Caribbean Stud Poker : Probability for winning with Royal Flush

Hello everybody,
My first time here, bad english... Please be patient and/or indulgent :)))
My question:

In the Caribbean Stud Poker table, for the player winning with a royal flush, I can see there are 16759740 combinations. As the total number of combinations is 19933230517200, it means the probability is 8.40794E-07.

But if I try to calculate by my own, I get a different result. Could you help me to find my mistake please ?
The odds of player royal are 4 in 2,598,960, which is 1 in 649,740 and the probability is 1.53908E-06
The odds of dealer royal is reduced to 3 in 1,533,939, or 1 in 511,313.
The odds of coincidental dealer royal is 1 in 332,220,508,620 (The product of the two prior values) and the probability of this happening (for a push hand) is 3.01E-12 (very weak)
So the probability for player winning with royal flush should be near 4/ 2,598,960 because the coincidence is weak: 1.53908E-06 - 3.01E-12 = 1.53907E-06 far from the expected 8.40794E-07
Where am I missing something ?
Thanks

Quote: Curious55

Hello everybody,
My first time here, bad english... Please be patient and/or indulgent :)))
My question:

In the Caribbean Stud Poker table, for the player winning with a royal flush, I can see there are 16759740 combinations. As the total number of combinations is 19933230517200, it means the probability is 8.40794E-07.

But if I try to calculate by my own, I get a different result. Could you help me to find my mistake please ?
The odds of player royal are 4 in 2,598,960, which is 1 in 649,740 and the probability is 1.53908E-06
The odds of dealer royal is reduced to 3 in 1,533,939, or 1 in 511,313.
The odds of coincidental dealer royal is 1 in 332,220,508,620 (The product of the two prior values) and the probability of this happening (for a push hand) is 3.01E-12 (very weak)
So the probability for player winning with royal flush should be near 4/ 2,598,960 because the coincidence is weak: 1.53908E-06 - 3.01E-12 = 1.53907E-06 far from the expected 8.40794E-07
Where am I missing something ?
Thanks

I assume the phrase "Player wins with royal flush" means getting the full 100x payout for the royal and not just 1 ante when dealer doesn't qualify. So you should take into account that there is a roughly 45% - 50% chance of dealer not qualifying against your royal (note that one A+K has already been removed from the deck to your royal, which means one less chance for dealer to make a A/K high qualifying hand).

As a side note I once got a royal which would have paid 20,000€ if dealer qualified. He didn't :(

Quote: Jufo81

I assume the phrase "Player wins with royal flush" means getting the full 100x payout for the royal and not just 1 ante when dealer doesn't qualify.

Of course, that's the point !
Ok, thank you very much Jufo...
Hope your next royal will pay ! :)

Quote: Curious55

Of course, that's the point !
Ok, thank you very much Jufo...
Hope your next royal will pay ! :)

If you want to know the exact probability of dealer qualifying/winning/losing against any player hand, you can use the calculator below where you put player's cards and dealers first card:

http://www.reviewpokerrooms.com/casino-games/caribbean-stud-poker/strategy-calculator.html

For example if player has Hearts royal Ah Kh Qh Jh Th and Dealer has 5 of clubs (5c) the calc says exactly 45.0832% probability for dealer not qualifying.

Quote: Jufo81

If you want to know the exact probability of dealer qualifying/winning/losing against any player hand, you can use the calculator below :
http://www.reviewpokerrooms.com/casino-games/caribbean-stud-poker/strategy-calculator.html

Ooooooooooh, I didn't know .
Thanks a lot again !!
I hope your next two royals will pay, lol

Quote: Curious55

Ooooooooooh, I didn't know .
Thanks a lot again !!
I hope your next two royals will pay, lol

Yeah me too, but you don't get them every day lol. Actually I got my royal in Oasis Poker where you can replace one of your cards by paying one ante. Other than the possibility to replace one card, the rules and payouts are exactly the same as in Caribbean Stud.

Quote: Jufo81

Yeah me too, but you don't get them every day lol. Actually I got my royal in Oasis Poker where you can replace one of your cards by paying one ante.

Do you know if any calculator for such a game exists (with possibility of drawing one card) ?
It would be very nice !

Quote: Curious55

Do you know if any calculator for such a game exists (with possibility of drawing one card) ?
It would be very nice !

