Royal Possibilities
Take a 9/6 JoB game (I’ll choose this because of its simplest pay table)
It has a “perfect strategy”
Of all the ~2.6million starting hands, assuming perfect play could possibly result in a Royal..or to ask another way, how many will never result in a Royal (eg hold a pair or 3 of a kind). Yet all of the “discard all” can result in at least three of the Royals (you’d never throw a pair of T’s just one of them).
How much does this change as you go up the JoB ladder-BP, DB, DDB etc. and how about Wild games…not including a wild Royal…just looking at chances for the Jackpot.
Obviously “we” play a strategy table to trend a more positive EV and lower variance in order to make our money last until that Royal shows up…but is that prolonged because we often throw away that chance for the Royal by making an optimal play?
There is a thread about a strategy to maximize your chances of getting a royal. I'm not sure it answers your questions. But obviously you'd only want to hold royal cards even sacrificing quads.
Quote: AlanMendelsonFirst of all, perfect play means disregarding the chance for a royal. In the most extreme example if dealt quads perfect play says hold the quads and that makes a royal impossible.
There is a thread about a strategy to maximize your chances of getting a royal. I'm not sure it answers your questions. But obviously you'd only want to hold royal cards even sacrificing quads.
link to original post
Allan
Exactly my point. I am not looking to maximize my Royals.
My point is just as you said it. Hold quad. No Royal.
It’s purely a statistics question.
Using perfect strategy. How many deals if played correctly could result in a Royal?
This is not meant to improve my play or win more or get better comps.
Just a math nerd question.
Quote: shantytown
Using perfect strategy. How many deals if played correctly could result in a Royal?
link to original post
If dealt 4 to the royal, 1/47 could result in a royal.
If dealt 3 to the royal, your chance is 1/1081.
If dealt 2 to the royal, it's 1/16215.
If dealt one royal card it's 1/178365.
On a complete redraw it's 1 in 383,484.75
Quote: AlanMendelson
On a complete redraw it's 1 in 383,484.75
link to original post
That number scares me as I have had quite a few Royals on a re-deal. I may have gambled a lot.
Quote: DRichQuote: AlanMendelson
On a complete redraw it's 1 in 383,484.75
link to original post
That number scares me as I have had quite a few Royals on a re-deal. I may have gambled a lot.
link to original post
It happened to me once.
Ironically, I've been dealt royals at least five times (I have photos of four of them).
Years ago before cell phone cameras I was dealt a royal on a 50 play machine. But the casino had no camera and the convenience store in the hotel wasn't open yet.
Quote: DRichQuote: AlanMendelson
On a complete redraw it's 1 in 383,484.75
link to original post
That number scares me as I have had quite a few Royals on a re-deal. I may have gambled a lot.
link to original post
It’s probably how I am asking that is confusing. But this is not my question. Let me try to clarify with examples.
Obviously any of the 2.6mil starting hands could after the draw result in a Royal depending on what you hold or don’t hold.
Deal AhTcTd3c5s you COULD hold just the Ace and have a shot at a Royal. Same if you dump all the cards or just hold one of the Tens. Right?
But no. You wouldn’t. Based on the strategy table you hold only the pair therefore nullifying the chance of this hand making a Royal because you followed VP EV strategy.
So. Take the full “list” of initial deals.
Apply the optimal strategy.
I don’t care the chance of getting the Royal.
Either yes or no. Based on the hold could the end result be a Royal. Even if the smallest chance.
What is the %/proportion of Yes to No.
Does this make it more clear what I am asking?
Quote: shantytownSo. Take the full “list” of initial deals.
Apply the optimal strategy.
I don’t care the chance of getting the Royal.
Either yes or no. Based on the hold could the end result be a Royal. Even if the smallest chance.
What is the %/proportion of Yes to No.
Does this make it more clear what I am asking?
link to original post
Yes...and, in 9/6 Jacks or Better, I get 721,420 possible deals where you can still get a Royal Flush after the discard.
Quote: ThatDonGuyQuote: shantytownSo. Take the full “list” of initial deals.
Apply the optimal strategy.
I don’t care the chance of getting the Royal.
Either yes or no. Based on the hold could the end result be a Royal. Even if the smallest chance.
What is the %/proportion of Yes to No.
Does this make it more clear what I am asking?
link to original post
Yes...and, in 9/6 Jacks or Better, I get 721,420 possible deals where you can still get a Royal Flush after the discard.
link to original post
Mind if I ask how you calculated that. (For their math nerd?) thanks in advance
Quote: shantytownQuote: ThatDonGuyQuote: shantytownSo. Take the full “list” of initial deals.
Apply the optimal strategy.
I don’t care the chance of getting the Royal.
Either yes or no. Based on the hold could the end result be a Royal. Even if the smallest chance.
What is the %/proportion of Yes to No.
