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2nd Chance Royal

Introduction

The first I heard of 2nd Chance Royal was in 2006, when the inventors approached me and hired to me do a math analysis of the game. At some point after that, they partnered with IGT to market the game. Four years and two months later, it made its debut at the Santa Fe Station in Las Vegas. This should give other game inventors an idea of how long it takes to get a game into a casino, if they are lucky enough to ever get that far.

 

 

 

The thrust of 2nd Chance Royal is that if the player makes a max 5-coin bet and has four to a royal after the draw, then he gets a second draw. This is free — no additional bet required. Of course, nothing is truly free in Vegas, so the casino may depress the pay table a bit to pay for this feature. The player will be paid for all wins on both the first and second draw. However, a royal on the second draw pays 200 times the players bet, not the normal 800.

 

 

Strategy

Sorry, you're on your own when it comes to strategy. The 2nd Chance feature is worth about 1.5%, so the player should obviously be more inclined to go for royals than conventional video poker.

Return

I'm happy to report that my 2006 math agreed exactly with that of IGT's mathematicians. The following table was kindly provided by IGT, showing the expected return of all available games and pay tables, assuming optimal strategy. In 2006, I analyzed six of these.

 

 

2nd Chance Royal Return Summary

Game Return
JACKS OR BETTER POKER - 8/6 99.96%
JACKS OR BETTER POKER - 8/5 98.82%
JACKS OR BETTER POKER - 7/5 97.67%
JACKS OR BETTER POKER - 6/5 96.52%
BONUS POKER - 7/5 99.51%
BONUS POKER - 6/5 98.37%
DOUBLE BONUS POKER - 9/6/5 99.37%
DOUBLE BONUS POKER - 9/6/4 97.95%
DOUBLE BONUS POKER - 8/5 95.69%
DOUBLE DOUBLE BONUS POKER - 9/5 99.37%
DOUBLE DOUBLE BONUS POKER - 8/5 98.28%
DOUBLE DOUBLE BONUS POKER - 7/5 97.22%
DOUBLE DOUBLE BONUS POKER - 6/5 96.17%
DEUCES WILD POKER - 20/12/10/4/4/3 99.06%
DEUCES WILD POKER - 20/12/9/4/4/3 98.54%
DEUCES WILD POKER - 25/16/13/4/3/2 98.31%
DEUCES WILD POKER - 20/10/8/4/4/3 97.46%
DEUCES WILD POKER - 25/15/10/4/3/2 96.38%
BONUS POKER DELUXE - 8/6 100.08%
BONUS POKER DELUXE - 8/5 98.90%
BONUS POKER DELUXE - 7/5 97.77%
BONUS POKER DELUXE - 6/5 96.88%
SUPER ACES BONUS POKER - 6/5 99.30%
SUPER DOUBLE DOUBLE BONUS POKER - 7/5 100.08%
SUPER DOUBLE DOUBLE BONUS POKER - 6/5 99.17%
TRIPLE DOUBLE BONUS POKER - 9/6 99.70%
TRIPLE DOUBLE BONUS POKER - 9/5 98.53%
TRIPLE DOUBLE BONUS POKER - 8/5 97.48%
TRIPLE DOUBLE BONUS POKER - 7/5 96.43%
DEUCES WILD BONUS POKER - 10/4/3/3 99.07%
DEUCES WILD BONUS POKER - 12/4/3/2 97.84%
DEUCES WILD BONUS POKER - 10/4/3/2 96.97%

 

The following were the pay tables at the Santa Fe Station at the 25¢ and $1 denominations. The best pay table is the 9-6 Triple Double Bonus at the $1 denomination.

