Handling Sequential Royals in an Video Poker Analyzer
October 12th, 2020 at 6:50:12 PM
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A friend sent me a link to an article about the recent monster sequential royal jackpot at Red Rock Casino. The payoff for the progressive jackpot was 120,680 bets versus 800 bets for a normal royal. It appears to be a reversible royal jackpot.
I was able to estimate the contribution of the progressive to the EV, but I could not get the exact EV with my current VP analyzer. I have not been gambling much due to the pandemic. Out of boredom, I decided to add Sequential Royal Flushes (SRF) to my video poker analyzer. At first, I thought this might slow the analysis considerably. There are 120 different ways the cards can be dealt, so any dealt hand containing royal cards in one or more suits will have different expectation values (EVs) for holding the same cards, depending on the order of the dealt cards. I quickly realized that only holds that qualify as ordinary royal draws need to be handled specially. This is only four of 32 possible holds for the most complicated case of a dealt hand containing royal cards of four different suits.
My program takes 1.2 seconds to analyze the EV for a standard VP game using a single thread, and another 0.3 seconds to create a very accurate strategy for the game. I can analyze games with SRFs in 3.6 seconds. Therefore, games including SRFs only take three times longer than non-SRF games.
I use C++ to code my analyzer and emscripten to connect the C++ code to my html GUI. I suppose every VP analyzer has a loop like this:
This fills an array, double holdEV[32], with the EV resulting from each of the 32 possible draws. The variable, dt, is the draw-type index. The index, t, is the type of paying hand, and pay[t] holds the pay-table for this particular game. The array numHeld[dt] holds the number of cards held for each draw type. The store instance holds the precomputed number of ways of drawing each type of paying hand. The divNumHeld array tells me how many ways I can draw to a hand for a particular draw type and number of jokers (in this case, zero jokers).
To handle SRFs, I add a bit of code inside this loop that only triggers if it is possible to draw an ordinary RF (t == isRF):
This added code considers the 120 possible ways the hand can be dealt, checks if the order makes this draw type a SRF draw, and then stores the number of normal SRFs (nSeqRF) and reverse-order SRFs (rSeqRF) that are possible for each ordering and draw type. I had carefully defined the 120 hand orders such that the 120th ordering is the opposite of the first ordering, the same for the 119th and 2nd orderings, etc. Therefore, I don't need to have a separate function that decides whether a reverse SRF is possible. If the Nth ordering makes a SRF possible, the the (120 - N)th ordering makes a reverse SRF possible.
Note that out of 32 draw types, only between one and four draws can yield any royals. So, the ord-loop code rarely executes, and the isSeq(ord, dt) function usually returns false, so the inner code rarely executes. This preliminary calculation doesn't take as much time as I expected.
The part of the analysis that takes extra time is finding the right hold for each order. For normal video poker, I search the holdEV[dt] to find the draw type that gives the highest EV. With SRFs, I simply do this 120 times for the different orders after adjusting the holdEV[dt] for the numbers of SRFS (nSRF = nSeqRF +rSeqRF). This adjustment is only a function of the number of cards held and nSRF, so I use a precomputed array to adjust the holdEV array and avoid some floating point operations:
I then determine the hold with the highest EV for all 120 orderings of the dealt cards, and use the average of all these 120 EVs for the EV of the hand.
I could speed the code up more by only doing the best-EV recalculation for orderings of the dealt hands where any SRFS are possible. For example, if a dealt hand has only one RF card, then only 24 of the 120 orderings have a SRF draw and another 24 have a reverse SRF draw. (If the RF card is a queen, the same 24 orderings pertain to both.) When there are four or five cards to the RF, only two of the 120 orderings have SRF draws. All of the other orderings have the same strategy and EV. I prefer to keep the code simple, and I just burn the CPU cycles to loop over all 120 orderings.
As a sanity check, I set the SRF payoff to 1,000,000 for one. This means all SRF draws holding 3 to 5 cards are the best hold, even holding three to a RF versus holding trips. It is easy to calculate the cycles for SRFs when you always draw for them. I calculated the cycle for drawing a SRF holding 3, 4, or 5 cards and put these in the last column. These exactly match the cycles that came out of my VP analyzer:
For a SRF paying 10,000 for one, the cycle for getting a SRF holding three cards is slightly longer than the example above. For holding 4 or 5 cards, the cycles are still the same, as I would expect:
If I run the program for a range of SRF payoff values, I can see how the EV increases with the value of the SRF,
The frequency of SRFs is one in 2,414,162 when they pay the same as a normal royal. This drops to one in 1,479,061 when the SRF pays 120,000 for one and the player makes all the right strategy changes.
I was able to estimate the contribution of the progressive to the EV, but I could not get the exact EV with my current VP analyzer. I have not been gambling much due to the pandemic. Out of boredom, I decided to add Sequential Royal Flushes (SRF) to my video poker analyzer. At first, I thought this might slow the analysis considerably. There are 120 different ways the cards can be dealt, so any dealt hand containing royal cards in one or more suits will have different expectation values (EVs) for holding the same cards, depending on the order of the dealt cards. I quickly realized that only holds that qualify as ordinary royal draws need to be handled specially. This is only four of 32 possible holds for the most complicated case of a dealt hand containing royal cards of four different suits.
My program takes 1.2 seconds to analyze the EV for a standard VP game using a single thread, and another 0.3 seconds to create a very accurate strategy for the game. I can analyze games with SRFs in 3.6 seconds. Therefore, games including SRFs only take three times longer than non-SRF games.
