Mathematical Analysis
Of The
Following
Casino
Game
June 3, 2003
| INSTALLATIONS (Land) |
Mathematical Analysis
Of The
Following
Casino
Game
June 3, 2003
RULES
Following are the rules for Ca$ino Omaha™
1. Players make an Ante wager and an optional side bet Triple+.
2.
3
cards are dealt face down to each player and the dealer,
In addition 3 flop (community)
cards are dealt face up.
The players may examine their own
3
cards but sharing information should be discouraged.
All players and dealer may use the
3
flop
(community)
cards as part of their hand.
3.
Each player must decide to either fold or call.
If the player folds he gives up
his cards and his Ante. If the player call the Call bet must be
double
the Ante.
4.
The dealer will then deal
2
more community cards, form a total of
5.
The dealer will also turn over his own
3
cards.
5. Each player and the dealer will form the best 5-card poker hand by using exactly 2 of his own 3 cards and 3 of the 5 flop (community) cards.
6. Each player will compare his best hand against the dealer’s best hand.
7.
The dealer must have a pair of
7s
or better to qualify.
If the
dealer
does not
qualify then the ante will pay according to the AnteWin® pay table below and
the call bet will push.
If the dealer qualifies:
(a) Dealer beats the player, then the player will lose both the Ante and Call.
(b) Player beats the dealer,
then the ante
will pay according to the AnteWin® pay table shown below, and the call bet
will pay 1 to 1.
8. If the dealer qualifies and the player ties the dealer then both ante and call bets will push.
|
Table 1: AnteWin®
Pay Table |
|
|
Hand |
Pays |
|
Royal Flush |
100 to 1 |
|
Straight flush |
50 to 1 |
|
Four of a kind |
10 to 1 |
|
Full house |
2 to 1 |
|
Flush |
2 to 1 |
|
Straight or less |
1 to 1 |
9.
Side bet:
Triple+
The
Triple Bonus
will be scored according to the best poker hand consisting
of 2 out of player’s 3 own cards and the first 3 flop (community) cards. This
bet will pay even if the player folds and is acted on before the last 2 flop
(community) cards are deal.
There are 4 pay tables available for the Triple Bonus
side bet,
which are indicated in table 2.
Table 2 are the pay tables available for the Triple Bonus side bet.
|
Table 2: Triple+
Pay Tables |
||||
|
Hand |
Pay Table A |
Pay Table B |
Pay Table C |
Pay Table D |
|
Royal flush |
1000 to 1 |
200 to 1 |
15 to 1 |
100 to 1 |
|
Straight flush |
500 to 1 |
100 to 1 |
15 to 1 |
50 to 1 |
|
Four of a kind |
100 to 1 |
80 to 1 |
15 to 1 |
40 to 1 |
|
Full house |
30 to 1 |
40 to 1 |
15 to 1 |
30 to 1 |
|
Flush |
25 to 1 |
25 to 1 |
15 to 1 |
20 to 1 |
|
Straight |
15 to 1 |
15 to 1 |
15 to 1 |
15 to 1 |
|
Three of a kind |
10 to 1 |
10 to 1 |
15 to 1 |
12 to 1 |
|
Two pair or less |
Loss |
Loss |
Loss | Loss |
--------------------------------------------------------------------------------------------------------------------------------------------
ANALYSIS
Table 3 shows a
breakdown of every significant outcome, the number of combinations, the
probability, and contribution to the total return – based on optimal player
strategy.