No, I haven't seen any calculator for that game. But the strategy when you should replace one card is given here (it's not completely precise):
https://wizardofodds.com/oasispoker

After the card switch decision you would simply follow the same strategy as in Caribbean Stud.

If you scroll down the above link it also shows you the probability to get each hand (royal etc.). As you see, getting a Royal in Oasis is much easier because you can switch one card to get it.

Quote: Jufo81

But the strategy when you should replace one card is given here (it's not completely precise):
https://wizardofodds.com/oasispoker

What impressive the Wizard of odds site is !
Thanks to him.

Quote: Curious55

What impressive the Wizard of odds site is !
Thanks to him.

That's true. If you really wanted to play with perfect strategy with best odds, you should also add the following rules to the switching strategy:

Switch one player card also with:
-Four to a flush with a 22 to 66 pair when the dealer's upcard is a greater rank than the pair
-Four to an outside straight with a 22 to 33 pair when the dealer's upcard is a greater rank than all cards in the straight
-Four to an inside straight without pair when the high card of the straight is 10 or greater, and the dealer's upcard is lesser rank than all cards in the straight

These rules were listed at another site.

Quote: Jufo81

These rules were listed at another site.

Thanks.

Hi,
I have a doubt about these pokers rules (Caribbean and Oasis) for the comparisons of the close hands, push or not :

If dealer and player have both flush hand, is it a push or do we rank the hands by the higher card ?
If dealer and player have both straight hand, is it a push or do we rank the hands by the higher card ?
If dealer and player have both full house , is it a push or do we rank the hands by the higher 3-cards and then the higher pair ?
If dealer and player have both two pairs hand, is it a push or do we rank the hands by the higher pair, then the better second and then the better card (kicker) ?
If dealer and player have both one pair hand, is it a push or do we rank the hands by the higher pair and then by the higher card (kicker) ?
If dealer and player have both AK hand, is it a push or do we rank the hands by the third, then fourth , then fifth card?

Quote: Curious55

Thanks.

Hi,
I have a doubt about these pokers rules (Caribbean and Oasis) for the comparisons of the close hands, push or not :

If dealer and player have both flush hand, is it a push or do we rank the hands by the higher card ?
If dealer and player have both straight hand, is it a push or do we rank the hands by the higher card ?
If dealer and player have both full house , is it a push or do we rank the hands by the higher 3-cards and then the higher pair ?
If dealer and player have both two pairs hand, is it a push or do we rank the hands by the higher pair, then the better second and then the better card (kicker) ?
If dealer and player have both one pair hand, is it a push or do we rank the hands by the higher pair and then by the higher card (kicker) ?
If dealer and player have both AK hand, is it a push or do we rank the hands by the third, then fourth , then fifth card?

The answer to all those questions is the latter one. You compare all five cards starting from the highest combination, if necessary. Push only occurs when all five cards are of identical value between dealer and player.

Quote: Jufo81

The answer to all those questions is the latter one. You compare all five cards starting from the highest combination, if necessary. Push only occurs when all five cards are of identical value between dealer and player.

Thank you again !

Quote: Jufo81

you can use the calculator below where you put player's cards and dealers first card:

http://www.reviewpokerrooms.com/casino-games/caribbean-stud-poker/strategy-calculator.html

Hi,
Does anybody know how this calculator can give all these results so fast ?
Is it directly from an already established table ?

Quote: Curious55

Hi,
Does anybody know how this calculator can give all these results so fast ?
Is it directly from an already established table ?

It will cycle through all combinations.

Quote: WizardofEngland

It will cycle through all combinations.

Thank you.
I have just programmed it with Visual Basic for Excel but it takes several hours for every player's hand and dealer's upcard :((
(For every dealer's upcard, we have 163185 possible dealer's hands )
With this online calculator, it takes less than one second !

Quote: Curious55

Thank you Wizard
I have just programmed it with Visual Basic for Excel but it takes aseveral hours for every player's hand and dealer's upcard . :((
(For every dealer's upcard, we have 163185 possible dealer's hands )
With this online calculator, it takes less than one second !

sorry for repetition...

Could it be possible to get one example of the calculation of the combinations in the Caribbean stud table ?
For example , how can we calculate there are 43805516100 combinations for the player winning with a straight ?
Are these calculations already described and detailed anywhere?
Thanks.

Trying ....

I don't understand what happens with this thread. :(
Bug ?

Quote: Curious55

I don't understand what happens with this thread. :(
Bug ?

No one is interested in your questions? Ask something else?