I use C++ to code my analyzer and emscripten to connect the C++ code to my html GUI. I suppose every VP analyzer has a loop like this:
for (dt = 0; dt < 32; dt++) {
pay = 0.0;
nh = numHeld[dt];
for (t = 0; t < maxType; t++) {
n = tCount(dt, t);
if (n > 0) {
pay += pays[t] * n;
}
}
holdEV[dt] = pay / divNumHeld[numJoker][nh];
}This fills an array, double holdEV[32], with the EV resulting from each of the 32 possible draws. The variable, dt, is the draw-type index. The index, t, is the type of paying hand, and pay[t] holds the pay-table for this particular game. The array numHeld[dt] holds the number of cards held for each draw type. The store instance holds the precomputed number of ways of drawing each type of paying hand. The divNumHeld array tells me how many ways I can draw to a hand for a particular draw type and number of jokers (in this case, zero jokers).
To handle SRFs, I add a bit of code inside this loop that only triggers if it is possible to draw an ordinary RF (t == isRF):
for (dt = 0; dt < 32; dt++) {
pay = 0.0;
nh = numHeld[dt];
for (t = 0; t < maxType; t++) {
n = tCount(dt, t);
if (n > 0) {
pay += pays[t] * n;
if (t == isRF) {
for (int ord = 0; ord < 120; ord++) {
if Seq(ord, dt)) {
nSeqRF[ord][dt] += n;
rSeqRF[119 - ord][dt] += n;
}
}
}
}
}
holdEV[dt] = pay / divNumHeld[numJoker][nh];
}This added code considers the 120 possible ways the hand can be dealt, checks if the order makes this draw type a SRF draw, and then stores the number of normal SRFs (nSeqRF) and reverse-order SRFs (rSeqRF) that are possible for each ordering and draw type. I had carefully defined the 120 hand orders such that the 120th ordering is the opposite of the first ordering, the same for the 119th and 2nd orderings, etc. Therefore, I don't need to have a separate function that decides whether a reverse SRF is possible. If the Nth ordering makes a SRF possible, the the (120 - N)th ordering makes a reverse SRF possible.
Note that out of 32 draw types, only between one and four draws can yield any royals. So, the ord-loop code rarely executes, and the isSeq(ord, dt) function usually returns false, so the inner code rarely executes. This preliminary calculation doesn't take as much time as I expected.
The part of the analysis that takes extra time is finding the right hold for each order. For normal video poker, I search the holdEV[dt] to find the draw type that gives the highest EV. With SRFs, I simply do this 120 times for the different orders after adjusting the holdEV[dt] for the numbers of SRFS (nSRF = nSeqRF +rSeqRF). This adjustment is only a function of the number of cards held and nSRF, so I use a precomputed array to adjust the holdEV array and avoid some floating point operations:
holdEV[dt] += extraSeqEV[nh][nSRF];
I then determine the hold with the highest EV for all 120 orderings of the dealt cards, and use the average of all these 120 EVs for the EV of the hand.
I could speed the code up more by only doing the best-EV recalculation for orderings of the dealt hands where any SRFS are possible. For example, if a dealt hand has only one RF card, then only 24 of the 120 orderings have a SRF draw and another 24 have a reverse SRF draw. (If the RF card is a queen, the same 24 orderings pertain to both.) When there are four or five cards to the RF, only two of the 120 orderings have SRF draws. All of the other orderings have the same strategy and EV. I prefer to keep the code simple, and I just burn the CPU cycles to loop over all 120 orderings.
As a sanity check, I set the SRF payoff to 1,000,000 for one. This means all SRF draws holding 3 to 5 cards are the best hold, even holding three to a RF versus holding trips. It is easy to calculate the cycles for SRFs when you always draw for them. I calculated the cycle for drawing a SRF holding 3, 4, or 5 cards and put these in the last column. These exactly match the cycles that came out of my VP analyzer:
| # Held | Probability | % of SRF | SRF Cycle | Calc Cycle |
|---|---|---|---|---|
| 0 | 0.00000000093 | 0.13% | 1073294773 | = |
| 1 | 0.00000005405 | 7.50% | 18500741 | = |
| 2 | 0.00000025547 | 35.44% | 3914397 | = |
| 3 | 0.00000025651 | 35.58% | 3898440 | 3898440 |
| 4 | 0.00000012826 | 17.79% | 7796880 | 7796880 |
| 5 | 0.00000002565 | 3.56% | 38984400 | 38984400 |
SeqRF pays 1,000,000 for 1
For a SRF paying 10,000 for one, the cycle for getting a SRF holding three cards is slightly longer than the example above. For holding 4 or 5 cards, the cycles are still the same, as I would expect:
| # Held | Probability | % of SRF | SRF Cycle | Calc Cycle |
|---|---|---|---|---|
| 0 | 0.00000000119 | 0.22% | 838235093 | = |
| 1 | 0.00000001414 | 2.66% | 70710440 | = |
| 2 | 0.00000011205 | 21.10% | 8924322 | = |
| 3 | 0.00000024987 | 47.04% | 4002102 | 3898440 |
| 4 | 0.00000012826 | 24.15% | 7796880 | 7830200 |
| 5 | 0.00000002565 | 4.83% | 38984400 | 38984400 |
SeqRF pays 10,000 for 1
If I run the program for a range of SRF payoff values, I can see how the EV increases with the value of the SRF,
| SRF Pays | SRF EV | Game EV | SRF Cycle |
|---|---|---|---|
| 800 | 0.03% | 96.87% | 2414162 |
| 5000 | 0.26% | 97.07% | 1935832 |
| 10000 | 0.54% | 97.34% | 1863686 |
| 15000 | 0.81% | 97.61% | 1851058 |
| 20000 | 1.08% | 97.88% | 1846229 |
| 40000 | 2.35% | 99.00% | 1701320 |
| 60000 | 3.65% | 100.19% | 1645224 |
| 80000 | 4.90% | 101.42% | 1633834 |
| 100000 | 6.48% | 102.65% | 1543558 |
| 120000 | 8.11% | 103.97% | 1479061 |
The frequency of SRFs is one in 2,414,162 when they pay the same as a normal royal. This drops to one in 1,479,061 when the SRF pays 120,000 for one and the player makes all the right strategy changes.