 

25 Cent Santa Fe Station Pay Tables

Game Pay Table Return
Bonus Poker Deluxe 7-5 97.77%
Bonus Poker 6-5 98.37%
Bonus Deuces 25-12-4-3-2 97.84%
Deuces Wild 25-16-13-4-3-2 98.31%
Double Bonus 9-6-4 97.95%
Double Double Bonus 8-5 98.28%
Super Double Double Bonus 6-5 99.17%
Triple Double Bonus 9-5 98.53%

 

 

$1 Santa Fe Station Pay Tables

Game Pay Table Return
Bonus Poker Deluxe 8-5 98.90%
Bonus Poker 7-5 99.51%
Bonus Deuces 25-10-4-3-3 99.07%
Deuces Wild 20-12-10-4-4-3-2 99.06%
Double Bonus 9-6-4 97.95%
Double Double Bonus 9-5 99.37%
Super Double Double Bonus 6-5 99.17%
Triple Double Bonus 9-6 99.70%

 

The following six return tables I did in 2006. The top rows are for hands with four to a royal after the first draw. The "event" is what the player had on the first and second draws respectively. For example, if the player had 2h Th Jh Qh Kh after the first draw, and Ks Th Jh Qh Kh after the second draw, then the event would be indicated as "Flush -> Pair." The lower rows are for hands where the player did not have four to a royal after the first draw.

 

7-5 Jacks or Better — 2nd Chance Royal

Event Pays Combinations Probability Return
St flush->Royal 250 714,320,808 0.000000 0.000107
St flush->Str flush 100 0 0.000000 0.000000
St flush->Flush 55 4,861,484,064 0.000003 0.000160
St flush->Straight 54 4,102,696,740 0.000002 0.000132
St flush->Pair 51 6,259,051,032 0.000004 0.000191
St flush->Nothing 50 16,349,344,572 0.000010 0.000488
Flush ->Royal 205 16,822,740,960 0.000010 0.002060
Flush ->Str flush 55 2,870,195,688 0.000002 0.000094
Flush ->Flush 10 109,706,174,952 0.000066 0.000655
Flush ->Straight 9 56,284,754,832 0.000034 0.000303
Flush ->Pair 6 156,852,688,692 0.000094 0.000562
Flush ->Nothing 5 409,556,271,564 0.000245 0.001223
Straight->Royal 204 8,144,383,692 0.000005 0.000992
Straight->Str flush 54 2,526,112,752 0.000002 0.000081
Straight->Flush 9 59,511,381,732 0.000036 0.000320
Straight->Straight 8 22,806,715,944 0.000014 0.000109
Straight->Pair 5 74,815,496,388 0.000045 0.000223
Straight->Nothing 4 195,514,505,412 0.000117 0.000467
Pair ->Royal 201 24,401,319,480 0.000015 0.002929
Pair ->Str flush 51 4,354,973,892 0.000003 0.000133
Pair ->Flush 6 182,197,551,480 0.000109 0.000653
Pair ->Straight 5 82,623,134,268 0.000049 0.000247
Pair ->Pair 2 209,171,193,168 0.000125 0.000250
Pair ->Nothing 1 590,728,815,504 0.000353 0.000353
Nothing ->Royal 200 53,595,021,636 0.000032 0.006402
Nothing ->Str flush 50 9,579,586,080 0.000006 0.000286
Nothing ->Flush 5 400,318,657,416 0.000239 0.001195
Nothing ->Straight 4 180,484,232,028 0.000108 0.000431
Nothing ->Pair 1 496,813,605,192 0.000297 0.000297
Nothing ->Nothing 0 1,255,382,291,808 0.000750 0.000000
Royal Flush 800 50,974,846,272 0.000030 0.024355
Straight Flush 50 148,341,682,944 0.000089 0.004430
Four of a kind 25 3,934,692,381,312 0.002350 0.058748
Full House 7 19,205,184,535,488 0.011470 0.080290
Flush 5 17,966,298,283,920 0.010730 0.053650
Straight 4 18,444,057,812,208 0.011015 0.044062
Three of a kind 3 123,903,222,310,224 0.073999 0.221997
Two pair 2 215,599,195,914,480 0.128763 0.257525
Pair 1 352,111,195,951,200 0.210292 0.210292
Nonpaying hand 0 918,390,851,024,976 0.548492 0.000000
Total 0 1,674,391,363,444,800 1.000000 0.976691

 

 