The number in the lower
right cell
of table 3 shows a
house edge of
2.33%
|
Table 3: breakdown
of every significant outcome |
||||
|
Event |
Combinations |
Probability |
Pays |
Return |
|
Dealer qualifies, player wins with straight or less |
1,228,477,331,537,950 |
0.220106 |
3 |
0.660317 |
|
Dealer qualifies, player wins with flush |
145,139,512,840,088 |
0.026005 |
4 |
0.104018 |
|
Dealer qualifies, player wins with full house |
163,914,672,830,796 |
0.029369 |
4 |
0.117474 |
|
Dealer qualifies, player wins with four of a kind |
15,081,271,619,280 |
0.002702 |
12 |
0.032425 |
|
Dealer qualifies, player wins with straight flush |
1,688,786,599,264 |
0.000303 |
52 |
0.015734 |
|
Dealer qualifies, player wins with royal flush |
212,742,187,532 |
0.000038 |
102 |
0.003888 |
|
Dealer doesn’t qualify, player has straight or less |
895,363,506,870,640 |
0.160422 |
1 |
0.160422 |
|
Dealer doesn’t qualify, player has flush |
46,189,460,117,352 |
0.008276 |
2 |
0.016551 |
|
Dealer doesn’t qualify, player has full house |
20,799,848,717,340 |
0.003727 |
2 |
0.007453 |
|
Dealer doesn’t qualify, player has four of a kind |
906,957,913,068 |
0.000162 |
10 |
0.001625 |
|
Dealer doesn’t qualify, player has straight flush |
490,651,135,784 |
0.000088 |
50 |
0.004395 |
|
Dealer doesn’t qualify, player has royal flush |
44,950,822,404 |
0.000008 |
100 |
0.000805 |
|
Dealer beats player |
1,691,564,573,202,400 |
0.303077 |
-3 |
-0.909231 |
|
Player folds |
1,335,182,111,247,600 |
0.239224 |
-1 |
-0.239224 |
|
Tie |
36,248,167,174,500 |
0.006495 |
0 |
0.000000 |
|
Total |
5,581,304,544,816,000 |
1.000000 |
|
-0.023345 |
The total number of combinations equals the number of ways to choose sets of 3, 3, 2, and 3 out of 52. In other words combin(52,3)*combin(49,3)*combin(46,2)*combin(44,3) = 5,581,304,544,816,000.
The number in the lower right cell of table 3 shows the player can expect to
lose 2.33% for each initial wager made. In other words the house edge is
2.33%, defined as the ratio of expected player loss to the initial wager.
The average wager in this game is 2.52 units. The ratio of expected player loss
to total amount bet, what I define as the “element of risk” is 0.93%.
As in most
poker based games the optimal strategy in this game is complex. It would also be
difficult to put a basic strategy into words.
According to table 3 the player should fold 23.92% of the time, so the player
should fold on about the weakest 24% of hands.
--------------------------------------------------------------------------------------------------------------------------------------------
Triple+
Analysis
Table 4 shows
the number of combinations for each outcome of the
Triple+
side bet,
the probability, and contribution to the total return for Pay Table A
The lower right cell of table 4 shows a house edge of 6.60%
|
Table 4: Triple+ Pay Table A |
||||
| Hand | Combinations | Probability | Pays | Return |
|
Royal flush |
1880 |
0.000005 |
1000 |
0.004617 |
|
Straight flush |
16776 |
0.000041 |
500 |
0.020601 |
|
Four of a kind |
234624 |
0.000576 |
100 |
0.057623 |
|
Full house |
1676064 |
0.004116 |
30 |
0.123491 |
|
Flush |
2126344 |
0.005222 |
25 |
0.130556 |
|
Straight |
4321584 |
0.010614 |
15 |
0.159205 |
|
Three of a kind |
15448576 |
0.037941 |
10 |
0.379413 |
|
Two pair |
40936896 |
0.100540 |
-1 |
-0.100540 |
|
Pair |
206304520 |
0.506679 |
-1 |
-0.506679 |
|
Nothing |
136103136 |
0.334266 |
-1 |
-0.334266 |
|
Total |
407170400 |
1.000000 |
|
-0.065978 |