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October 12th, 2020 at 6:55:16 PM
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What is the full pay table for the game?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
October 12th, 2020 at 7:05:04 PM
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Quote: WizardWhat is the full pay table for the game?
The first two tables were computed for 9/6 JoB.
The article I read said the big SRF was hit while playing 6/5 Bonus Poker. Below is the pay table for the third table in the original post.
| Hand Name | Payout | Probability | Cycle | % of Return |
|---|---|---|---|---|
| SeqRF | 120680 | 0.00000067616 | 1478930.96 | 8.159948% |
| Royal | 800 | 0.00002557266 | 39104.2679 | 2.045812% |
| Str_Flush | 50 | 0.00010474637 | 9546.87064 | 0.523732% |
| Quads | 25 | 0.00162391036 | 615.797537 | 4.059776% |
| QuadA | 80 | 0.00020138345 | 4965.65125 | 1.611068% |
| Quad234 | 40 | 0.00052530403 | 1903.65948 | 2.101216% |
| Full_House | 6 | 0.01134779636 | 88.1228361 | 6.808678% |
| Flush | 5 | 0.01075497391 | 92.9802348 | 5.377487% |
| Straight | 4 | 0.01105209506 | 90.4805826 | 4.420838% |
| Trips | 3 | 0.07393991086 | 13.5244956 | 22.181973% |
| Two_Pair | 2 | 0.12854140450 | 7.77959447 | 25.708281% |
| JOB | 1 | 0.21019485503 | 4.75749038 | 21.019486% |
| Nada | 0 | 0.55168737125 | 1.81262079 | 0.000000% |
Last edited by: Mental on Oct 12, 2020
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October 12th, 2020 at 7:43:12 PM
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My eyebrows are raised.
Assuming no strategy changes from 6-5 bonus poker, I get an average royal win of 2,798 and a return of 101.83%.
What denom is this game?
I may personally check it out tomorrow.
Assuming no strategy changes from 6-5 bonus poker, I get an average royal win of 2,798 and a return of 101.83%.
What denom is this game?
I may personally check it out tomorrow.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
October 12th, 2020 at 7:45:26 PM
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Quote: WizardMy eyebrows are raised.
Assuming no strategy changes from 6-5 bonus poker, I get an average royal win of 2,798 and a return of 101.83%.
What denom is this game?
I may personally check it out tomorrow.
I believe he is referring to the one that hit at Red Rock that is 0.25 denom.
At my age, a "Life In Prison" sentence is not much of a deterrent.
October 12th, 2020 at 8:27:56 PM
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Quote: DRichI believe he is referring to the one that hit at Red Rock that is 0.25 denom.
Right. I am sorry that I could not figure out how to link the article here. /news/local-news/gambler-hits-150k-jackpot-at-red-rock-casino-resort/ I am only inferring from the picture in the article that this is a reversible royal game, but maybe it is one-way with AKQJT order or even wrap-around.
The return for a one-way sequential royal is 100.43374301%
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October 12th, 2020 at 8:38:20 PM
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Oops. I wrote: "For a SRF paying 10,000 for one, the cycle for getting a SRF holding three cards is slightly longer than the example above. For holding 4 or 5 cards, the cycles are still the same, as I would expect:" but I originally meant to write this about the 100,000 SRF.
It is not true for the 10,000 SRF. For a 10,000 SRF, only the SRF cycle numbers for a five card hold are the same. You don't break up a dealt straight flush, 9TJQK, so the cycle is longer for hitting 4-card SRF draws, than for a strategy where you always draw for SRFs.
| # Held | Probability | % of SRF | SRF Cycle | Calc Cycle |
|---|---|---|---|---|
| 0 | 0.00000000060 | 0.19% | 1676470186 | = |
| 1 | 0.00000001282 | 4.06% | 78032126 | = |
| 2 | 0.00000009671 | 30.67% | 10339768 | = |
| 3 | 0.00000012824 | 40.67% | 7797601 | 7796880 |
| 4 | 0.00000006413 | 20.34% | 15593760 | 15593760 |
| 5 | 0.00000001283 | 4.07% | 77968800 | 77968800 |
SeqRF pays 100,000
It is not true for the 10,000 SRF. For a 10,000 SRF, only the SRF cycle numbers for a five card hold are the same. You don't break up a dealt straight flush, 9TJQK, so the cycle is longer for hitting 4-card SRF draws, than for a strategy where you always draw for SRFs.
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October 13th, 2020 at 12:57:30 PM
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Back in June I got a call regarding this Progressive with veg details, other than some super inflated percentage. Since there were lots of unknown details to me, I thought there could be more to this, perhaps a progressive dump or something. I do appropriate the effort but after more information I tried to explain why this was not something I was interested in. the person seemed annoyed that I wasn't interested. I tried to explain why but they seem to be focused on the large number and small bet.
