5high
5high
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Joined: Jun 8, 2024
June 8th, 2024 at 9:20:39 PM permalink
So Iíve been playing a jacks or better video poker game with a progressive jackpot. It also has the option to play multiple hands.

I like playing it when the jackpot reaches the break even threshold and the machine has over 100% RTP.

Itís a $1 machine so you need to bet $5 per hand in order for the jackpot to be live and right now with the current progressive the payable is this:

Roughly $8800 Royal (resets at $4000 after being hit)
$250 Straight flush
$125 Quads
$40 Full house
$25 Flush
$20 Straight
$15 3 of a kind
$10 2 pair
$5 Jacks or better

With this current progressive and this payable the RTP is 100.08% if you play only 1 hand at a time and use an optimal strategy but thatís where my question comes in.

This machine gives you the option to play 1/3/5/10/25/50/100 hands at once and I am curious how playing more than one hand affects the RTP.

Of course the EV changes because if you play 100 hands at once and are dealt 4 to a royal for example and are lucky enough to hit multiple royals in one spin the progressive is only paid out for the first royal and after that the next royals are paid out $4000 for eachÖ

Does anyone know how to solve this and figure out how much EV is being lost by playing multiple hands ? I just want to know if itís still profitable to play 10 or even more hands at once as the progressive gets higher.

The base RTP when the royal flush resets to $4000 is 97.3% and with the progressive at $8800 it's about 100.08% RTP with an optimized strategy. With where it sits now how much EV do you lose by playing 10 or 25 hands and if it gets higher like maybe it reaches 100.50% rtp does it ever become worth it to play 100 hands at once or is the EV loss too much still??

Sorry for the long winded question, I know this is probably pretty simple to solve but for me it feels so complicated :')
Thanks for any help!
Mental
Mental
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5high
June 9th, 2024 at 4:13:13 AM permalink
I wouldn't say it is a simple problem. The part that is simple is the fraction of RFs that come from dealt Royals. This happens once in 649,740 = (52 choose 5) / 4 hands. So if you are playing N-play, then (N-1) RFs per 649,740 dealt hands will be paid at 800 to 1. You could use combinatorics to solve the problem for 4-card RF draws, too.

But, the easiest way I know to get a good answer to the general question you raise is via Monte Carlo simulation. I am already set up to do that for any number of hands, N. I just set N in my program and it produces a histogram for me.


0, 0.262005440638
1, 0.155447378952
2, 0.159159837012
3, 0.137299893034
4, 0.074138424601
5, 0.074141206483
6, 0.062010007849
7, 0.008590770924
8, 0.009769134577
9, 0.016852575646
10, 0.007190726291
11, 0.002677551790
12, 0.015184046696
13, 0.000482877766
14, 0.003293729030
15, 0.002148855696
16, 0.000064460400
17, 0.000029404069
18, 0.000965624711
19, 0.000226371318
20, 0.000000003848
21, 0.000000007695
24, 0.001474893804
25, 0.001299542894
26, 0.000083556500
27, 0.001116373472
28, 0.000640660110
29, 0.000267622434
30, 0.000128101240
31, 0.002214177979
32, 0.000000126974
33, 0.000034448395
34, 0.000015956383
35, 0.000011189091
36, 0.000302040047
37, 0.000000003848
41, 0.000010207929
50, 0.000156778096
51, 0.000029942746
52, 0.000010365685
53, 0.000106627266
54, 0.000023274694
55, 0.000043894481
56, 0.000003043525
57, 0.000000277034
58, 0.000008761197
59, 0.000004609536
60, 0.000004120879
61, 0.000000003848
75, 0.000241642811
100, 0.000003062764
101, 0.000000138517
102, 0.000000007695
104, 0.000000427094
105, 0.000000673346
150, 0.000013890171
800, 0.000026387478
801, 0.000020946840
802, 0.000006298673
803, 0.000001523686
804, 0.000003101240
805, 0.000006845046
806, 0.000002235510
807, 0.000000088497
808, 0.000000207775
809, 0.000000607935
810, 0.000000700280
826, 0.000000003848
850, 0.000000092345
851, 0.000000053868
854, 0.000000015391
855, 0.000000015391
1600, 0.000000303968
1601, 0.000000134669
1604, 0.000000050020
1605, 0.000000069258
1650, 0.000000007695
2400, 0.000001539077