6-5 Bonus Poker — 2nd Chance Royal

Event Pays Combinations Probability Return
St flush->Royal 250 714,803,808 0.000000 0.000107
St flush->Str flush 100 0 0.000000 0.000000
St flush->Flush 55 4,865,017,776 0.000003 0.000160
St flush->Straight 54 4,105,110,732 0.000002 0.000132
St flush->Pair 51 6,262,914,024 0.000004 0.000191
St flush->Nothing 50 16,360,939,596 0.000010 0.000489
Flush ->Royal 205 16,827,666,384 0.000010 0.002060
Flush ->Str flush 55 2,869,120,488 0.000002 0.000094
Flush ->Flush 10 109,744,994,040 0.000066 0.000655
Flush ->Straight 9 56,294,928,744 0.000034 0.000303
Flush ->Pair 6 156,899,699,460 0.000094 0.000562
Flush ->Nothing 5 409,671,317,124 0.000245 0.001223
Straight->Royal 204 8,145,000,840 0.000005 0.000992
Straight->Str flush 54 2,525,763,984 0.000002 0.000081
Straight->Flush 9 59,518,046,628 0.000036 0.000320
Straight->Straight 8 22,806,939,720 0.000014 0.000109
Straight->Pair 5 74,822,403,708 0.000045 0.000223
Straight->Nothing 4 195,526,360,752 0.000117 0.000467
Pair ->Royal 201 24,405,334,848 0.000015 0.002930
Pair ->Str flush 51 4,352,967,972 0.000003 0.000133
Pair ->Flush 6 182,234,779,944 0.000109 0.000653
Pair ->Straight 5 82,628,421,732 0.000049 0.000247
Pair ->Pair 2 209,206,496,856 0.000125 0.000250
Pair ->Nothing 1 590,822,102,376 0.000353 0.000353
Nothing ->Royal 200 53,604,894,744 0.000032 0.006403
Nothing ->Str flush 50 9,575,924,016 0.000006 0.000286
Nothing ->Flush 5 400,413,678,048 0.000239 0.001196
Nothing ->Straight 4 180,500,942,400 0.000108 0.000431
Nothing ->Pair 1 496,908,758,880 0.000297 0.000297
Nothing ->Nothing 0 1,255,595,961,096 0.000750 0.000000
Royal Flush 800 50,979,917,520 0.000030 0.024357
Straight Flush 50 147,300,776,784 0.000088 0.004399
Four A 80 329,176,528,128 0.000197 0.015728
Four 2-4 40 882,720,584,928 0.000527 0.021088
Four 5-K 25 2,730,865,834,368 0.001631 0.040774
Full House 6 19,032,059,150,496 0.011367 0.068199
Flush 5 17,947,143,920,160 0.010719 0.053593
Straight 4 18,431,099,993,376 0.011008 0.044031
Three of a kind 3 124,077,856,066,416 0.074103 0.222310
Two pair 2 215,613,096,595,344 0.128771 0.257542
Pair   352,229,647,908,096 0.210363 0.210363
Nonpaying hand   918,281,204,878,464 0.548427 0.000000
0   1,674,391,363,444,800 1.000000 0.983730

 

 