♪♪Now you swear and kick and beg us That you're not a gamblin' man Then you find you're back in Vegas With a handle in your hand♪♪ Your black cards can make you money So you hide them when you're able In the land of casinos and money You must put them on the table♪♪ You go back Jack do it again roulette wheels turinin' 'round and 'round♪♪ You go back Jack do it again♪♪
October 13th, 2020 at 2:38:58 PM
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Quote: AxelWolfBack in June I got a call regarding this Progressive with veg details, other than some super inflated percentage. Since there were lots of unknown details to me, I thought there could be more to this, perhaps a progressive dump or something. I do appropriate the effort but after more information I tried to explain why this was not something I was interested in. the person seemed annoyed that I wasn't interested. I tried to explain why but they seem to be focused on the large number and small bet.
I also know few of the details. Was this a two-way SRF game? Single machine or linked bank? What is the meter rate? If I assume 2% meter. 10K bet meter reset level, and 1.8M RSRF cycle, then this only went a bit over three cycles before hitting. If the meter is much slower, then it is less likely that the progressive got to 120K bets by ordinary accumulation.
The variance is 9869 * $1.252 =15,420 $2. That is slightly lower variance than playing FPDW at $25 per hand (16,147 $2). I would have played some if I was in the neighborhood. Many folks don't have the bankroll for it, but the hourly EV is okay.
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October 13th, 2020 at 8:08:32 PM
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Hello,
Thank you for the posts. Are using the terms "sequential" and "reversible" as synonyms in your results?
Thanks
Tony8216
Thank you for the posts. Are using the terms "sequential" and "reversible" as synonyms in your results?
Thanks
Tony8216
October 14th, 2020 at 8:25:48 AM
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It takes a lot of effort to be clear and comprehensive on terminology.Quote: Tony8216Hello,
Thank you for the posts. Are using the terms "sequential" and "reversible" as synonyms in your results?
I did define Sequential Royal Flushes as SRF, and I use it to generically describe any sequential RF type. I also used RSRF to refer specifically to a two-way SRF. Since there are even wraparound sequential RFs, I would use the acronym WSRF if I ever cared to work on these.
I will try to tag each table in any future posts with one-way SRF or two-way SRF to be clearer. I will also include the game and pay table.
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October 14th, 2020 at 10:57:59 AM
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Quote: MentalIt takes a lot of effort to be clear and comprehensive on terminology.
I did define Sequential Royal Flushes as SRF, and I use it to generically describe any sequential RF type. I also used RSRF to refer specifically to a two-way SRF. Since there are even wraparound sequential RFs, I would use the acronym WSRF if I ever cared to work on these.
I will try to tag each table in any future posts with one-way SRF or two-way SRF to be clearer. I will also include the game and pay table.
I agree with Mental's use of terminology. From a math point of view, it does not matter whether a payout is for TJQKA or AKQJT RFs; it simply matters whether the payout is for one ordered sequence or for two ordered sequences.
Its interesting that in a game with a payout for both TJQKA and AKQJT RFs, when you have a hand such as:
2d-3c-Qs-6h-7c you will be drawing to the Queen with two ways to make a SRF.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
October 14th, 2020 at 1:52:54 PM
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This was last night. I think this is the 3rd, maybe 4th time I have had this draw on the same machine over the course of a year.
http://imgur.com/gallery/8zPI3BY
http://imgur.com/gallery/8zPI3BY
October 15th, 2020 at 8:58:40 AM
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Quote: VegasriderThis was last night. I think this is the 3rd, maybe 4th time I have had this draw on the same machine over the course of a year.
Your SRF would have paid 15,846 bets. Your EV was 339 bets. Here are the average EVs for each sequential hold. Even though the Q counts for two SRF draws on a two-way RSRF game, it still gives a worse EV than a jack or ace held in a sequential position.
1 1 3 4 5 9 80 160 160 400 50 50 800 15846 : Double Double Bonus
| EV | Seq Hold |
|---|---|
| 0.323 | |
| 0.440 | J |
| 0.439 | Q |
| 0.432 | K |
| 0.464 | A |
| 0.599 | JT |
| 0.583 | QT |
| 0.567 | KT |
| 0.558 | AT |
| 0.711 | QJ |
| 0.696 | KJ |
| 0.687 | AJ |
| 0.696 | KQ |
| 0.687 | AQ |
| 0.687 | AK |
| 8.381 | QJT |
| 8.292 | KJT |
| 8.201 | AJT |
| 8.292 | KQT |
| 8.201 | AQT |
| 8.201 | AKT |
| 8.388 | KQJ |
| 8.297 | AQJ |
| 8.297 | AKJ |
| 8.297 | AKQ |
| 339.606 | KQJT |
| 338.419 | AQJT |
| 338.419 | AKJT |
| 338.419 | AKQT |
| 338.482 | AKQJ |
I noticed that the SRF progressive at Hard Rock was almost high enough to start holding a single ten in the right position. This would become the right play above 126,000 bets, and that progressive was already at 120,680 bets.
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October 15th, 2020 at 1:42:14 PM
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I created a table of the average values of the sequential RF holds assuming two-way SRFs. The top column is the payoff for a SRF starting at 800 and going to 120,000 bets. Therefore, the second column is equivalent to normal 6/5 BP with a standard 800 payout on a RF. These numbers should be the same as the values you would get from any VP analyzer for 6/5 BP.
For example, the table indicates that the value of a xxQJT hold averages to 1.463. (For 6/5 BP, the EV of TJQxx ranges from around 1.4191 to 1.4903, depending on penalty cards.)