The spoiler contains the output of my MC program for the base game strategy for N=3. The first column is the payoff and the second column is the probability of that payoff. For example, the 2400:1 payoff is the case of a dealt RF. The 1600:1 payoff is for two RFs and no win on the other line. The 1650:1 payoff is for holding TJQK and hitting two RFs and a SF, etc.

If you copy these results into a spreadsheet, you should be able to add formulas to sum the probabilities of all RFs. Hint: multiply the probability for all payoffs 16XX by two before summing. You should be able to create a second sum for the probabilities of RFs beyond one RF in a given hand. You can then convert that number into lost EV due to being paid only 800:1 for those RFs.

If you make strategy adjustments for the jackpot, then the number of RFs from holding zero to three cards will increase. The number of RFs from holding four to a RF will not change at all. Likewise for dealt RFs. So, I will not redo the calculations with different values for the RF jackpot. There would be trivial differences due to multiple RFRs holding three or fewer cards.

I did the MC calculation for 100 play and saw a number of hands hit nine RFs. This output is too large to fit in a spoiler (almost 4000 lines). I doubt that you will be playing much 100-play at $500 per hand just to win a $8000 jackpot. Here are the results for 10-play.

0, 0.025684870102
1, 0.030150521747
2, 0.075132472220
3, 0.085339889802
4, 0.101253951581
5, 0.103167259211
6, 0.078546457044
7, 0.065313086773
8, 0.055454928125
9, 0.035141194940
10, 0.039539773602
11, 0.033059381445
12, 0.035825053098
13, 0.032867870225
14, 0.028334172130
15, 0.026177798042
16, 0.018975640256
17, 0.011339581987
18, 0.007546418567
19, 0.005394269246
20, 0.025363941731
21, 0.003798211592
22, 0.002278957737
23, 0.001364834395
24, 0.000866392711
25, 0.001362052513
26, 0.018746621726
27, 0.001121456275
28, 0.000832406039
29, 0.000901629883
30, 0.008310174070
31, 0.000834864715
32, 0.008431695755
33, 0.000746263890
34, 0.000663115246
35, 0.005670398929
36, 0.000899609844
37, 0.000851840736
38, 0.002666751316
39, 0.000551555237
40, 0.005747922246
41, 0.000249669098
42, 0.000158982824
43, 0.000127231662
44, 0.000399471327
45, 0.000335118663
46, 0.000044987226
47, 0.000026225875
48, 0.000014898267
49, 0.000009084403
50, 0.002079043156
51, 0.000047888386
52, 0.003417043741
53, 0.000045187306
54, 0.000074075784
55, 0.000116381168
56, 0.000053725336
57, 0.002106534922
58, 0.000041358851
59, 0.000066457352
60, 0.000076130452
61, 0.000030943146
62, 0.000584422230
63, 0.000028665312
64, 0.000040423862
65, 0.000033836612
66, 0.000010646566
67, 0.000098004586
68, 0.000014144119
69, 0.000017464678
70, 0.000011627728
71, 0.000003666851
72, 0.000013043679
73, 0.000005113584