Table 5 shows the number of combinations for each outcome of the
Triple+
sidebet, the
probability, and contribution to the total return for Pay Table B
The lower right cell of table 5 shows a house edge of 5.6%
|
Table
5: Triple+ Pay Table B |
||||
| Hand | Combinations | Probability | Pays | Return |
|
Royal flush |
1880 |
0.000005 |
200 | 0.001 |
|
Straight flush |
16776 |
0.000041 |
100 | 0.0041 |
|
Four of a kind |
234624 |
0.000576 |
80 | 0.04608 |
|
Full house |
1676064 |
0.004116 |
40 | 0.16464 |
|
Flush |
2126344 |
0.005222 |
25 | 0.13055 |
|
Straight |
4321584 |
0.010614 |
15 | 0.15921 |
|
Three of a kind |
15448576 |
0.037941 |
10 | 0.37941 |
|
Two pair |
40936896 |
0.100540 |
-1 | -0.10054 |
|
Pair |
206304520 |
0.506679 |
-1 | -0.506679 |
|
Nothing |
136103136 |
0.334266 |
-1 | -0.334266 |
|
Total |
407170400 |
1.000000 |
-0.056495 | |
Table 6 shows the number of combinations for each outcome of the
Triple+
side bet, the
probability, and contribution to the total return for Pay Table C
The lower right cell of table 6 shows a house edge of 6.37%
|
Table 6:
Triple+ Pay Table C |
||||
|
Hand |
Combinations |
Probability |
Pays |
Return |
|
Royal flush |
1880 |
0.000005 |
15 |
0.000069 |
|
Straight flush |
16776 |
0.000041 |
15 |
0.000618 |
|
Four of a kind |
234624 |
0.000576 |
15 |
0.008643 |
|
Full house |
1676064 |
0.004116 |
15 |
0.061746 |
|
Flush |
2126344 |
0.005222 |
15 |
0.078334 |
|
Straight |
4321584 |
0.010614 |
15 |
0.159205 |
|
Three of a kind |
15448576 |
0.037941 |
15 |
0.569120 |
|
Two pair |
40936896 |
0.100540 |
-1 |
-0.100540 |
|
Pair |
206304520 |
0.506679 |
-1 |
-0.506679 |
|
Nothing |
136103136 |
0.334266 |
-1 |
-0.334266 |
|
Total |
407170400 |
1.000000 |
|
-0.063749 |
Table 7 shows the number of combinations for each outcome of the
Triple+
side bet, the
probability, and contribution to the total return for Pay Table D
The lower right cell of table 7 shows a house edge of 7.35%
|
Table 7:
Triple+ Pay Table D |
||||
|
Hand |
Combinations |
Probability |
Pays |
Return |
|
Royal flush |
1880 |
0.000005 |
100 |
0.000462 |
|
Straight flush |
16776 |
0.000041 |
50 |
0.002060 |
|
Four of a kind |
234624 |
0.000576 |
40 |
0.023049 |
|
Full house |
1676064 |
0.004116 |
30 |
0.123491 |
|
Flush |
2126344 |
0.005222 |
20 |
0.104445 |
|
Straight |
4321584 |
0.010614 |
15 |
0.159205 |
|
Three of a kind |
15448576 |
0.037941 |
12 |
0.455296 |
|
Two pair |
40936896 |
0.100540 |
-1 |
-0.100540 |
|
Pair |
206304520 |
0.506679 |
-1 |
-0.506679 |
|
Nothing |
136103136 |
0.334266 |
-1 |
-0.334266 |
|
Total |
407170400 |
1.000000 |
|
-0.073476 |
--------------------------------------------------------------------------------------------------------------------------------------------
Conclusion
This analysis
was conducted using a combinational approach that analyzed every possible
combination of player cards, community cards, and dealer cards.
This was a very difficult and time consuming analysis. A random simulation,
although slow, showed the house edge of 2.33% was reasonable.
If you or any interested party have any questions about this analysis please contact me by any of the means below.
E-mail: mail@gamingmath.com
Phone:
702-804-0015
Postal mail: 9200 Sienna Vista Dr., Las Vegas, NV 89117
Thank you for choosing me to analyze Ca$ino Omaha™ and I wish you success with it.
Regards,
US Patents: 6,206,373 & 6,637,747
South Africa Patent: 2001/2907
Australia Patent: 773436
© 2003-2010
Stephen Au-Yeung.
All rights reserved