Also, AxxxT is not a hold if the SRF pays 800, but AxxxT in the sequential positions is better than holding just an ace when the SRF pays 5000. ('-nan' means 'not a number'. Since there are zero cases, the average is undefined)
You could use this table to find the strategy break points for the two-way SRF progressive at different levels by interpolating the table. There are very few strategy break points for 4-card SRF holds, so I omitted these. Also, these are average EV values that don't consider penalty cards.
1 2 3 4 5 6 40 80 25 50 800 ???: Bonus Poker
For example, the table indicates that the value of a xxQJT hold averages to 1.463. (For 6/5 BP, the EV of TJQxx ranges from around 1.4191 to 1.4903, depending on penalty cards.)
Also, AxxxT is not a hold if the SRF pays 800, but AxxxT in the sequential positions is better than holding just an ace when the SRF pays 5000. ('-nan' means 'not a number'. Since there are zero cases, the average is undefined)
You could use this table to find the strategy break points for the two-way SRF progressive at different levels by interpolating the table. There are very few strategy break points for 4-card SRF holds, so I omitted these. Also, these are average EV values that don't consider penalty cards.
1 2 3 4 5 6 40 80 25 50 800 ???: Bonus Poker
| Seq Hold | 800 | 5000 | 10000 | 15000 | 20000 | 30000 | 40000 | 60000 | 80000 | 100000 | 120000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.356 | 0.356 | 0.356 | 0.356 | 0.356 | 0.357 | 0.357 | 0.358 | 0.359 | 0.359 | 0.360 | |
| J | 0.474 | 0.475 | 0.476 | 0.477 | 0.478 | 0.480 | 0.483 | 0.486 | 0.490 | 0.493 | 0.496 |
| Q | 0.470 | 0.472 | 0.474 | 0.476 | 0.479 | 0.483 | 0.486 | 0.493 | 0.500 | 0.508 | 0.516 |
| K | 0.465 | 0.466 | 0.467 | 0.469 | 0.470 | 0.472 | 0.474 | 0.479 | 0.484 | 0.487 | 0.491 |
| A | 0.478 | 0.479 | 0.480 | 0.481 | 0.482 | 0.483 | 0.484 | 0.486 | 0.489 | 0.494 | 0.498 |
| JT | 0.488 | 0.526 | 0.577 | 0.627 | 0.679 | 0.781 | 0.878 | 1.083 | 1.288 | 1.494 | 1.689 |
| QT | 0.474 | 0.512 | 0.561 | 0.612 | 0.663 | 0.766 | 0.863 | 1.068 | 1.273 | 1.479 | 1.674 |
| KT | 0.464 | 0.497 | 0.545 | 0.596 | 0.647 | 0.750 | 0.848 | 1.053 | 1.258 | 1.464 | 1.658 |
| AT | -nan | 0.485 | 0.533 | 0.583 | 0.635 | 0.737 | 0.836 | 1.040 | 1.246 | 1.451 | 1.657 |
| QJ | 0.596 | 0.638 | 0.689 | 0.740 | 0.792 | 0.894 | 0.997 | 1.201 | 1.406 | 1.591 | 1.797 |
| KJ | 0.580 | 0.622 | 0.673 | 0.724 | 0.776 | 0.879 | 0.982 | 1.186 | 1.391 | 1.576 | 1.782 |
| AJ | 0.567 | 0.610 | 0.661 | 0.712 | 0.764 | 0.867 | 0.970 | 1.174 | 1.379 | 1.574 | 1.769 |
| KQ | 0.580 | 0.622 | 0.673 | 0.724 | 0.776 | 0.879 | 0.982 | 1.186 | 1.391 | 1.576 | 1.782 |
| AQ | 0.567 | 0.610 | 0.661 | 0.712 | 0.764 | 0.867 | 0.970 | 1.174 | 1.379 | 1.574 | 1.769 |
| AK | 0.567 | 0.610 | 0.661 | 0.712 | 0.764 | 0.867 | 0.970 | 1.174 | 1.379 | 1.574 | 1.769 |
| QJT | 1.463 | 3.394 | 5.700 | 8.012 | 10.325 | 14.950 | 19.576 | 28.826 | 38.077 | 47.328 | 56.578 |
| KJT | 1.370 | 3.300 | 5.611 | 7.923 | 10.236 | 14.861 | 19.487 | 28.737 | 37.988 | 47.239 | 56.489 |
| AJT | 1.275 | 3.208 | 5.520 | 7.832 | 10.145 | 14.770 | 19.395 | 28.646 | 37.897 | 47.148 | 56.398 |
| KQT | 1.370 | 3.300 | 5.611 | 7.923 | 10.236 | 14.861 | 19.487 | 28.737 | 37.988 | 47.239 | 56.489 |
| AQT | 1.275 | 3.208 | 5.520 | 7.832 | 10.145 | 14.770 | 19.395 | 28.646 | 37.897 | 47.148 | 56.398 |
| AKT | 1.275 | 3.208 | 5.520 | 7.832 | 10.145 | 14.770 | 19.395 | 28.646 | 37.897 | 47.148 | 56.398 |
| KQJ | 1.470 | 3.397 | 5.707 | 8.020 | 10.333 | 14.958 | 19.583 | 28.834 | 38.085 | 47.335 | 56.586 |
| AQJ | 1.376 | 3.305 | 5.616 | 7.929 | 10.241 | 14.867 | 19.492 | 28.743 | 37.994 | 47.244 | 56.495 |
| AKJ | 1.376 | 3.305 | 5.616 | 7.929 | 10.241 | 14.867 | 19.492 | 28.743 | 37.994 | 47.244 | 56.495 |
| AKQ | 1.376 | 3.305 | 5.616 | 7.929 | 10.241 | 14.867 | 19.492 | 28.743 | 37.994 | 47.244 | 56.495 |
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October 15th, 2020 at 5:01:15 PM
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One difficulty with a lot of these types of analysis is what to assume about the discarded cards.