7-5 Bonus Poker — 2nd Chance Royal

Event Pays Combinations Probability Return
St flush->Royal 250 714,180,192 0.000000 0.000107
St flush->Str flush 100 0 0.000000 0.000000
St flush->Flush 55 4,860,650,448 0.000003 0.000160
St flush->Straight 54 4,102,002,732 0.000002 0.000132
St flush->Pair 51 6,257,937,192 0.000004 0.000191
St flush->Nothing 50 16,345,928,460 0.000010 0.000488
Flush ->Royal 205 16,827,672,096 0.000010 0.002060
Flush ->Str flush 55 2,869,203,144 0.000002 0.000094
Flush ->Flush 10 109,743,986,712 0.000066 0.000655
Flush ->Straight 9 56,296,054,344 0.000034 0.000303
Flush ->Pair 6 156,899,731,716 0.000094 0.000562
Flush ->Nothing 5 409,669,691,220 0.000245 0.001223
Straight->Royal 204 8,146,414,056 0.000005 0.000993
Straight->Str flush 54 2,525,834,544 0.000002 0.000081
Straight->Flush 9 59,528,795,940 0.000036 0.000320
Straight->Straight 8 22,810,298,376 0.000014 0.000109
Straight->Pair 5 74,835,033,948 0.000045 0.000223
Straight->Nothing 4 195,560,120,688 0.000117 0.000467
Pair ->Royal 201 24,409,387,008 0.000015 0.002930
Pair ->Str flush 51 4,353,883,236 0.000003 0.000133
Pair ->Flush 6 182,265,036,072 0.000109 0.000653
Pair ->Straight 5 82,643,565,924 0.000049 0.000247
Pair ->Pair 2 209,241,603,480 0.000125 0.000250
Pair ->Nothing 1 590,914,685,160 0.000353 0.000353
Nothing ->Royal 200 53,609,203,944 0.000032 0.006403
Nothing ->Str flush 50 9,576,169,968 0.000006 0.000286
Nothing ->Flush 5 400,444,473,456 0.000239 0.001196
Nothing ->Straight 4 180,515,306,400 0.000108 0.000431
Nothing ->Pair 1 496,947,127,392 0.000297 0.000297
Nothing ->Nothing 0 1,255,696,424,424 0.000750 0.000000
Royal Flush 800 50,982,344,784 0.000030 0.024359
Straight Flush 50 146,615,161,392 0.000088 0.004378
Four A 80 321,281,505,216 0.000192 0.015350
Four 2-4 40 882,721,798,560 0.000527 0.021088
Four 5-K 25 2,730,929,289,984 0.001631 0.040775
Full House 7 19,206,597,463,200 0.011471 0.080296
Flush 5 17,931,593,479,968 0.010709 0.053547
Straight 4 18,423,450,124,128 0.011003 0.044012
Three of a kind 3 123,915,851,971,632 0.074007 0.222020
Two pair 2 215,621,230,877,136 0.128776 0.257552
Pair 1 352,335,346,586,496 0.210426 0.210426
Nonpaying hand 0 918,186,152,440,032 0.548370 0.000000
Total   1,674,391,363,444,800 1.000000 0.995149

 

 

7-5 Bonus Poker Deluxe — 2nd Chance Royal

Event Pays Combinations Probability Return
St flush->Royal 250 747,271,788 0.000000 0.000112
St flush->Str flush 100 0 0.000000 0.000000
St flush->Flush 55 5,096,461,080 0.000003 0.000167
St flush->Straight 54 4,304,213,796 0.000003 0.000139
St flush->Pair 51 6,546,204,204 0.000004 0.000199
St flush->Nothing 50 17,083,472,508 0.000010 0.000510
Flush ->Royal 205 16,908,482,220 0.000010 0.002070
Flush ->Str flush 55 2,880,130,656 0.000002 0.000095
Flush ->Flush 10 110,445,021,144 0.000066 0.000660
Flush ->Straight 9 56,599,312,632 0.000034 0.000304
Flush ->Pair 6 157,484,976,240 0.000094 0.000564
Flush ->Nothing 5 411,884,915,940 0.000246 0.001230
Straight->Royal 204 8,111,467,848 0.000005 0.000988
Straight->Str flush 54 2,518,077,696 0.000002 0.000081
Straight->Flush 9 59,340,646,500 0.000035 0.000319
Straight->Straight 8 22,715,385,312 0.000014 0.000109
Straight->Pair 5 74,417,791,104 0.000044 0.000222
Straight->Nothing 4 194,894,605,332 0.000116 0.000466
Pair ->Royal 201 24,633,332,856 0.000015 0.002957
Pair ->Str flush 51 4,413,612,960 0.000003 0.000134
Pair ->Flush 6 184,121,115,252 0.000110 0.000660
Pair ->Straight 5 83,557,337,688 0.000050 0.000250
Pair ->Pair 2 210,999,016,272 0.000126 0.000252
Pair ->Nothing 1 596,981,959,884 0.000357 0.000357
Nothing ->Royal 200 53,504,847,336 0.000032 0.006391
Nothing ->Str flush 50 9,561,348,864 0.000006 0.000286
Nothing ->Flush 5 400,152,291,120 0.000239 0.001195
Nothing ->Straight 4 180,361,028,484 0.000108 0.000431
Nothing ->Pair 1 495,285,250,140 0.000296 0.000296
Nothing ->Nothing 0 1,254,128,344,872 0.000749 0.000000
Royal Flush 800 51,147,204,192 0.000031 0.024437
Straight Flush 50 150,843,716,352 0.000090 0.004504
Four of a kind 80 4,041,340,553,376 0.002414 0.193089
Full House 7 16,076,668,888,224 0.009602 0.067211
Flush 5 18,046,806,856,848 0.010778 0.053891
Straight 4 21,115,325,908,656 0.012611 0.050443
Three of a kind 3 127,897,042,056,336 0.076384 0.229153
Two pair 1 182,950,375,314,672 0.109264 0.109264
Pair 1 375,449,316,387,264 0.224230 0.224230
Nonpaying hand 0 923,962,818,637,152 0.551820 0.000000
Total   1,674,391,363,444,800 1.000000 0.977665