74, 0.000722866070
75, 0.000002885770
76, 0.000001046572
77, 0.000001608336
78, 0.000001204328
79, 0.000390637024
80, 0.001441184166
81, 0.000000242405
82, 0.000000257795
83, 0.000000250100
84, 0.000092875612
85, 0.000000165451
86, 0.000000069258
87, 0.000000042325
88, 0.000000023086
89, 0.000012939791
90, 0.000000023086
91, 0.000000015391
93, 0.000000003848
94, 0.000001065811
96, 0.000089851325
99, 0.000000053868
100, 0.000004140118
101, 0.000043436605
102, 0.000000688737
103, 0.000000353988
104, 0.000003843845
105, 0.000007087450
106, 0.000010500354
107, 0.000000900360
108, 0.000002100840
109, 0.000006063964
110, 0.000006237110
111, 0.000002501000
112, 0.000001058116
113, 0.000002627974
114, 0.000004159356
115, 0.000003274387
116, 0.000000796472
117, 0.000000834949
118, 0.000008968972
119, 0.000001689137
120, 0.000000954228
121, 0.000000330902
122, 0.000000300120
123, 0.000003470619
124, 0.000000450180
125, 0.000000188537
126, 0.000000069258
127, 0.000000050020
128, 0.000000654108
129, 0.000000050020
130, 0.000000015391
131, 0.000000007695
132, 0.000000003848
133, 0.000000034629
134, 0.000000003848
140, 0.000000346292
145, 0.000000150060
150, 0.000000411703
151, 0.000000057715
152, 0.000000026934
153, 0.000000011543
154, 0.000000461723
155, 0.000000696432
156, 0.000000119278
157, 0.000000046172
158, 0.000000250100
159, 0.000000477114
160, 0.000000523286
161, 0.000000076954
162, 0.000000065411
163, 0.000000277034
164, 0.000000373226
165, 0.000000215471
166, 0.000000030782
167, 0.000000061563
168, 0.000000107735
169, 0.000000107735
170, 0.000000080802
171, 0.000000023086
172, 0.000000015391
173, 0.000000042325
174, 0.000000023086
175, 0.000000007695
200, 0.000000034629
201, 0.000000007695
204, 0.000000038477
205, 0.000000057715
206, 0.000000007695
207, 0.000000003848
208, 0.000000019238
209, 0.000000019238
210, 0.000000034629
212, 0.000000003848
213, 0.000000019238
214, 0.000000026934
215, 0.000000011543
218, 0.000000003848
250, 0.000240099886
254, 0.000000003848
260, 0.000000003848
500, 0.000013851695
800, 0.000004147813
801, 0.000013336104
802, 0.000021497060
803, 0.000023301628
804, 0.000021643273
805, 0.000019634777
806, 0.000018576661
807, 0.000016768246
808, 0.000012555022
809, 0.000008795826
810, 0.000009015145
811, 0.000008868932
812, 0.000006683443
813, 0.000004513344
814, 0.000003616831
815, 0.000004455628
816, 0.000004101641
817, 0.000002916551
818, 0.000001900760
819, 0.000001754548
820, 0.000001985410
821, 0.000001542925
822, 0.000000923446
823, 0.000000577154
824, 0.000000607935
825, 0.000000588697
826, 0.000000357835
827, 0.000000230862
828, 0.000000157755
829, 0.000000084649
830, 0.000000096192
831, 0.000000073106
832, 0.000000023086
833, 0.000000019238
834, 0.000000019238
835, 0.000000007695
850, 0.000000042325
851, 0.000000080802