Obviously, holding a Qs in a sequential position is affected if one of the cards you discard is a 10s.
But the return on a Qs is also affected if you discard a 10d or an 8 spades - or both. Or discarding 2 suited cards, decreasing the chances of a flush even further.
For the numbers being provided what are you assuming about the discards?
Edit: I just noticed the comment about "average EV with no penalty cards." I guess that's a reasonable way to simplify the analysis, but the fact that you have also discarded 2-4 cards that are known not to be Royal cards may not be trivial. And for a decision like whether to draw to Ks or KsTs with Ts-2h-4c-Ks-7d, you need to take into account that no royal is possible if drawing to the King.
Obviously, holding a Qs in a sequential position is affected if one of the cards you discard is a 10s.
But the return on a Qs is also affected if you discard a 10d or an 8 spades - or both. Or discarding 2 suited cards, decreasing the chances of a flush even further.
For the numbers being provided what are you assuming about the discards?
Edit: I just noticed the comment about "average EV with no penalty cards." I guess that's a reasonable way to simplify the analysis, but the fact that you have also discarded 2-4 cards that are known not to be Royal cards may not be trivial. And for a decision like whether to draw to Ks or KsTs with Ts-2h-4c-Ks-7d, you need to take into account that no royal is possible if drawing to the King.
Last edited by: gordonm888 on Oct 15, 2020
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
October 15th, 2020 at 6:00:28 PM
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Quote: gordonm888One difficulty with a lot of these types of analysis is what to assume about the discarded cards.
Obviously, holding a Qs in a sequential position is affected if one of the cards you discard is a 10s.
But the return on a Qs is also affected if you discard a 10d or an 8 spades - or both. Or discarding 2 suited cards, decreasing the chances of a flush even further.
For the numbers being provided what are you assuming about the discards?
I am averaging over only those hands where the hold listed in the left column is the best hold for the given deal. This eliminates holding Qs and discarding Ts. Under no circumstance would the single Qs be the best hold if another spade royal card is present in the dealt hand. be the best hold. I realize it would be very deceptive to average the EV over all holding of a bare Qs without regard to the discards, since the worst penalties (RF penalties) would lower the average EV.
For purposes of this table, I am averaging over all penalty situations that are not bad enough to prevent the sequential RF hold from being the best hold. Near the strategy break points, when the SRF hold is very close to another hold, this means my average will be slightly higher than the generic average. When I am not near strategy break points, every penalty situation is included in the average, except that RF penalty cards are never included in the EV averages.
For example, the EV average for the AxxxT hold doesn't average over very many single flush penalty situations of hands that also contain a king, and includes no double flush penalty situation. The EV average for xxQJT includes every penalty situation under the sun, except for RF penalty cards.
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October 15th, 2020 at 6:09:01 PM
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Quote: MentalI am averaging over only those hands where the hold listed in the left column is the best hold for the given deal. This eliminates holding Qs and discarding Ts. Under no circumstance would the single Qs be the best hold if another spade royal card is present in the dealt hand. be the best hold. I realize it would be very deceptive to average the EV over all holding of a bare Qs without regard to the discards, since the worst penalties (RF penalties) would lower the average EV.
For purposes of this table, I am averaging over all penalty situations that are not bad enough to prevent the sequential RF hold from being the best hold. Near the strategy break points, when the SRF hold is very close to another hold, this means my average will be slightly higher than the generic average. When I am not near strategy break points, every penalty situation is included in the average, except that RF penalty cards are never included in the EV averages.
For example, the EV average for the AxxxT hold doesn't average over very many single flush penalty situations of hands that also contain a king, and includes no double flush penalty situation. The EV average for xxQJT includes every penalty situation under the sun, except for RF penalty cards.
Excellent methodology. You've put a lot of thought into that -as well as a lot of work. I'm impressed.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
October 15th, 2020 at 6:15:55 PM
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Also, my own VP strategy generator breaks hands down by flush penalties, straight penalties, and high-card penalties. It just isn't possible to present all this information in a simple table. Also, a 6/5 bonus progressive is almost always very negative EV and unplayable or very positive where penalties are fairly unimportant compared to the huge variance of the game.
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October 15th, 2020 at 6:24:23 PM
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Quote: gordonm888Edit: I just noticed the comment about "average EV with no penalty cards." I guess that's a reasonable way to simplify the analysis, but the fact that you have also discarded 2-4 cards that are known not to be Royal cards may not be trivial. And for a decision like whether to draw to Ks or KsTs with Ts-2h-4c-Ks-7d, you need to take into account that no royal is possible if drawing to the King.
I absolutely agree. Of course, once you discard the Ts, any ordinary EV calculator will give you the correct EV and the best hold with respect to other possible holds.
The question only comes up if the K and T are in sequential order. I can't imagine that you would ever hold a bare K. The conflict would be with KJ-offsuit, etc.
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October 15th, 2020 at 7:31:27 PM
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Assuming the SQRF resets at $12,500. And max bet is 1.25, when or what jackpot dollar amount does the game become playable? According to Bob Dancer, no amount.
October 15th, 2020 at 8:13:02 PM
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Quote: VegasriderAssuming the SQRF resets at $12,500. And max bet is 1.25, when or what jackpot dollar amount does the game become playable? According to Bob Dancer, no amount.