Quote: Jufo81

No one is interested in your questions? Ask something else?

Ok, thank you. I thought there was a bug...
I will try to find by myself.

Quote: Curious55

Ok, thank you. I thought there was a bug...
I will try to find by myself.

I can always try to help with Caribbean Stud questions but unfortunately I haven't looked into how the number of combinations is calculated.

Quote: Curious55

Could it be possible to get one example of the calculation of the combinations in the Caribbean stud table ?
For example , how can we calculate there are 43805516100 combinations for the player winning with a straight ?
Are these calculations already described and detailed anywhere?
Thanks.

I looked a bit into his. I don't know if this helps you but here's what I got:

First, the total number of combinations in Caribbean Stud:

[In Excel, use formula: COMBIN(number, numbers chosen)]

Possible Player Hands = Combin(52,5) = 2598960 (Choose any 5 cards from 52 card deck)
Possible Dealer hands = Combin(47,5) = 1533939 (Choose any 5 cards from remaining 47 cards)
There are five different possibilities for dealer's upcard from dealer's five cards. So the total number of combinations is:

Combin(52,5)*Combin(47,5)*5 = 19933230517200

This matches with the total number of combinations in Wizard of Odd's table: https://wizardofodds.com/caribbeanstud

You asked for the number of combinations for player's winning straight:

There are 10 possibilities for the rank of the straight (ie. straight's highest card is either 5,6,7,...,A)

Then there are four possibilites for the suit of each card in the straight:

So the total number of combinations that makes a straight: 10*4*4*4*4*4 = 10240.

However four of these combinations are royals (one for each suit) and 36 of them are straight flushes (there are 9 different straight flushes multiplied by four different suits), so by substracting these, the number of combinations for straight is:

10240 - 36 -4 = 10210.
See https://wizardofodds.com/poker , and there table "5 Card Stud" which agrees with the number of combinations for a straight.

Now including all Dealer hand combinations (see above): Combin(47,5)*5 = 7669695, there are

10210*7669695 = 78307585950 combinations in total that includes a player straight.

The part that I didn't calculate is that 43805516100 out of these 78307585950 combinations end up being the ones where player wins with a straight. The remaining 78307585950 - 43805516100 = 34502069850 combinations are either the ones where dealer doesn't qualify (player still wins but only gets paid his ante bet) or dealer's hand beats player's straight. I don't know if there is an easy way to arrive at those 43805516100 combinations other than to use a computer that goes through all possible combinations.

Not sure if this helped you or not...

What an answer!

Quote: Jufo81

Not sure if this helped you or not...

Yes and ... No :))

Quote: Jufo81

So the total number of combinations is:
Combin(52,5)*Combin(47,5)*5 = 19933230517200

I know that. We can also consider other point of view as:
Combin(52,6)*Combin(46,4)*6 = 19933230517200 or Combin(51,5)*Combin(46,4)*52= 19933230517200

Quote: Jufo81

There are 10 possibilities for the rank of the straight (ie. straight's highest card is either 5,6,7,...,A)
Then there are four possibilites for the suit of each card in the straight:
So the total number of combinations that makes a straight: 10*4*4*4*4*4 = 10240.
etc...

Yes, I agree....
The same with 3744 Full houses or 54912 three of a kind etc...

Quote: Jufo81

The part that I didn't calculate is that 43805516100 out of these 78307585950 combinations end up being the ones where player wins with a straight. The remaining 78307585950 - 43805516100 = 34502069850 combinations are the ones where dealer doesn't qualify (player still wins but only gets paid his ante bet)

Yes, it was my question.

Quote: Jufo81

or dealer's hand beats player's straight.

No, I think these ones are included in the 2726592727512 hands the dealers wins

Quote: Jufo81

I don't know if there is an easy way to arrive at those 43805516100 combinations other than to use a computer that goes through all possible combinations.

I think you are right because these numbers depend on the strategy.
The Wizard said :"The following table shows the all possible outcomes, assuming the player follows optimal strategy ".
Any different strategy (for example, an easier one concerning the AK hands) should have given other numbers of combinations and so, I think a computer is needed.

Jufo81, thanks a lot for helping !!

Quote: Curious55

I think you are right because these numbers depend on the strategy.
The Wizard said :"The following table shows the all possible outcomes, assuming the player follows optimal strategy ".
Any different strategy (for example, an easier one concerning the AK hands) should have given other numbers of combinations and so, I think a computer is needed.

