Introduction
I thought I would also write an explanation about Omaha Hold’em in the same spirit as I did about Short Deck Hold’em, but Omaha Hold’em has a very large number of hand combinations. Unlike Texas Hold’em, there is probably no table for which hands to enter with pre-flop. Hands that are likely to be high-ranking straights or flushes tend to be strong, and it is assumed that players decide which hands to enter with based on intuition.
In Texas Hold’em, a hand combination can be made by drawing two cards from a deck of 52 cards, so there are \(\small {}_{52}C_2=1326\) possible combinations. There are essentially 169 possible hand combinations, and the combination is:
- Pocket Pair…13×6=78
- Off-suit…78×12=936
- Suited…78×4=312.
I thought about classifying hands in the same way for Omaha Hold’em, but I found it quite difficult. Since a hand combination is the combination you get when you draw 4 cards from 52 cards, there are \(\small {}_{52}C_4=270725\) possible combinations. To sum up, if my calculations are correct, there are essentially 16432 possible hand combinations. You might think that it’s impossible to classify something like that, but I did some calculations to kill time, so I thought I’d post it here (how bored am I…).
Classify your hand as follows and calculate the number of combinations for each.
- Four of a Kind in the Hand (XXXX)
- Three of a Kind in the Hand (XXXY)
- Two Pairs in the Hand (XXYY)
- One Pair in the Hand (XXYZ)
- All Cards Have Different Rank (XYZW)
In addition, to represent suits, we assign the symbols s and u to cards of the same suit in a row, and o to cards of different suits. For example, if the first and third cards are of the same suit and the other two are of different suits, we will represent them as soso.
Four of a Kind in the Hand
It is rare that all the cards you are dealt are of the same rank, and it wouldn’t be a very strong hand. There are 13 possible hand combinations, from AAAA to 2222, and 13 / 270725 possible card combinations (combos). Inevitably, all suits are different, so the expression is XXXX = XXXXoooo.
Three of a Kind in the Hand
In XXXY, there are \(\small {}_{13}P_2=156\) possible combinations of ranks for the combinations of X and Y (the hand changes when the order is changed, so the calculation is done using permutations). You can distinguish between hands with and without cards of matching suits in X and Y. Therefore, there are 156 possible hands, XXXYo (=XXXYoooo), and 156 possible hands, XXXYs (=XXXYsoos), for a total of 312 possible hands. The number of combos is 156×16=2496 / 270725, since there are \(\small {}_{4}C_3\times{}_{4}C_1=4\times4=16\) combinations of X’s suit and Y’s suit. Offsuit and suited have a 50/50 probability and are 1248 combos each.
Two Pairs in the Hand
XXYY is a combination of X and Y, and there are \(\small {}_{13}C_2=78\) possible combinations of ranks (the hand is the same even if the order is changed, so the calculation is done by combination). The number of combos is 78×36=2808 / 270725, since there are \(\small {}_{4}C_2\times{}_{4}C_2=6\times6=36\) combinations of X’s suit and Y’s suit. There are three suit combinations: XXYYo (=XXYYoooo), where all suits are different; XXYYs (=XXXYsoso), where there is one suited; and XXYYss (=XXXYsusu), where there are two suited. Each combo number is:
- XXYYo…78×6=468
- XXYYs…78×24=1872
- XXYYss…78×6=468.
One Pair in the Hand
In XXYZ, the combinations of X, Y, and Z have \(\small {}_{13}C_1\times {}_{12}C_2=13 \times 66 = 858\) possible rank combinations. XX is unrelated to the rank strength of YZ, but Y>Z must be true, so the calculation is as shown here. Since there are \(\small 6 \times 4 \times 4 = 96 \) possible combinations of suits, the number of combos is \(\small 858 \times 96 = 82368 / 270725\). This means that there is roughly a 1/3 chance that you will have a pair in your hand.