The quarter game has roughly the same variance as FPDW at $25 per hand. Imagine that I could play quarter FPDW and earned less than a penny a hand, but somehow, the variance was the same as playing $25 per hand. I would not touch that FPDW game with a ten foot pole. I don't really enjoy playing VP. Unless the hourly rate is really good, I would not have any interest in that imaginary game. The reward has to match the risk.
I usually play around 1200 hands per hour, but checking for any SRF draws both ways would slow this down. The progressive at $150K level is worth less than $50 per hour. When I have played games with a similar amount of variance per hour, I have usually had an earn rate of $75 per hour or higher. These sort of opportunities were getting scarce, so more of my play lately has been non-VP opportunities. If I needed to earn money and the Red Rock progressive was the only game in town, I think I would play it above $125K, but I have a rather large bankroll and there is no way this game would harm it much.
I did hear the GWAE podcast. IIRC, Dancer answered the question without mentioning the meter rate. For some progressives, this can be 2%. Variance is variance whether you can lock up the progressive with a team or not. However, in order to capture meter movement in your own EV, you need to lock up the progressive -- possibly, for a very long time. There are a lot of factors, but I think this game had to be getting close to the tipping point for some Vegas team. There is a significant chance that a team could lose $100K taking down the progressive starting at $150K. This isn't really that bad. It is not easy to make any decent money gambling without risking a 100K loss.
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October 16th, 2020 at 1:01:18 PM
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I was curious to see if an increase in the sequential royal progressive added more value to some low variance games versus higher variance games. There turns out to be a small affect. In these tables, I calculated the EV for 11 payoffs for the two-way SRF from zero to 100,000 bets. I then subtracted the EV of the zero SRF game and presented the resulting difference as a percentage. JOB gains the least, 5.79% and TDB gains the most, 5.85%.
I was thinking that on a progressive machine with multiple game choices, the best game at a lower SRF payoff might be different than the best game choice at higher levels. I don't think that will ever be the case. The strategy changes cost you a bit less on high variance games because throwing away two-pair on DDB or trips on TDB to go for the SRF is not as costly as it is on a JOB game. This difference in costs amounts to only 0.06%.
A 10,000 bet increase in the SRF payout adds around 0.5% to the EV at lower levels and adds more than 0.6% to the EV once the SRF payoff is high enough.
I was thinking that on a progressive machine with multiple game choices, the best game at a lower SRF payoff might be different than the best game choice at higher levels. I don't think that will ever be the case. The strategy changes cost you a bit less on high variance games because throwing away two-pair on DDB or trips on TDB to go for the SRF is not as costly as it is on a JOB game. This difference in costs amounts to only 0.06%.
A 10,000 bet increase in the SRF payout adds around 0.5% to the EV at lower levels and adds more than 0.6% to the EV once the SRF payoff is high enough.
1 2 3 4 5 8 25 50 800 : Jacks or Better
| 0 | 10000 | 20000 | 30000 | 40000 | 50000 | 60000 | 70000 | 80000 | 90000 | 100000 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.00 | 0.50 | 1.04 | 1.59 | 2.16 | 2.75 | 3.35 | 3.96 | 4.57 | 5.18 | 5.79 |
1 2 3 4 5 6 40 80 25 50 800 : Bonus Poker
| 0 | 10000 | 20000 | 30000 | 40000 | 50000 | 60000 | 70000 | 80000 | 90000 | 100000 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.00 | 0.50 | 1.04 | 1.59 | 2.17 | 2.76 | 3.36 | 3.97 | 4.58 | 5.19 | 5.81 |
1 1 3 4 5 8 50 80 160 160 400 50 800 : Double Double Bonus
| 0 | 10000 | 20000 | 30000 | 40000 | 50000 | 60000 | 70000 | 80000 | 90000 | 100000 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.00 | 0.50 | 1.05 | 1.60 | 2.19 | 2.78 | 3.37 | 3.98 | 4.59 | 5.20 | 5.84 |
1 1 2 4 6 8 80 400 160 800 50 50 800 : Triple Double Bonus
| 0 | 10000 | 20000 | 30000 | 40000 | 50000 | 60000 | 70000 | 80000 | 90000 | 100000 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.00 | 0.49 | 1.05 | 1.62 | 2.20 | 2.79 | 3.38 | 3.97 | 4.57 | 5.20 | 5.85 |
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October 17th, 2020 at 5:34:08 PM
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I paid a visit to the Red Rock to confirm this game. The screen above shows the jackpot as of about 1PM on Oct 17.
The same jackpot is available on six different base games. The following table shows those games with the return for all royals paying 800x the bet amount.
| Base game | Pay table | Return |
| Bonus Poker Deluxe | 6-5 | 0.953611 |
| Jacks or Better | 6-5 | 0.949961 |
| Triple Double Bonus | 7-5 | 0.949178 |
| Double Double Bonus | 6-5 | 0.946569 |
| Double Bonus | 8-5 | 0.941897 |
| Bonus Poker | 10-8-5-3-1 | 0.941829 |
As you can see, Bonus Poker Deluxe offers the best value.
The next table shows the return with optimal strategy, including card position for 8-6 Bonus Poker Deluxe. The lower right cell shows a return of 105.22%!
