Outline
The purpose of this article is to create a function that displays the hand ranges corresponding to a specified VPIP in Omaha Hold’em in order of the probability of winning, just as we have done for Texas Hold’em and Short Deck Hold’em. As mentioned in the previous post, there are 16,432 possible combinations of cards in pre-flop in Omaha Hold’em, so it would be difficult to display them, and it would be unrealistic for a human to remember each and every card. Therefore, let’s consider a way to display them with some simplification.
As I mentioned at the end of my last post, if you divide your hand based on the rank of the suited cards, the combinations of cards in your hand can become quite complicated. On the other hand, it can be assumed that it does not have much impact on the probability of winning, because flash clashes are rare and only one high-ranked card can decide the outcome of a kicker match. Therefore, we will group suited cards together and not consider what rank combination they are made of. There are 4 types of XXYZs, 11 types of XYZWs, and 3 types of XYZWss, but the winning probability of suited cards in each category will be represented by the following hands.
- XXYZs…XX[YZo]os
- XYZWs…XYZWsoos
- XYZWss…XYZWssuu
Configuring Tables
Based on the assumptions in the previous section, divide your hand into the following three categories:
- If there is more than one pair
- If you have no pairs and the two highest cards in your hand are 9 or higher
- If you have no pairs and the two lowest cards in your hand are 7 or lower
The suited cards are “o”, “s”, and “ss”. The more suited cards you have, the higher your chances of winning. Therefore, we will represent the hands that are entered into each square by color coding them into three types: enter “ss” only, enter “s” and “ss” only, or enter all. Note that this eliminates the need to prepare separate cells for “o”, “s”, and “ss” in the table. The vertical and horizontal axes of each table are as follows.
- vThe 13 ranks of paired cards (AA to 22) are represented on the horizontal axis, and the combinations of two cards \(\small 13+{}_{13}C_2 = 13 + 78 = 91\) are represented on the vertical axis to represent rank combinations. Note that the vertical axis also includes pocket pairs. For example, AAKK has two squares, KKAA and AAKK, but one of them (top right) is unused.
- Arrange the combinations of the two highest-ranking cards on the horizontal axis. The 15 numbers on the horizontal axis are AK, AQ, AJ, AT, A9, KQ, KJ, KT, K9, QJ, QT, Q9, JT, J9, and T9. The vertical axis is arranged as follows: QJ, QT, …, Q2, JT, J9, …, J2, …. Squares that would form pairs or squares that would have a higher rank on the vertical axis than on the horizontal axis should not be used.
- Arrange the combinations of two cards with the lowest ranking on the horizontal axis. The 15 numbers on the horizontal axis are 76, 75, 74, 73, 72, 65, 64, 63, 62, 54, 53, 52, 43, 42, and 32. The vertical axis is arranged as follows: A8, A7, …, A2, Q8, Q7, …, Q2, …. Squares that form pairs or squares that have a lower rank on the vertical axis than the horizontal axis should not be used.
This expression method can express all rank combinations. Although it is a huge table, I think it is much easier to read than 858 or 715 combinations lined up in a row. Also, since the probability of winning is relatively low in Table 3, if your VPIP is low, it may be easier to understand that there will generally be no problem if you only look at Tables 1 and 2.
Implementation
I created a table based on the above principles, so I’ve included a link to it. The probability calculations were slow to converge, so I ran 100,000 Monte Carlo simulations, which took more than a week to complete… I’ll provide an overview of Omaha Hold’em next time.
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