Jufo81, thanks a lot for helping !!

The numbers of combinations of course depend on playing decisions [raise/fold] (and in Oasis Poker additional decision to buy a replacement card). But in a situation where player has a pat straight, the decision is obviously to raise every time since the probability to lose is much less than 1%. So the reason a computer is needed in calculating the probability for "player winning with straight" is not the strategy but rather the fact that the composition of player's exact cards have a small effect on dealer's probability of qualifying and also dealer's probability to beat player's hand (a King-high player straight would have lower chances to get beaten by dealer than, say, Six-high straight).

Quote: Jufo81

But in a situation where player has a pat straight, the decision is obviously to raise every time since the probability to lose is much less than 1%. So the reason a computer is needed in calculating the probability for "player winning with straight" is not the strategy

You are right for the straight hands and...I wasn't clear.
I would like to talk about AK hands. If we change the strategy for AK hands , the number of combinations where "player wins with pair or less" will change.
Anyway and actually, that doesn't answer if computer is necessary or not.
Of course a computer will help but doesn't get calculations impossible.

Quote: Curious55

You are right for the straight hands and...I wasn't clear.
I would like to talk about AK hands. If we change the strategy for AK hands , the number of combinations where "player wins with pair or less" will change.
Anyway and actually, that doesn't answer if computer is necessary or not.
Of course a computer will help but doesn't get calculations impossible.

AK high hands are interesting because these are the marginal plays between raising and folding. Some people never play AK hands saying that they are "too risky" whereas I play AK hands optimally because it is always cool to beat dealer's AKJ high with AKQ high and people are like "OMG what the hell happened" at the table when that happens.

But I don't see how analyzing AK hands would be any simpler than analyzing player straights. If you look at the optimal AK hand decision chart (https://wizardofodds.com/caribbean/when-to-stand-on-ace-king.html) I think each cell in that table has required computer simulation to go through possible combinations to determine the decision for each cell. So I don't see how you can get the precise result without a computer simulation. If you do find a way, let us know.

Hello Jufo,

Quote: Jufo81

For example if player has Hearts royal Ah Kh Qh Jh Th and Dealer has 5 of clubs (5c) the calc says exactly 45.0832% probability for dealer not qualifying.

And ....the general probability (player has Hearts royal Ah Kh Qh Jh Th and dealer's card is unknown) is 0.453700571 (see below...)
I looked for this number for answering my initial question.
I would like to check there actually were 16759740 combinations for Player winning with royal flush .

Both player or dealer have 2598960 different possible hands.
If we know the player's five cards, the dealer has (47,5) or 1533939 possible hands .
It means that if , among the 2598960 hands, I remove all these with Ah or Kh or Qh or Jh or Th, it will be only 1533939 hands left.

But what about the 1135260 high-card hands (out of 2598960) that disqualify the dealer?
There are 329 Ace-High hands with 1020 colour combinations for a total of 335580
There are 329 K-High hands with 1020 colour combinations for a total of 335580
There are 209 Q-High hands with 1020 colour combinations for a total of 213180
There are 125 J-High hands with 1020 colour combinations for a total of 127500
There are 69 T-High hands with 1020 colour combinations for a total of 70380
There are 34 9-High hands with 1020 colour combinations for a total of 34680
There are 14 8-High hands with 1020 colour combinations for a total of 14280
There are 4 7-High hands with 1020 colour combinations for a total of 4080

What happens if I remove all Ah or Kh or Qh or Jh or Th? We will get:
For the Ace-High hands, for exemple, we obtain:
321 hands with 8 colour combinations
429 hands with 84 colour combinations
573 hands with 168 colour combinations
765 hands with 69 colour combinations
The total of Ace-High hands is 187653.

The total of Ace-High hands is 187653
The total of K-High hands is 188097
The total of Q-High hands is 129309
The total of J-High hands is 85065
The total of T-High hands is 52785
The total of 9-High hands is 34680
The total of 8-High hands is 14280
The total of 7-High hands is 4080
Finally:
We get 695949 High-card that disqualify the dealer. the probability for dealer not qualifying is 0.453700571
Among the 1533939 dealer's hands : 3 will push, 695949 will not qualify and therefore 837987 will lose.

So the Hearts royal Ah Kh Qh Jh Th will win 837987 times.
As there are 4 suits and 5 dealer cards, there are 20*837987 combinations or 16759740 combinations for Player winning with royal flush .