The mathematics of Draw Poker: Probabilities

Among all the variations of poker, for draw poker (or classical poker) the probabilities involved are the hardest to compute. If the odds for your own hand were partially figured out before, the odds for opponents' hands were not. Players were only able to figure out odds for opponents' hands in particular gaming situations, by applying some basic combinatorial calculus and algebraic cleverness. No formulas to cover all gaming situations and return numerical values when inputting parameters were figured out before, due to the assumed heaviness of such formulas.

Well, these formulas have been finally worked out! They are the outcome of one year of study, math work and tests. We found the right probability model in which to apply the theory and quantify the card distributions in order to work out the draw poker probability formulas. They were built with an enough large range of variables to cover all possible situations. Thus, we can know the mathematical odds of one of more of your opponents holding a certain card formation and a formation higher than yours, at every stage of the game.

The results were gathered and the outcome is the book (Draw Poker Odds: The Mathematics of Classical Poker) (see the Books section) and a software (Draw Poker Odds Calculator 1.1).

The software is based on the formulas from the book and is the first poker odds calculator that uses compact probability formulas instead of partial simulations. See the software section for more details.

The draw poker most often practiced is one with 52 cards, without wild cards, and with the ability to discard up to five cards. We will present the probabilities involved in this type of game.
     We are interested only in the card distributions that generate the predicted events. Additional information like the way your opponents discard or their psychological profile is not taken into account.
     The following types of probabilities are calculated:
     – Initial probabilities on the first card distribution for your own hand;
     – Prediction probabilities after first card distribution and before the second for your own hand;
     – Prediction probabilities for opponents’ hands.

The events whose probability is to be measured are the occurrence of the various combinations of cards (of size 1, 2, 3, 4 or 5) in your own hand or your opponents’ hands.

In this section we present the probabilities of receiving various valuable formations from the first five cards dealt.

In this table, "at least one pair" means you can also achieve two pair, three of a kind, a full house or four of a kind; "at least two pair" means you can also achieve three of a kind, a full house or four of a kind and the like.

2) Prediction probabilities after first card distribution and before the second for your own hand

These are the probabilities of receiving various card combinations at the second distribution, after you have been dealt the first five cards and you have discarded (or not).

There are 17 cases of possible types of dealt combinations that require making a decision to keep or discard cards the most (one pair + high card, one pair + three suited, one pair + three from a straight, ...., two pair + three suited, ..., three from a straight + three suited, four suited + four from a straight, etc.)
All these cases are presented in detail along with the associated odds and conclusions in the book Draw Poker Odds. We present here only one of these cases:

5) One pair + three suited (the suit does not contain any paired card)   (example: 7♥ 7♠ 8♣ J♣ K♣)
     The player has two playing options to achieve a valuable formation: holding only the pair and discarding the suited cards, or holding the suited cards and discarding the paired cards.
      A) Held: (77)   Discarded: (8♣ J♣ K♣)   Goal: two pair, three of a kind, full house or four of a kind
     The probabilities of achieving these target formations are
     - two pair: 15.98519%
     - three of a kind: 11.43385%
     - full house: 1.01757%
     - four of a kind: 0.24668%,
totaling 28.68329%.
      B) Held: (8♣ J♣ K♣)   Discarded: (77)   Goal: flush
     The number of all possible combinations for the second distribution is C(47, 2) = 1081.
     The favorable combinations are (♣♣), numbering C(13-3, 2) = 45, so the probability of achieving a flush is
P = 45/1081 = 0.0416281 = 4.16281%.
     Let us also calculate the probability of achieving three of a kind in subcase B:
     The favorable combinations are (xx), x being any of the held cards, numbering 3C(3, 2) = 9.
     The probability of achieving three of a kind is P = 9/1081 = 0.0083256 = 0.83256%.

Conclusions:
   – The probability of achieving a flush in subcase B is lower than the probability of achieving three of a kind or better in subcase A and has a low value itself.
   – The probability of achieving three of a kind in subcase B is lower than the probability of achieving three of a kind in subcase A.
     Therefore, holding the pair and discarding the suited cards is recommended with respect to the goal of achieving the target formations.