 

 

9-6-4 Double Bonus — 2nd Chance Royal

Event Pays Combinations Probability Return
St flush->Royal 250 748,246,440 0.000000 0.000112
St flush->Str flush 100 0 0.000000 0.000000
St flush->Flush 56 5,103,772,272 0.000003 0.000171
St flush->Straight 54 4,308,802,296 0.000003 0.000139
St flush->Pair 51 6,551,208,756 0.000004 0.000200
St flush->Nothing 50 17,110,281,276 0.000010 0.000511
Flush ->Royal 206 17,313,845,220 0.000010 0.002130
Flush ->Str flush 56 2,907,466,776 0.000002 0.000097
Flush ->Flush 12 113,052,479,424 0.000068 0.000810
Flush ->Straight 10 57,741,508,428 0.000034 0.000345
Flush ->Pair 7 161,200,537,404 0.000096 0.000674
Flush ->Nothing 6 421,466,856,588 0.000252 0.001510
Straight->Royal 204 7,991,440,500 0.000005 0.000974
Straight->Str flush 54 2,428,345,284 0.000001 0.000078
Straight->Flush 10 58,542,640,368 0.000035 0.000350
Straight->Straight 8 22,168,798,320 0.000013 0.000106
Straight->Pair 5 73,317,157,164 0.000044 0.000219
Straight->Nothing 4 192,077,195,436 0.000115 0.000459
Pair ->Royal 201 24,085,774,404 0.000014 0.002891
Pair ->Str flush 51 4,317,151,140 0.000003 0.000131
Pair ->Flush 7 180,157,577,604 0.000108 0.000753
Pair ->Straight 5 81,527,146,560 0.000049 0.000243
Pair ->Pair 2 205,810,243,032 0.000123 0.000246
Pair ->Nothing 1 584,064,432,828 0.000349 0.000349
Nothing ->Royal 200 54,536,327,436 0.000033 0.006514
Nothing ->Str flush 50 9,561,078,468 0.000006 0.000286
Nothing ->Flush 6 407,683,681,308 0.000243 0.001461
Nothing ->Straight 4 182,987,799,372 0.000109 0.000437
Nothing ->Pair 1 504,605,660,904 0.000301 0.000301
Nothing ->Nothing 0 1,277,540,468,880 0.000763 0.000000
Royal Flush 800 51,492,395,808 0.000031 0.024602
Straight Flush 50 153,799,733,808 0.000092 0.004593
Four A 160 378,848,925,504 0.000226 0.036202
Four 2-4 80 882,842,208,192 0.000527 0.042181
Four 5-K 50 2,721,183,304,896 0.001625 0.081259
Full House 9 18,072,340,816,320 0.010793 0.097140
Flush 6 18,325,035,200,304 0.010944 0.065666
Straight 4 20,940,967,776,960 0.012507 0.050026
Three of a kind 3 125,240,622,359,328 0.074798 0.224393
Two pair 1 204,900,815,046,816 0.122373 0.122373
Pair 1 349,285,753,210,464 0.208605 0.208605
Nonpaying hand 0 928,756,754,542,512 0.554683 0.000000
Total   1,674,391,363,444,800 1.000000 0.979537

