Ultimate 4 of a Kind Bonus Poker

Last update: July 22, 2008

Introduction

Ultimate 4 of a Kind Bonus Poker is a video poker variant I noticed at the Red Rock casino in Las Vegas in June 2008. It is another in a rash of recent games that offers the player larger wins for four of a kinds for a sixth coin bet.

Rules

For 1 to 5 coins bet, the game plays like ordinary video poker. If the player bets a 6th coin, he will be eligible to play a bonus game if he gets any natural four of a kind on the draw.

In the bonus game, the player picks from 1 to 3 cards from a 53-card deck. Four aces will earn 3 picks, Four 2’s-4’s will earn two picks, and four 5’s to K’s will earn one pick. If the player gets a four of a kind on the deal, then he will be awarded with an extra pick. The prizes associated with the 53-card bonus deck are as follows.

Joker: 3996
Aces: 400
2’s – 4’s: 300
5’s – K’s: 200

When the joker is revealed, the player's total bonus win immediately jumps to 3996 credits and the bonus ends.

Summary

The following table shows the 5 and 6 coin returns for the pay tables available at the Red Rock in June 2008. My video poker program is not easily modified to analyze the wild cards games, so those are filled in with question marks for now.

Ultimate 4 of a Kind Bonus Poker — Summary

Game Pay Table 5 Coins Bet 6 Coins Bet

Double Double Bonus 7-5 95.71% 98.68%

Bonus Poker Deluxe 6-5 95.36% 98.5%

Jacks or Better 7-5 96.15% 98.28%

Double Bonus 9-5 95.27% 98.01%

Bonus Poker 10-8 94.18% 96.38%

Bonus Deuces 10-4-4-3-3 97.36% ?

Joker Poker 17-7-5-3 97.36% ?

Bonus Analysis

The following table shows the possible outcomes of each bonus pick, assuming the player does not pick the joker. The lower right cell shows the average non-joker pick is worth 238.46 credits.

Bonus Probabilities — No Joker

Card Pays Number Probability Return

Aces 400 4 0.076923 30.769231

2 to 4 300 12 0.230769 69.230769

5 to K 200 36 0.692308 138.461538

Total 52 1 238.461538

The expected win from p bonus picks is (p/53)×3996 + ((53-p)/53)×238.46. The next table shows this expected bonus win by number of picks.

Expected Bonus Win by Number of Picks

Picks Win

4 1183.442671

3 901.079826

2 609.718433

1 309.358491

Bonus Poker Deluxe

The following table shows the wins return for Bonus Poker Deluxe. The “Base Pay” column shows the standard video poker win per coin bet, based on five coins bet. The “Bonus” column shows the bonus win, divided by 5, to be consistent with the Base Pay column. The return is the product of the total win, the probability, and 5/6. The reason for multiplying by 5/6 is because the player must bet a six coins to be eligible for the bonus. The lower right cell shows a return of 97.37%.

"6/5" Bonus Poker Deluxe

Hand Base Pay Bonus Total Combinations Probability Return

Royal flush 800 0 800 495429996 0.000025 0.01657

Straight flush 50 0 50 2054514540 0.000103 0.004295

Four aces on deal 80 236.69 316.69 4332670368 0.000018 0.004874

Four 2-4 on deal 80 180.22 260.22 9849862080 0.000055 0.012015

Four 5-K on deal 80 121.94 201.94 31215382332 0.000166 0.027973

Four aces on draw 80 180.22 260.22 165716748396 0.000217 0.047134

Four 2-4 on draw 80 121.94 201.94 201525295044 0.000494 0.083158

Four 5-K on draw 80 61.87 141.87 249481016712 0.001566 0.185142

Full house 6 0 6 165716748396 0.008314 0.041568

Flush 5 0 5 201525295044 0.01011 0.042125

Straight 4 0 4 249481016712 0.012516 0.041719

Three of a kind 3 0 3 1580289703032 0.079279 0.198198

Two pair 1 0 1 2199859576848 0.110361 0.091968

Jacks or better 1 0 1 4502373663204 0.225873 0.188227

Nothing 0 0 0 10981250764968 0.550902 0

Total 19933230517200 1 0.984965

The return for ordinary “6/5” Bonus Poker Deluxe is is 95.36%. So, the player should bet the full six coins.

Double Bonus

For an explanation of this table, please see the explanation in the Bonus Poker Deluxe section.

"9/5" Double Bonus

Hand Base Pay Bonus Total Combinations Probability Return

Royal flush 800 0 800 496337520 0.000025 0.0166

Straight flush 50 0 50 2005684116 0.000101 0.004193

Four aces on deal 160 236.69 396.69 4371289872 0.000018 0.006105

Four 2-4 on deal 80 180.22 260.22 9858654204 0.000055 0.012015

Four 5-K on deal 50 121.94 171.94 29300643588 0.000166 0.023817

Four aces on draw 160 180.22 340.22 211642918272 0.000219 0.062174

Four 2-4 on draw 80 121.94 201.94 204735812904 0.000495 0.083232

Four 5-K on draw 50 61.87 111.87 237125397372 0.00147 0.137037

Full house 9 0 9 211642918272 0.010618 0.079632

Flush 5 0 5 204735812904 0.010271 0.042796

Straight 4 0 4 237125397372 0.011896 0.039653

Three of a kind 3 0 3 1522028976792 0.076356 0.190891

Two pair 1 0 1 2478372668424 0.124334 0.103611

Jacks or better 1 0 1 4265356495356 0.213982 0.178318

Nothing 0 0 0 10963149749100 0.549994 0

Total 19933230517200 1 0.980074

The return for ordinary "9/5" Double Bonus is is 95.27%. So, the player should bet the full six coins.