These are the probabilities for one or many of your opponents to hold various card formations at any moment of the game, function of the value or/and symbol distributions of your own seen cards (the first five dealt plus their replacements).
     We present here only the probabilities of one or more opponents holding three of a kind as a final (or initial) hand, as a function of the information given by your own seen cards (the first five dealt plus their replacements).
     The set of parameters the formula depends on are:
          n = number of opponents (n could be 1, 2, 3, or 4);
          c = number of seen cards (c could be 5, 6, 7, 8, 9 or 10);

We are not interested in how many cards of each value in a distribution, but in a cumulative representation: how many values are represented by four cards, how many by three, and so on, how many by zero. This is the hypothesis for any calculus of the probability of the occurrence of a card formation made of values in your opponents’ hands.
     We call value distribution any finite sequence of numbers of values from the seen cards, in the cumulative form.
     For example, if the seen cards are 2♣ 3♠ 3♥ 5♦ 5♣ 5♥ 8♦ J♠ Q♥, their value distribution is 3-2-1-1-1-1-0-0-0-0, meaning there are six represented values, in this way: three cards of one value (5), two cards of another value (3), one card of another value (2), one card of another value (8), one card of another value (J) and one card of another value (Q); the rest of the values (4, 6, 7, 9, 10, K, A) are not represented.
     As a convention, we remove the zeroes from this denotation, so the above value distribution is denoted by 3-2-1-1-1-1.
     If a maximum number of five replacements is allowed (c could be maximum 10), there are 82 possible value distributions.
     The next table notes the probabilities of one and at least one of your opponents holding a three of a kind formation, corresponding to the value distribution of your own seen cards.
     The numerical values are returned by the formula of number of favorable card combinations for one oponent to hold three of a kind and some formulas of a classical probability repartition.
     The existence of a dash in the column n = 5 indicates that such case is impossible (if five-card replacement is alowed – so c could be up to 10, the maximum number of players can be maximum five, so you may play against a maximum of four opponents).