 

 

7-5 Double Double Bonus — 2nd Chance Royal

Event Pays Combinations Probability Return
St flush->Royal 250 743,391,912 0.000000 0.000111
St flush->Str flush 100 0 0.000000 0.000000
St flush->Flush 55 5,071,643,532 0.000003 0.000167
St flush->Straight 54 4,285,486,836 0.000003 0.000138
St flush->Pair 51 6,511,382,844 0.000004 0.000198
St flush->Nothing 50 16,995,376,332 0.000010 0.000508
Flush ->Royal 205 16,462,823,748 0.000010 0.002016
Flush ->Str flush 55 2,863,418,436 0.000002 0.000094
Flush ->Flush 10 107,412,844,584 0.000064 0.000642
Flush ->Straight 9 55,182,128,304 0.000033 0.000297
Flush ->Pair 6 153,872,441,916 0.000092 0.000551
Flush ->Nothing 5 400,122,152,676 0.000239 0.001195
Straight->Royal 204 7,640,388,540 0.000005 0.000931
Straight->Str flush 54 2,403,196,356 0.000001 0.000078
Straight->Flush 9 55,911,072,576 0.000033 0.000301
Straight->Straight 8 21,437,225,688 0.000013 0.000102
Straight->Pair 5 70,342,577,388 0.000042 0.000210
Straight->Nothing 4 183,267,419,580 0.000109 0.000438
Pair ->Royal 201 23,070,035,400 0.000014 0.002769
Pair ->Str flush 51 4,268,389,140 0.000003 0.000130
Pair ->Flush 6 172,587,325,068 0.000103 0.000618
Pair ->Straight 5 78,519,005,424 0.000047 0.000234
Pair ->Pair 2 197,945,757,228 0.000118 0.000236
Pair ->Nothing 1 558,988,391,628 0.000334 0.000334
Nothing ->Royal 200 51,917,207,520 0.000031 0.006201
Nothing ->Str flush 50 9,472,873,428 0.000006 0.000283
Nothing ->Flush 5 387,788,400,756 0.000232 0.001158
Nothing ->Straight 4 175,243,426,092 0.000105 0.000419
Nothing ->Pair 1 482,344,863,156 0.000288 0.000288
Nothing ->Nothing 0 1,213,995,297,624 0.000725 0.000000
Royal Flush 800 49,390,211,808 0.000029 0.023598
Straight Flush 50 147,793,555,728 0.000088 0.004413
Four A + 2-4 400 102,390,364,944 0.000061 0.024460
Four 2-4 + A-4 160 245,545,316,352 0.000147 0.023464
Four A 160 289,047,565,296 0.000173 0.027621
Four 2-4 80 661,410,285,984 0.000395 0.031601
Four 5-K 50 2,719,373,649,552 0.001624 0.081205
Full House 7 17,588,592,259,056 0.010504 0.073531
Flush 5 18,083,246,981,184 0.010800 0.053999
Straight 4 20,929,476,155,616 0.012500 0.049999
Three of a kind 3 126,137,734,929,360 0.075333 0.226000
Two pair 1 205,357,562,459,280 0.122646 0.122646
Pair 1 349,901,722,206,288 0.208972 0.208972
Nonpaying hand 0 927,711,411,560,640 0.554059 0.000000
Total   1,674,391,363,444,800 1.000000 0.972157

 

Acknowledgement

  • Vegas Royalty Enterprises: The game inventors, for letting me know about the Santa Fe placement and help with this page.
  • IGT: The game makers, for sharing the return summary table above and help with this page.

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Written by: Michael Shackleford