Does this make it more clear what I am asking?
link to original post
Yes...and, in 9/6 Jacks or Better, I get 721,420 possible deals where you can still get a Royal Flush after the discard.
link to original post
Mind if I ask how you calculated that. (For their math nerd?) thanks in advance
link to original post
Brute force computing. Specifically, I took my existing app that generates a full strategy for any VP table, and had it count the number of hands where, after discarding for the best play, either all five cards were discarded or all of the held cards were part of the same Royal Flush. (I did make one assumption, which I am pretty sure is correct; if you discard cards and the rest are all part of the same Royal, then none of the other cards in that Royal were discarded.)
Quote: ThatDonGuyQuote: shantytownQuote: ThatDonGuyQuote: shantytownSo. Take the full “list” of initial deals.
Apply the optimal strategy.
I don’t care the chance of getting the Royal.
Either yes or no. Based on the hold could the end result be a Royal. Even if the smallest chance.
What is the %/proportion of Yes to No.
Does this make it more clear what I am asking?
link to original post
Yes...and, in 9/6 Jacks or Better, I get 721,420 possible deals where you can still get a Royal Flush after the discard.
link to original post
Mind if I ask how you calculated that. (For their math nerd?) thanks in advance
link to original post
Brute force computing. Specifically, I took my existing app that generates a full strategy for any VP table, and had it count the number of hands where, after discarding for the best play, either all five cards were discarded or all of the held cards were part of the same Royal Flush. (I did make one assumption, which I am pretty sure is correct; if you discard cards and the rest are all part of the same Royal, then none of the other cards in that Royal were discarded.)
link to original post
Thanks
What app do you use?
Also. I assume that number goes down (ie gets worse) as you go up the JoB ladder to BP DDB etc as they add more non Royal paying combos?
Does anyone have the link for that thread?Quote: AlanMendelsonFirst of all, perfect play means disregarding the chance for a royal. In the most extreme example if dealt quads perfect play says hold the quads and that makes a royal impossible.
There is a thread about a strategy to maximize your chances of getting a royal. I'm not sure it answers your questions. But obviously you'd only want to hold royal cards even sacrificing quads.
link to original post
Quote: shantytownQuote: ThatDonGuyQuote: shantytownMind if I ask how you calculated that. (For their math nerd?) thanks in advance
link to original post
Brute force computing. Specifically, I took my existing app that generates a full strategy for any VP table, and had it count the number of hands where, after discarding for the best play, either all five cards were discarded or all of the held cards were part of the same Royal Flush. (I did make one assumption, which I am pretty sure is correct; if you discard cards and the rest are all part of the same Royal, then none of the other cards in that Royal were discarded.)
link to original post
Thanks
What app do you use?
Also. I assume that number goes down (ie gets worse) as you go up the JoB ladder to BP DDB etc as they add more non Royal paying combos?
link to original post
I wrote my own in Visual C#. I think the methodology has been posted here before; it's somewhat detailed.
9/7/5 Double Bonus has 547,420 hands where you can make a Royal after perfect play discarding.
On the other hand, 9/6 DDB has 873,520.
Quote: oldbudmanJust to clarify these numbers, is the 1/1081 chance of a royal based on being dealt 3 to a royal or holding 3 to a royal? Since there are times you discard a 3 card royal and hold a high pair for instance.
link to original post
Holding.
When holding 3 to a royal, QKA of hearts for example, the probability of getting either a 10 or J of hearts for the next drawn card is 2 / 47. And the probability of getting the royal on the second drawn card is 1 / 46.
So, the probability of getting a royal holding three cards is: (2 / 47) * (1 / 46) = 1 / 1081
Quote: ThatDonGuy
Brute force computing. Specifically, I took my existing app that generates a full strategy for any VP table, and had it count the number of hands where, after discarding for the best play, either all five cards were discarded or all of the held cards were part of the same Royal Flush. (I did make one assumption, which I am pretty sure is correct; if you discard cards and the rest are all part of the same Royal, then none of the other cards in that Royal were discarded.)
link to original post
I was able to verify your number of 721,400 for 9-6 Jacks or Better using Video Poker for Winners' Perfect Play Occurrences in its Strategy Analysis. I do think the assumption that you made is violated when a singleton Ace is held from dealt hands with Ace-Ten suited plus 3 cards from deuce thru nine. To get an accurate count, we cannot include 987s that would make 4-card open-ended straights nor instances when the 3 suited low cards could make a 3-card straight flush hold. I calculated 11,556 such hands, so that number should be subtracted from 721,400.
My calculation was essentially by hand so there might have been a logic error. Details follow. There are 8 choose 3 minus 1 = 55 sets of 3 small cards from deuce thru nine. With a 2-1-1-1 suit distribution, there are 24 suit combinations. With a 2-2-1 suit distribution, there are 24 suit combinations multiplied by 3 ways to choose the 2-small-card suit. With a 3-1-1 suit distribution, there are 24 suit combinations multiplied by 3 ways to choose the small card suited with the ace-ten. With a 3-2 ATx-yz suit distribution, there are 12 suit combinations multiplied by 3 ways to choose the small card suited with the ace-ten. And there are 12 suit combinations for a 2-3 AT-xyz hand, but 27 of the rank distributions make 3-card straight flushes, so there are only 28 rank combinations of this form. So, we get 27 x (24+72+72+36) plus 28 x (24+72+72+36+12) = 11,556.
