On Estimating Card Equity. Part 1
As should be apparent by now, this Hold’em math site has a big focus on expected value, since that is the primary math factor for making poker decisions. To handle both betting and calling decisions, the EV equation involves, at minimum, these four factors: pot size, bet/raise/call size, fold equity, and card equity. Generally, the pot size is a given and the bet values are either known or are to be determined. The fold and card equity values, however, have to be estimated. This post deals with estimating the card equity, the chance you will win the hand (or tie) if it goes to showdown.
We present 6 ways card equity can be estimated. They are as follows:
1. Outs 2. Standard Confrontations 3. Combinatorics 4. Data 5. Equity Calculator 6. Bayes
This post will deal with the first three ways to be followed by a second post for the last three.
Example: You hold A5hh and the flop is 7h 9s Jh. You have the nut flush draw with 9 outs. If the bet was all-in, your estimated showdown equity is 4*9 = 36%. If not an all-in bet, your equity estimate is 2*9 = 18%, the chance you will hit on the turn. The exact values are:
Showdown, Flop to River: 9/47 + 38/47 * 9/46 = 34.9%
Flop to Turn: 9/47 = 19.1% Turn to River: 9/46 = 19.6%
For this particular example, if an ace falls on the turn or river that could produce a win. So, there are two more possible outs but because of the low kicker and a pair is not often a winner, you might consider that to be one more out.
The two tables in this section present the winning chances for many types of matchups or confrontations – e.g., a pair vs. 2 undercards. The first table shows pre-flop matchups in the form of favorite to underdog and gives both probability and odds for values. The second table gives the showdown equity (no folds) for when the matchup is on the flop and for when the matchup is after the turn and going into the river. In both tables, the results are approximate because of possible suit or rank interactions or rank value extremes.
If we had to recommend one table to memorize, it would probably be this one for it encapsulates so many familiar situations and it is not too extensive. There may be some surprises here. Consider the first two matchups – pair vs. 2 undercards and pair vs. lower pair. The probabilities are very close – 0.82 for the former and 0.81 for the latter. The reason for the closeness is that if you have a lower pair, there are only two outs remaining for improvement. With two undercards, you can still get trips, but you also have better chances for a flush or straight. So, while three of a kind is less likely for the undercard case, that is almost compensated for by the better flush and straight chances for the undercards. The so-called race situation is one where there is a pair vs. two over-cards, such as 88 vs. AK. We show a range for the pair equity because so much depends of rank/suit interactions. The pair is usually a slight favorite but not always. For example, with 3ª3§ vs. Q¨J¨, the suited connector has 52% equity, primarily due to additional equity given by the flush and straight potential.
Example: Having a higher pair has an approximate equity of 82%. Exact results for all cases depend on rank and suit interactions.
Example: A dominated hand occurs when one rank is held by both players so that the player with the higher rank for the second card is ahead. KQ dominates KJ and is about a 2.7 to 1 favorite.
Equity: Hand Matchups - On the Flop; Going into River
This is a follow-up to the pre-flop matchup table showing equities for flop and turn activity. It includes drawing and made hands that occur after the flop and turn cards are dealt. As an example, if you flop a flush, against a set you have 66% equity. If the same situation remains after the turn, your equity improves to 77%. As in the previous table, suit and rank interactions can have an effect so all entries are approximate. Note that for some cases, such as flush vs. 2 pair, your equity improves with each street, while for others such as top 2 pair vs underset the equity decreases. In general, hands with less than 50% equity after the flop will have decreased equity on the river if they do not improve. The opposite is true for hands with greater than 50% equity after the flop.
Example: If you have an open-end straight draw facing a set after the flop, your chance of winning the hand is 26%.
Example: We know that an under-pair against 2 over-cards (race) is roughly 50-50 pre-flop. But if the flop doesn’t improve either hand the pair equity increases to about 75% and further increases to 86% after the turn. From the 4-2 rule, the over-cards have 6 outs or about a 4*6= 24% chance after the flop if all-in. If the same situation stays going into the river, the chance reduces to 2*6= 12%, approximately.
If you know what will win the hand and can figure out how many ways or combinations that can be achieved then the win probability is equal to
eq = Number of success combinations / Total number of combinations
This is simply a more accurate estimate than the 4-2 rule. It is much more complete in its applicability though requiring much more input. Let’s do an example.
Example: Suppose after a river ace is dealt, villain goes all-in with a board of T K 2 T A, rainbow. You have A K for two pair, aces over kings. The range you put villain on after the turn is {TT+, AT+}. The combos (hands) that beat you (villain hits a set or better) are AT, TT, KK and AA with AK a tie. Pairs have 6 combos and non-pairs have 12 combos (4 * 3) but you have to account for card removal, namely the cards in your hand and those on the board. Here is the count for villain combos that will beat you:
TT-1, JJ-6, QQ-6, KK-1, AA-1, AT-4, AJ-8, AQ-8, AK-4.
We now see that villain’s range has 15 pairs plus 24 non-pairs for a total of 39 combinations. Of those, 7 beat you and 4 tie. We assign half the ties as a win. Therefore, the number of winning combos you have is 39-7-2 = 30. The total number of combos is 39, therefore your equity is
With equity over 50%, your call has positive EV. The river situation makes this calculation exact. A similar calculation can be done on an earlier street but then the calculation is only a first-cut evaluation and folding aspects also have to be considered.
The greater input requirement using combinatorics compared to outs is having to estimate villain’s range, where specific hands and their frequency have to be determined. For the outs method, you usually only have to estimate what cards you need for a win, and that is typically easy to do. Here you cannot do combos without specifics on villain’s range.