852, 0.000000069258
853, 0.000000073106
854, 0.000000069258
855, 0.000000180842
856, 0.000000153908
857, 0.000000161603
858, 0.000000126974
859, 0.000000100040
860, 0.000000211623
861, 0.000000207775
862, 0.000000150060
863, 0.000000080802
864, 0.000000130822
865, 0.000000173146
866, 0.000000119278
867, 0.000000069258
868, 0.000000046172
869, 0.000000088497
870, 0.000000042325
871, 0.000000038477
872, 0.000000015391
873, 0.000000019238
874, 0.000000019238
875, 0.000000023086
876, 0.000000007695
878, 0.000000007695
901, 0.000000003848
902, 0.000000007695
903, 0.000000007695
905, 0.000000007695
906, 0.000000023086
907, 0.000000015391
908, 0.000000003848
909, 0.000000011543
910, 0.000000038477
911, 0.000000007695
912, 0.000000007695
913, 0.000000003848
914, 0.000000015391
916, 0.000000011543
917, 0.000000003848
918, 0.000000003848
919, 0.000000007695
920, 0.000000003848
973, 0.000000003848
1600, 0.000000073106
1601, 0.000000219318
1602, 0.000000261643
1603, 0.000000203928
1604, 0.000000180842
1605, 0.000000300120
1606, 0.000000484809
1607, 0.000000469419
1608, 0.000000380922
1609, 0.000000330902
1610, 0.000000519439
1611, 0.000000542525
1612, 0.000000438637
1613, 0.000000234709
1614, 0.000000261643
1615, 0.000000373226
1616, 0.000000319359
1617, 0.000000176994
1618, 0.000000092345
1619, 0.000000115431
1620, 0.000000192385
1621, 0.000000111583
1622, 0.000000065411
1623, 0.000000030782
1624, 0.000000053868
1625, 0.000000053868
1626, 0.000000023086
1627, 0.000000007695
1628, 0.000000003848
1629, 0.000000011543
1651, 0.000000003848
1652, 0.000000003848
1654, 0.000000007695
1656, 0.000000023086
1657, 0.000000019238
1658, 0.000000007695
1659, 0.000000015391
1660, 0.000000007695
1661, 0.000000015391
1662, 0.000000007695
1664, 0.000000011543
1666, 0.000000003848
1667, 0.000000003848
1670, 0.000000003848
1676, 0.000000003848
1712, 0.000000003848
1718, 0.000000003848
2400, 0.000000003848
2401, 0.000000030782
2402, 0.000000007695
2403, 0.000000015391
2404, 0.000000007695
2405, 0.000000023086
2406, 0.000000042325
2407, 0.000000050020
2408, 0.000000019238
2409, 0.000000015391
2410, 0.000000038477
2411, 0.000000030782
2412, 0.000000019238
2413, 0.000000007695
2414, 0.000000011543
2415, 0.000000019238
2416, 0.000000015391
2417, 0.000000011543
2418, 0.000000003848
2420, 0.000000003848
2421, 0.000000007695
2422, 0.000000003848
2425, 0.000000003848
2455, 0.000000007695
2456, 0.000000003848
2458, 0.000000003848
2459, 0.000000003848
2460, 0.000000003848
3206, 0.000000003848
3209, 0.000000003848
3215, 0.000000003848
8000, 0.000001539077
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ThatDonGuy
ThatDonGuy
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5high
June 9th, 2024 at 7:34:56 AM permalink
I get 1850 hands as the point where EV drops below 100%.