This pattern has a slightly more complicated suit combination, and requires three pieces of information: whether YZ is a suit, whether XY has a suit, and whether XZ has a suit. Let’s denote this as \(\small XX[YZs_1]s_2s_3 \), where \(\small s_1\) denotes whether YZ is a suit, \(\small s_2\) denotes whether XY has a suit, and \(\small s_3\) denotes whether XZ has a suit. This results in the following 6 possible suit combinations:
- XX[YZo]oo…858×12=10296
- XX[YZo]so…858×24=20592
- XX[YZo]os…858×24=20592
- XX[YZo]ss…858×12=10296
- XX[YZs]oo…858×12=10296
- XX[YZs]ss…858×12=10296
All Cards Have Different Rank
XYZW has \(\small {}_{13}C_4 = 715\) possible rank combinations. Without loss of generality, let’s say the rank strength is X>Y>Z>W. There are \(\small 4 \times 4 \times 4 \times 4 = 256 \) possible suit combinations, so the number of combos is \(\small 715 \times 256 = 183040 / 270725\). The suit combinations are complex, with 15 different patterns.
- Off-suit (oooo)…All four suits are different.
- Single suited (ssoo,soso,soos,osso,osos,ooss)
- Double suited (ssuu,susu, suus)
- Tri-suited (ssso, ssos, soss, osss)…Three cards of the same suit.
- Quad-suited (ssss)…All four cards are of the same suit.
The combo numbers for each are as follows:
- Off-suit…715×24=17160
- Single suited…715×24×6=102960
- Double suited…715×12×3=25740
- Tri-suited…715×12×4=34320
- Quad-suited…715×4=2860
Summary
The results of the calculations so far are summarized in the table below.
| Symbol | Number of Hands | Number of Combos |
| XXXX=XXXXoooo | 13 | 13 |
| XXXYo=XXXYoooo | 156 | 1248 |
| XXXYs=XXXYsoos | 156 | 1248 |
| XXYYo=XXYYoooo | 78 | 468 |
| XXYYs=XXYYsoso | 78 | 1872 |
| XXYYss=XXYYsusu | 78 | 468 |
| XX[YZo]oo=XXYZoooo | 858 | 10296 |
| XX[YZo]so=XXYZsoso | 858 | 20592 |
| XX[YZo]os=XXYZsoos | 858 | 20592 |
| XX[YZo]ss=XXYZsusu | 858 | 10296 |
| XX[YZs]oo=XXYZooss | 858 | 10296 |
| XX[YZs]ss=XXYZsoss | 858 | 10296 |
| XYZWoooo | 715 | 17160 |
| XYZWssoo | 715 | 17160 |
| XYZWsoso | 715 | 17160 |
| XYZWsoos | 715 | 17160 |
| XYZWosso | 715 | 17160 |
| XYZWosos | 715 | 17160 |
| XYZWooss | 715 | 17160 |
| XYZWssuu | 715 | 8580 |
| XYZWsusu | 715 | 8580 |
| XYZWsuus | 715 | 8580 |
| XYZWssso | 715 | 8580 |
| XYZWssos | 715 | 8580 |
| XYZWsoss | 715 | 8580 |
| XYZWosss | 715 | 8580 |
| XYZWssss | 715 | 2860 |
| Total | 16432 | 270725 |
Since there are nearly 100 times as many hand combinations as in Texas Hold’em, rather than memorizing the betting strategy for each hand, you will need to memorize the characteristic rules and decide on a betting strategy for each one. I will think about this later, but for now, let’s calculate the probability of winning for 16432 combinations…
In reality, when judging the strength of a hand, you might not care much about the suit combination of the cards, and might only distinguish whether the hand is off-suit, single suited, or double suited. In this case, XX[YZo]so, XX[YZo]os, XX[YZs]oo, and XX[YZs]ss could be combined into one called XXYZs, and the probability of winning could be calculated as the average of these, which could simplify the calculation. In this case, the symbols and combo numbers would be as follows. Since the probability of making a suited hand is about 90%, you probably won’t play many unsuited hands. We’ll consider how to display hand ranges corresponding to VPIP next time.
| Symbol | Number of Hands | Number of Combos |
| XXXX | 13 | 13 |
| XXXYo | 156 | 1248 |
| XXXYs | 156 | 1248 |
| XXYYo | 78 | 468 |
| XXYYs | 78 | 1872 |
| XXYYss | 78 | 468 |
| XXYZo | 858 | 10296 |
| XXYZs | 858 | 61776 |
| XXYZss | 858 | 10296 |
| XYZWo | 715 | 17160 |
| XYZWs | 715 | 140140 |
| XYZWss | 715 | 25740 |
Comments