This has an expected profit of 1.3 cents per hand. Assuming a playing speed of 1000 hands per hour, the expected win per hour is $13.05, not including the value of points earned.
| Outcome | Win | Combinations | Probability | Return |
|---|---|---|---|---|
| Reversible Royal Flush | 807,782 | 67,567,248 | 0.000001 | 0.109525 |
| Royal Flush | 4,000 | 2,531,111,347 | 0.000025 | 0.020317 |
| Straight Flush | 250 | 10,545,248,235 | 0.000106 | 0.005290 |
| Four of a Kind | 400 | 246,151,065,684 | 0.002470 | 0.197580 |
| Full House | 30 | 811,309,337,868 | 0.008140 | 0.048842 |
| Flush | 25 | 1,067,259,148,415 | 0.010708 | 0.053542 |
| Straight | 20 | 1,275,953,861,466 | 0.012802 | 0.051209 |
| Three of a Kind | 15 | 7,722,917,814,918 | 0.077488 | 0.232464 |
| Two Pair | 5 | 10,770,132,885,066 | 0.108062 | 0.108062 |
| Jacks or Better | 5 | 22,463,692,195,905 | 0.225389 | 0.225389 |
| All Other | 0 | 55,295,592,349,848 | 0.554808 | 0.000000 |
| Totals | 99,666,152,586,000 | 1.000000 | 1.052220 |
The next table shows the return with optimal strategy, NOT including strategy changes for card position. It is based on an average royal of 17,396 coins, which is a weighted average between 1/60 the jackpot and 59/60 a win of 4000 coins. The lower right cell shows a return of 103.56%.
| Outcome | Prize | Combinations | Probability | Return |
|---|---|---|---|---|
| Royal Flush | 17,396 | 52,825,312 | 0.000032 | 0.110643 |
| Straight Flush | 250 | 160,811,571 | 0.000097 | 0.004841 |
| Four of a Kind | 400 | 4,128,394,310 | 0.002485 | 0.198827 |
| Full House | 30 | 13,632,719,598 | 0.008207 | 0.049242 |
| Flush | 25 | 18,607,023,152 | 0.011202 | 0.056008 |
| Straight | 20 | 20,468,179,301 | 0.012322 | 0.049288 |
| Three of a Kind | 15 | 129,801,408,860 | 0.078142 | 0.234425 |
| Two Pair | 5 | 181,189,140,336 | 0.109078 | 0.109078 |
| Jacks or Better | 5 | 370,795,511,313 | 0.223223 | 0.223223 |
| All Other | 0 | 922,266,529,347 | 0.555213 | 0.000000 |
| Totals | 1,661,102,543,100 | 1.000000 | 1.035574 |
The next table shows the return with optimal strategy for non-progressive 8/6 Bonus Poker Deluxe. The lower right cell shows a return of 101.97%.
| Outcome | Prize | Combinations | Probability | Return |
|---|---|---|---|---|
| Royal Flush | 17,396 | 40,987,885 | 0.000025 | 0.085850 |
| Straight Flush | 250 | 179,539,017 | 0.000108 | 0.005404 |
| Four of a Kind | 400 | 4,136,976,035 | 0.002491 | 0.199240 |
| Full House | 30 | 13,653,938,163 | 0.008220 | 0.049319 |
| Flush | 25 | 18,041,709,548 | 0.010861 | 0.054306 |
| Straight | 20 | 21,688,438,418 | 0.013057 | 0.052227 |
| Three of a Kind | 15 | 129,936,095,954 | 0.078223 | 0.234668 |
| Two Pair | 5 | 181,027,999,728 | 0.108981 | 0.108981 |
| Jacks or Better | 5 | 381,597,343,986 | 0.229725 | 0.229725 |
| All Other | 0 | 910,799,514,366 | 0.548310 | 0.000000 |
| Total | 1,661,102,543,100 | 1.000000 | 1.019720 |
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
October 17th, 2020 at 9:45:44 PM
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Quote: WizardI paid a visit to the Red Rock to confirm this game. The screen above shows the jackpot as of about 1PM on Oct 17.
The meter must be fast to get this SRF payoff up to 161K bets or maybe the reset is very high. With these crappy pay tables, Red Rock could afford to reset it higher. My original post was in regard to a quarter machine with two-way SRF at Red Rock.
For this nickel game, I get Variance = 18177.5802 and EV = 105.33775096%
| HandName | Payout | Probability | Cycle | ROI % |
|---|---|---|---|---|
| SeqRF | 161556 | 0.00000069514 | 1438564.32 | 11.230363% |
| Royal | 800 | 0.00002556632 | 39113.9519 | 2.045306% |
| Str_Flush | 50 | 0.00010583650 | 9448.53591 | 0.529183% |
| Quads | 80 | 0.00247378310 | 404.239159 | 19.790265% |
| Full_House | 6 | 0.00815154215 | 122.676173 | 4.890925% |
| Flush | 5 | 0.01078311754 | 92.7375590 | 5.391559% |
| Straight | 4 | 0.01265402772 | 79.0262217 | 5.061611% |
| Trips | 3 | 0.07762031837 | 12.8832246 | 23.286096% |
| Two_Pair | 1 | 0.10819361217 | 9.24268984 | 10.819361% |
| JOB | 1 | 0.22293082507 | 4.48569640 | 22.293083% |
| Nada | 0 | 0.55706067591 | 1.79513659 | 0.000000% |
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October 19th, 2020 at 10:38:40 AM
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I am sorry for hijacking the thread with the reversible royal game. I think I'll make a separate thread for that.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
October 19th, 2020 at 6:30:19 PM
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Player Hits Rare Sequential Royal Flush for $217,000 Jackpot (on a $1 machine)
https://vitalvegas.com/player-hits-rare-sequential-royal-flush-for-217000-jackpot/
https://vitalvegas.com/player-hits-rare-sequential-royal-flush-for-217000-jackpot/