Here's how I calculated it:
Using the Wizard's VP strategy calculator, I set it to 8/5 Jacks or Better with a Royal payout of 1760 (representing a payout of 8800 on a bet of 5).
It came up with these numbers:
The probability of a Royal using modified basic strategy is 1 / 32,576
A non-Royal has an ER of 0.946793

One Royal will add 1760 / N to the EV
Each additional Royal adds 800 / N to the EV

Based on that, I calculated the total ER of an N-hand game as:
EV for 0 Royals = (32,575 / 32,576)^N x 0.946793
EV for 1 Royal = C(N,1) x 32,575^(N-1) / 32,576^N x (0.946793 + 1760 / N)
EV for 2 Royals = C(N,2) x 32,575^(N-2) / 32,576^N x (0.946793 + (1760 + 800) / N)
EV for 3 Royals = C(N,3) x 32,575^(N-3) / 32,576^N x (0.946793 + (1760 + 800 x 2) / N)
...
EV for N Royals = C(N,N) x 1 / 32,576^N x (0.946793 + (1760 + 800 x (N - 1) / N)

There may or may not be an "easy way" to calculate a closed-form solution, given that combinations are involved; I just plugged numbers into Excel.
Mental
Mental
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5high
June 9th, 2024 at 9:01:30 AM permalink
I decided to just count the numbers of extra RFs in my MC simulation. The number of 'extra RFs increases as you increase the number of hands played. I created a table for the results.
'# Extra' is the number of extra RFs after you have already hit the first RF on the multi-play hand.
'Hands w/ RF' is the number of hands that result in at least one RF.
'Total # RF' is the total number of RF in the simulation.
'Cycle' is just the RF cycle in the simulation. It is a sanity check.
'% Extra' is the percentage of RFs that are paid out at the 800:1 payoff.

The first table is for the base game strategy with the jackpot at the 800:1 reset value.
# Hands# ExtraHands w/ RFTotal # RFCycle% Extra
1065066506399470.0000%
39271837219299404004.8034%
1055185930464822400948.5125%
25209771404761614534024312.9926%
50605262631063236324015318.7021%


The percentage of extra RFs is the important result. If 10% of the RFs pay 800:1 instead of 1760:1, then you need to adjust the effective value of the jackpot to be 96 units lower than it actually is. After making this adjustment, you can just get the effective EV from any VP analyzer.

The second table is for the modified optimal strategy for the jackpot at the 1760:1 value.
# Hands# ExtraHands w/ RFTotal # RFCycle% Extra
1079007900328980.00%
39492324424193322283.92%
1055677394579512326867.00%
25214561779941994503257710.76%
50620263379713999973248715.51%

When you use the modified strategy, you get more single royals because you hold more singletons and two-card holds.
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Mental
Mental
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June 9th, 2024 at 9:23:13 AM permalink
Quote: ThatDonGuy

I get 1850 hands as the point where EV drops below 100%.

Here's how I calculated it:
Using the Wizard's VP strategy calculator, I set it to 8/5 Jacks or Better with a Royal payout of 1760 (representing a payout of 8800 on a bet of 5).
It came up with these numbers:
The probability of a Royal using modified basic strategy is 1 / 32,576
A non-Royal has an ER of 0.946793

One Royal will add 1760 / N to the EV
Each additional Royal adds 800 / N to the EV

Based on that, I calculated the total ER of an N-hand game as:
EV for 0 Royals = (32,575 / 32,576)^N x 0.946793
EV for 1 Royal = C(N,1) x 32,575^(N-1) / 32,576^N x (0.946793 + 1760 / N)
EV for 2 Royals = C(N,2) x 32,575^(N-2) / 32,576^N x (0.946793 + (1760 + 800) / N)
EV for 3 Royals = C(N,3) x 32,575^(N-3) / 32,576^N x (0.946793 + (1760 + 800 x 2) / N)
...
EV for N Royals = C(N,N) x 1 / 32,576^N x (0.946793 + (1760 + 800 x (N - 1) / N)

There may or may not be an "easy way" to calculate a closed-form solution, given that combinations are involved; I just plugged numbers into Excel.
link to original post

You can estimate the number of extra RFs just from dealt Royals. Let us say you are using a strategy that puts the RF cycle at 33,000 hands. If you play 100-play, then you will be dealt a RF every 649,740 starting hands or 99 extra RF every 649740*100 total hands. You expect to get 649740*100/33000 = 1969 RFs in this number of hands. This amounts about 5% extra RFs just from dealt RFs. If you add in the draws, the effective RF jackpot has to be significantly reduced already at 100 play.

The fraction of extra RFs on a 1850-play game would be rather large, so the effective RF payoff would be pretty low.

Twice, I hit two RFs from a 3-card hold on a 10-play machine. Once, I hit 4 RFs from a 4-card hold on a 10-play machine.
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5high
5high
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July 2nd, 2024 at 11:53:06 PM permalink
That's the answer I was looking for! Thanks for the help with this, appreciate it :)
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