Double Double Bonus

For an explanation of this table, please see the explanation in the Bonus Poker Deluxe section.

"7/5" Double Double Bonus

Hand Base Pay Bonus Total Combinations Probability Return

Royal flush 800 0 800 496286304 0.000025 0.016598

Straight flush 50 0 50 2011044216 0.000101 0.004204

4 aces + 2-4 on deal 400 236.69 636.69 92036340 0.000005 0.00245

4 2-4 + A-4 on deal 160 180.22 340.22 276109020 0.000014 0.003927

4 aces + 5-K on deal 160 236.69 396.69 276109020 0.000014 0.004579

4 2-4 + 5-K on deal 80 180.22 260.22 828327060 0.000042 0.009011

4 5-K on deal 50 121.94 171.94 3313308240 0.000166 0.023817

4 aces + 2-4 on draw 400 180.22 580.22 1148214852 0.000058 0.027852

4 2-4 + A-4 on draw 160 121.94 281.94 2690934876 0.000135 0.031718

4 aces + 5-K on draw 160 180.22 340.22 3223235496 0.000162 0.045845

4 2-4 + 5-K on draw 80 121.94 201.94 7167109416 0.00036 0.060508

4 5-K on draw 50 61.87 111.87 31153037664 0.001563 0.145701

Full house 7 0 7 165742198992 0.008315 0.048503

Flush 5 0 5 204735672540 0.010271 0.042796

Straight 4 0 4 239002991496 0.01199 0.039967

Three of a kind 3 0 3 1580975570820 0.079314 0.198284

Two pair 1 0 1 2202932423628 0.110516 0.092096

Jacks or better 1 0 1 4520003893488 0.226757 0.188964

Nothing 0 0 0 10967162013732 0.550195 0

Total 0 0 19933230517200 1 0.986821

The return for ordinary "7/5" Double Double Bonus is is 95.71%. So, the player should bet the full six coins.

Jacks or Better

For an explanation of this table, please see the explanation in the Bonus Poker Deluxe section.

"7/5" Jacks or Better

Hand Base Pay Bonus Total Combinations Probability Return

Royal flush 800 0 800 497941608 0.000025 0.016654

Straight flush 50 0 50 1984110324 0.0001 0.004147

Four aces on deal 25 236.69 261.69 3930382320 0.000018 0.004028

Four 2-4 on deal 25 180.22 205.22 9859193424 0.000055 0.009475

Four 5-K on deal 25 121.94 146.94 29423360772 0.000166 0.020354

Four aces on draw 25 180.22 205.22 223027715844 0.000197 0.03372

Four 2-4 on draw 25 121.94 146.94 199818112788 0.000495 0.060567

Four 5-K on draw 25 61.87 86.87 217867866324 0.001476 0.106859

Full house 7 0 7 223027715844 0.011189 0.065268

Flush 5 0 5 199818112788 0.010024 0.041768

Straight 4 0 4 217867866324 0.01093 0.036433

Three of a kind 3 0 3 1512158573688 0.075861 0.189653

Two pair 2 0 2 2602718111808 0.130572 0.21762

Jacks or better 1 0 1 4216077070884 0.21151 0.176258

Nothing 0 0 0 10911082187736 0.547382 0

Total 19933230517200 1 0.982804

The return for ordinary "7/5" Jacks or Better is is 96.15%. So, the player should bet the full six coins.

Bonus Poker

For an explanation of this table, please see the explanation in the Bonus Poker Deluxe section.

"10/8" Bonus Poker

Hand Base Pay Bonus Total Combinations Probability Return

Royal flush 800 0 800 397331376 0.00002 0.013289

Straight flush 50 0 50 2232922548 0.000112 0.004668

Four aces on deal 80 236.69 316.69 4116184116 0.000018 0.004874

Four 2-4 on deal 40 180.22 220.22 9677409204 0.000055 0.010168

Four 5-K on deal 25 121.94 146.94 28557904932 0.000166 0.020354

Four aces on draw 80 180.22 260.22 205672012524 0.000206 0.044779

Four 2-4 on draw 40 121.94 161.94 328137420456 0.000485 0.065519

Four 5-K on draw 25 61.87 86.87 280156665180 0.001433 0.103716

Full house 10 0 10 205672012524 0.010318 0.085984

Flush 8 0 8 328137420456 0.016462 0.109746

Straight 5 0 5 280156665180 0.014055 0.058561

Three of a kind 3 0 3 1460493243936 0.073269 0.183173

Two pair 1 0 1 2371585371048 0.118976 0.099147

Jacks or better 1 0 1 3822600300984 0.19177 0.159809

Nothing 0 0 0 11414817861216 0.572653 0

Total 19933230517200 1 0.963785

The return for ordinary "10/8" Bonus Poker is is 94.18%. So, the player should bet the full six coins.

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