGuy

David Fabian, like you, claimed to have discovered some theorem - in his case, a "one-page proof of Fermat's Last Theorem using only methods already known in Fermat's day." He offered to reveal it for $1 million; needless to say, he got no offers, although later he withdrew the offer when he realized that something he thought was a theorem was only a conjecture. About nine years later, he returned with a claim that he really had solved it, although the proof was now 10 pages long, and he wanted to know what was the best way to make money from it. When the best answer he got was pretty much, "Reveal the proof, then, if it holds up to scrutiny, go on the lecture circuit," he and his proof pretty much disappeared.

I will gladly eat my skepticism if you have a theorem that pretty much revolutionizes probability/combinatorics. It's just that I for one have been burned too many times by promises. Ever hear the stories of the people who (a) claimed to find two new Mersenne primes, only to claim that they both ended in 3 (fairly elementary mathematics shows that every Mersenne prime > 3 ends in 1 or 7), or (b) claimed to have a "universal file compressor" that was guaranteed to compress 100% of all files of a particular size and then restore them (again, fairly elementary mathematics shows that the total number of files < N bytes is one less than the number of files of N bytes, so it is impossible to compress every file of N bytes uniquely, and if they're not unique, then they can't be restored as there is no way of knowing which of the duplicates was the original)?

Ok, this makes perfect sense to me. I am sending you a message that contains 3 two hour videos where I go into great details about my theorem & even show proof that my theorem has multiple new discoveries in combinatorics. You shouldn't be skeptical after watching. In fact, I'll make you an offer that I won't make anyone else on WoV. I know your a talented mathematician, help me publish my work in exchange for me sharing it with you. I'll even give you credit as the publisher! Please don't share the links with anyone

Quote: USpapergames

Ok, this makes perfect sense to me. I am sending you a message that contains 3 two hour videos where I go into great details about my theorem & even show proof that my theorem has multiple new discoveries in combinatorics. You shouldn't be skeptical after watching. In fact, I'll make you an offer that I won't make anyone else on WoV. I know your a talented mathematician, help me publish my work in exchange for me sharing it with you. I'll even give you credit as the publisher! Please don't share the links with anyone

if you think it's good, just submit it to the Journal of Combinatorial Theory. I have 20 articles in professional mathematical journals, theorems with proofs and the like. It's a very straightforward procedure.

Quote: teliot

if you think it's good, just submit it to the journal of combinatorics. I have 20 articles in professional mathematical journals, theorems with proofs and the like. It's a very straightforward procedure.

No, formal publishing is anything but straightforward & it definitely doesn't help that I have a huge lack of time to publish, especially when you're worried about making money :/ One thing to note is that I have a language processing disorder and much of my research isn't even written in words.

It's funny that you think publishing is so easy when even Ramanujan need Hardy to publish his works! And it's not like Ramanujan was bad at English since he was an excellent writer.

Quote: gordonm888

The Wizard has published on the internet the combination math for hundreds and hundreds of games.

Many of us have published combination math on these forums.

For a private client, I have developed the combination math for 8-card poker, where each player must form two 3-card hands (with pairs, flushes, straights, straight flushes, etc) and one 2-card hand. Try that, try 3-card flushes out of 8 cards with multiple players..

What have you done USpapergames? Worked out the combinations to a game you invented and no one has ever heard of or played?Sorry, that is the mathematical equivalent of jacking off into a Kleenex.

I am aware of the Wizard's work which is why I thought maybe Shackleford would be jealous of my work since he knows I was able to solve the Royal Deck's hand ranking probabilities & have the game ready for the Cutting Edge Table Game Conference. But that's just not the case but rather he was upset at my calculus remarks. Nobody is taking away from Shackledford's success, not every game's probabilities can be solved using combinatorics anyway plus Shackleford has calculus skills that destroy mine. I'm not trying to become the best at game analysis, I'm just trying to become the best at combinatorics & game design & let me tell you that nobody is on my level when it comes to combinatorial probabilities. It's sad if you think I've only done math analysis on my own games...

Quote: ThatDonGuy

David Fabian, like you, claimed to have discovered some theorem - in his case, a "one-page proof of Fermat's Last Theorem using only methods already known in Fermat's day." He offered to reveal it for $1 million; needless to say, he got no offers, although later he withdrew the offer when he realized that something he thought was a theorem was only a conjecture. About nine years later, he returned with a claim that he really had solved it, although the proof was now 10 pages long, and he wanted to know what was the best way to make money from it. When the best answer he got was pretty much, "Reveal the proof, then, if it holds up to scrutiny, go on the lecture circuit," he and his proof pretty much disappeared.

I will gladly eat my skepticism if you have a theorem that pretty much revolutionizes probability/combinatorics. It's just that I for one have been burned too many times by promises. Ever hear the stories of the people who (a) claimed to find two new Mersenne primes, only to claim that they both ended in 3 (fairly elementary mathematics shows that every Mersenne prime > 3 ends in 1 or 7), or (b) claimed to have a "universal file compressor" that was guaranteed to compress 100% of all files of a particular size and then restore them (again, fairly elementary mathematics shows that the total number of files < N bytes is one less than the number of files of N bytes, so it is impossible to compress every file of N bytes uniquely, and if they're not unique, then they can't be restored as there is no way of knowing which of the duplicates was the original)?

I have a truly marvelous demonstration of this proposition, which this margin is too narrow to contain.

Quote: unJon

I have a truly marvelous demonstration of this proposition, which this margin is too narrow to contain.

Wow, sounds like that David Fabian guy really upset me. I just want to say I have no plans on letting people down.

Quote: USpapergames

Wow, sounds like that David Fabian guy really upset me. I just want to say I have no plans on letting people down.

I never heard of David Fabian. That’s just Fermat’s quote in the book (translated into English).

ETA: Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.

Quote: unJon

I never heard of David Fabian. That’s just Fermat’s quote in the book (translated into English).

ETA: Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.

Very nice :) Fermat wrote is several languages, which 1 is this?

Quote: USpapergames

It's sad if you think I've only done math analysis on my own games...

1. A person's reputation is based on what others think about them and not on what they think about themselves. You have published nothing and offered no proof that you have done anything.

2. You are the type of mathematician that, when he has a girl in bed, stands at the foot of that bed and tells the girl how good it is going to be. And keeps claiming and claiming and claiming how good it is going to be , , ,

And, after standing there for a awhile, he asks her "How was it?"

That's YOUR reputation.

Ouch!
Looks like something of a personal attack there, Gordon.
Please be more sensitive next time.

Quote: unJon

Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.

Thread tangent: I'd better put it in a spoiler box, in case somebody wants to read Simon Singh's The Code Book and try to solve the codes in it (BTW, the $10,000 prize was won some time ago):

Quote: gordonm888

1. A person's reputation is based on what others think about them and not on what they think about themselves. You have published nothing and offered no proof that you have done anything.

2. You are the type of mathematician that, when he has a girl in bed, stands at the foot of that bed and tells the girl how good it is going to be. And keeps claiming and claiming and claiming how good it is going to be , , ,

And, after standing there for a awhile, he asks her "How was it?"

That's YOUR reputation.

https://youtu.be/02xueo-GoXk

Thank you so much for your respond wizard! I am actually learning how to code now but Im still at lost in how to count the wins. While making the game was easy because i just randomized the drawing, making sure that the program runs all the possible permutation to count all the win and losse is harder. If you don't mind, may I ask how you did that part or show it? I'm still a beginner though, if this is annoying you can just ignore it. But any help from you will be greatly appreciated. Thank you