Today we continue with some of the misconceptions associated with Omaha Hi-Low. Again I will use the same article written by a top female professional player as the basis for my comments.
In the third paragraph of her article Jennifer states "some players overvalue any hand that contains A- 2." There is good reason why people value hands containing an A-2. First of all they contain an Ace.
Second they contain a deuce. Let me explain. A basic law of probability states if you can do one thing in A ways and another in B ways then they can combine in A times B ways. In Hold-Em where you are dealt 2 cards they may combine in 52 x 51 or 2,652 combinations. In a poker hand the order of the cards does not matter. [As]-[Kh] is the exact same hand as [Kh]-[As].
To calculate the number of unique hands we must divide the total number of combinations by factorial 2 since we are looking at two cards at a time. The factorial of a number is a series of numbers beginning with the number and multiplying by one less then the previous number down to 1. Factorial 4 equals 4x3x2x1 or 24 and factorial 2 = 2x1 or 2. For Hold- Em where we are dealing with a 2 card hand, we divide 2,652 by factorial 2 to arrive at 1,326 possible starting hands.
Now let's look at Omaha. You start with 4 cards so that equates to 52x51x50x49 possible combinations or 6,497,400. Since order does not matter and we are looking at 4 cards at a time we need to divide the combinations by factorial 4 or 24, as calculated above, for a total of 270,725 possible starting hands. Wait, we are not done yet! Since As-2s-Ad-2d is exactly the same hand as Af- 2f-Aa-2a, we need to adjust for such duplications and others. My database of starting hands contains 11,995 unique starting hands. Now that we got that out of the way lets look at some meaningful numbers. There are 1,934 profitable hands out of the 11,995 for a total of 16.13%. Of those, 1,934 hands 1,820 contain an ace. That is 94.11% of all the profitable starting hands contain an ace. Less then 6% do not. There are 825 possible hands that contain an A-2. Only 12 of them are unprofitable! If we were to only play every possible starting hand containing an A-2 we would be wrong less then 1.5% of the time. I will go out on a limb here and state that most Omaha Hi-Low players will have to improve their game in order to reach the point where they only play unprofitable hands less then 1.5% of the time. Indeed people play hands containing A-2. Simply put, it is the best two card starting combination. Jennifer then gives an example of an overplayed A-2 hand of [Ah][2c][8s][Jd] and states it "isn't all that great." With a net win of $0.60 vs. $23.03 for the AdKs2d4 it is certainly a lesser value hand. It is still a very playable hand.
Here is how the two hands compare. Note that the [Ad][Ks][2d][4s] is superior in every category. Is this hand "unlikely to make a decent high."? It is true that you will win more high hands as a percentage of hands dealt with [Ad][Ks][2d][4s]. This is primarily due to the excellent flush draws this hand possesses. What if we were to take a look at the win rate measured as a percent of the hands played?
The chart on the right shows the high hands won and lost with each of these hands as a percentage of the hands each player played. Note the huge discrepancy in the flush category. The [Ad][Ks][2d][4s] wins 24.2% of the high hands played with a flush. The [Ah][2c][8s][Jd] has zero flush potential. This is significant in explaining the difference between the high only win rates of 6.3% vs. 5.6%. In all of the other categories except for a Pair, the [Ah][2c][8s][Jd] actually wins more high hands then does the [Ad][Ks][2d][4s]. One explanation for this is that our tight player is careful in pursuing his high when a flush is possible. Look at the straight numbers. The [Ah][2c][8s][Jd] beats the [Ad][Ks][2d][4s] hands down. They both have the same straight potential with the A-2. A-K and A-J are virtually the same. The difference lies with the 8 and the 4.
The A-4 may combine with a 2-3-5 to make a straight. The 2-4 will make a straight with A-3-5 or 3-5-6. The A-2 can not make these 3 straights. The [Ah][2c][8s][Jd] will also have the nut low with the 3-5-6. The 8 in [Ah][2c][8s][Jd] may combine with the J to form only two straights with 7-9-T or 9-T-Q. One less then the A-4, 2-4 combination may produce. So why is there a large discrepancy in win percentages?
The [Ad][Ks][2d][4s] will have a difficult decision to continue with a flop of 7-9-T rainbow or suited in anything other then diamonds or clubs. It does not have a low draw or a high draw. A flop of T-Q-K which gives both hands a draw to the nut high straight offers the [Ah][2c][8s][Jd] twice as many straight possibilities as it can make a straight with a 9 or a J. Basically without the low draw there are more hands that favor the [Ah][2c][8s][Jd] then the [Ad][Ks][2d][4s]. So what have we learned? Most hands containing an A-2 are profitable.
Less then 1.5% are not. Only 16.13% of all the Omaha Hi-Lo starting hands are profitable and more then 94% of them contain an Ace. Next time I will divert from analyzing this article to answer a question posed by a tournament director.