There are also 1440 hands when a King is held while the suited ten is discarded. These occur when a 2 thru 8 is suited with the KT and a 9 is present. These come in 3-1-1 and 3-2 suit distributions with 24 and 12 combinations. There are 7 choose 2 minus 1 = 20 such rank combinations and either of the low cards from 2 thru 8 could be suited with the KT. 36 x 20 x 2 = 1440.
So, the accounting would be 721,400 - 11,556 - 1440 = 708,404.
Quote: drrockI was able to verify your number of 721,400 for 9-6 Jacks or Better using Video Poker for Winners' Perfect Play Occurrences in its Strategy Analysis. I do think the assumption that you made is violated when a singleton Ace is held from dealt hands with Ace-Ten suited plus 3 cards from deuce thru nine.
link to original post
You got me on that one...
I adjusted my code, and now get 708,280 for 9/6 Jacks or Better when excluding hands where card(s) in a Royal were kept but at least one card in the same Royal was discarded.
Quote: DRichThat number scares me as I have had quite a few Royals on a re-deal. I may have gambled a lot.
link to original post
Indeed, you must have. That is called a "dirty royal," or so I'm told. I've only seen them when playing a lot of hands at once, like 26 or 39 (who those exact numbers Wiz?) and even, then not that many.
Quote: WizardQuote: DRichThat number scares me as I have had quite a few Royals on a re-deal. I may have gambled a lot.
link to original post
Indeed, you must have. That is called a "dirty royal," or so I'm told. I've only seen them when playing a lot of hands at once, like 26 or 39 (who those exact numbers Wiz?) and even, then not that many.
link to original post
We always called a dirty royal as a royal with a wild card.
My curiosity will patiently sit in the corner as long as it needs to.
Quote: DieterI've heard "royal the hard way" and "underdog".
$1 9/6?
My curiosity will patiently sit in the corner as long as it needs to.
link to original post
By playing a lower number of hands different dealt combinations will not cause a taxable handpay. For example 26 hands of a dealt full house on 9/6 JOB pays $1170.
27 hands would pay $1215 which would cause a taxable hand pay.
Quote: DRichQuote: Dieter$1 9/6?
My curiosity will patiently sit in the corner as long as it needs to.
link to original post
By playing a lower number of hands different dealt combinations will not cause a taxable handpay. For example 26 hands of a dealt full house on 9/6 JOB pays $1170.
27 hands would pay $1215 which would cause a taxable hand pay.
link to original post
Well, yeah, but I have a hunch there's a Paul Harvey tagline still lurking.
Quote: DRichwhich would cause a taxable hand pay.
link to original post
As opposed to a taxable non hand pay?
Lowering your bet to avoid a W2G does not eliminate the tax liability. But you knew that.
It saves you from tipping and the time it takes to mess with the paperwork.Quote: AlanMendelsonQuote: DRichwhich would cause a taxable hand pay.
link to original post
As opposed to a taxable non hand pay?
Lowering your bet to avoid a W2G does not eliminate the tax liability. But you knew that.
link to original post
Not only do I know, I think you are mistaken! I think you mean 38 hands -- dealt quads on a $0.25 JoB game pays $1187.50. With 39 hands it pays $1218.75 and is a taxable.Quote: WizardQuote: DRichThat number scares me as I have had quite a few Royals on a re-deal. I may have gambled a lot.
link to original post
Indeed, you must have. That is called a "dirty royal," or so I'm told. I've only seen them when playing a lot of hands at once, like 26 or 39 (who those exact numbers Wiz?) and even, then not that many.
link to original post
Years ago I designed a VP variant that had a payline for 4-to-a-Royal. I never tried to sell it, but it seemed like a good idea at the time (and something to pass the time between gigs).Quote: oldbudmanWife and I keep track of our dealt 4 to a Royal hands, and after this last trip I am up to 148 chances and counting since a drew the 5th card to complete the Royal. Variance at it's ugliest. Luckily I have hit some Royals holding 3 during this streak so I have not been shut out.
link to original post
He said something like most royals come on 2 card draws in part because you're more likely to be dealt three to the royal than 4 to the royal.
Quote: AlanMendelsonSeveral years ago on another forum a player didn't hit a one card draw until 227 (or so) hands were played. But he hit royals on 2 card draws more often.
He said something like most royals come on 2 card draws in part because you're more likely to be dealt three to the royal than 4 to the royal.
link to original post
If I recall, you can expect five to seven three to a Royal an hour and one four to a royal every five hours, so you get about thirty 3TR for every one 4TR.