Distribution	n = 1	n = 2	n = 3	n = 4	n = 5
4-1	2.46685%	4.87285%	7.21950%	9.50826%	11.74056%
3-2	2.41522%	4.77211%	7.07207%	9.31649%	11.50670%
3-1-1	2.31339%	4.57326%	6.78086%	8.93739%	11.04403%
2-2-1	2.25850%	4.46599%	6.62363%	8.73254%	10.79382%
2-1-1-1	2.15621%	4.26594%	6.33017%	8.34989%	10.32607%
1-1-1-1-1	2.05367%	4.06516%	6.03534%	7.96507%	9.85517%
4-2	2.56793%	5.06992%	7.50766%	9.88280%	12.19696%
4-1-1	2.45996%	4.85941%	7.19983%	9.48268%	11.70938%
3-3	2.57114%	5.07617%	7.51680%	9.89468%	12.21142%
3-2-1	2.40466%	4.75150%	7.04191%	9.27724%	11.45882%
3-1-1-1	2.29604%	4.53936%	6.73117%	8.87266%	10.96498%
2-2-2	2.34615%	4.63727%	6.87463%	9.05949%	11.19311%
2-2-1-1	2.23731%	4.42456%	6.56288%	8.65337%	10.69708%
2-1-1-1-1	2.12817%	4.21105%	6.24961%	8.24478%	10.19750%
1-1-1-1-1-1	2.01874%	3.99673%	5.93480%	7.83374%	9.69434%
4-3	2.73704%	5.39916%	7.98843%	10.50682%	12.95629%
4-2-1	2.56024%	5.05494%	7.48577%	9.85436%	12.16232%
4-1-1-1	2.44492%	4.83006%	7.15689%	9.42683%	11.64128%
3-3-1	2.56360%	5.06148%	7.49533%	9.86678%	12.17744%
3-2-2	2.50131%	4.94006%	7.31781%	9.63609%	11.89638%
3-2-1-1	2.38558%	4.71425%	6.98736%	9.20626%	11.37222%
3-1-1-1-1	2.26951%	4.48752%	6.65520%	8.77368%	10.84408%
2-2-2-1	2.32288%	4.59180%	6.80803%	8.97277%	11.08723%
2-2-1-1-1	2.20657%	4.36446%	6.47473%	8.53844%	10.55661%
2-1-1-1-1-1	2.08994%	4.13620%	6.13970%	8.10132%	10.02195%
1-1-1-1-1-1-1	1.97298%	3.90703%	5.80292%	7.66141%	9.48323%
4-4	2.91711%	5.74912%	8.49852%	11.16772%	13.75906%
4-3-1	2.73294%	5.39120%	7.97681%	10.49176%	12.93798%
4-2-2	2.66665%	5.26219%	7.78851%	10.24748%	12.64087%
4-2-1-1	2.54363%	5.02256%	7.43843%	9.79286%	12.08740%
4-1-1-1-1	2.42024%	4.78191%	7.08642%	9.33516%	11.52947%
3-3-2	2.67033%	5.26936%	7.79898%	10.26106%	12.65740%
3-3-1-1	2.54713%	5.02938%	7.44840%	9.80582%	12.10319%
3-2-2-1	2.48028%	4.89904%	7.25781%	9.55808%	11.80130%
3-2-1-1-1	2.35643%	4.65733%	6.90402%	9.09776%	11.23982%
3-1-1-1-1-1	2.23221%	4.41460%	6.54827%	8.63432%	10.67380%
2-2-2-2	2.41324%	4.76825%	7.06643%	9.30914%	11.49774%
2-2-2-1-1	2.28912%	4.52584%	6.71136%	8.84685%	10.93346%
2-2-1-1-1-1	2.16462%	4.28240%	6.35433%	8.38141%	10.36462%
2-1-1-1-1-1-1	2.03976%	4.03792%	5.99533%	7.91281%	9.79117%
1-1-1-1-1-1-1-1	1.91453%	3.79242%	5.63435%	7.44102%	9.21310%
4-4-1	2.91711%	5.74912%	8.49852%	11.16773%	-
4-3-2	2.85062%	5.61998%	8.31040%	10.92413%	-
4-3-1-1	2.71952%	5.36508%	7.93869%	10.44232%	-
4-2-2-1	2.64825%	5.22637%	7.73622%	10.17960%	-
4-2-1-1-1	2.51642%	4.96952%	7.36089%	9.69208%	-
4-1-1-1-1-1	2.38417%	4.71151%	6.98335%	9.20104%	-
3-3-3	2.85477%	5.62805%	8.32217%	10.93937%	-
3-3-2-1	2.65209%	5.23385%	7.74715%	10.19379%	-
3-3-1-1-1	2.52006%	4.97661%	7.37125%	9.70556%	-
3-2-2-2	2.58052%	5.09444%	7.54350%	9.92937%	-
3-2-2-1-1	2.44817%	4.83640%	7.16617%	9.43890%	-
3-2-1-1-1-1	2.31540%	4.57719%	6.78662%	8.94489%	-
3-1-1-1-1-1-1	2.18222%	4.31682%	6.40484%	8.44730%	-
2-2-2-2-1	2.37607%	4.69568%	6.96019%	9.17088%	-
2-2-2-1-1-1	2.24299%	4.43568%	6.57918%	8.67461%	-
2-2-1-1-1-1-1	2.10950%	4.17450%	6.19594%	8.17475%	-
2-1-1-1-1-1-1-1	1.97559%	3.91215%	5.81046%	7.67127%	-
1-1-1-1-1-1-1-1-1	1.84127%	3.64863%	5.42272%	7.16415%	-
4-4-2	3.04702%	6.00119%	8.86536%	11.64226%	-
4-4-1-1	2.90736%	5.73020%	8.47097%	11.13206%	-
4-3-3	3.05172%	6.01031%	8.87862%	11.65940%	-
4-3-2-1	2.83565%	5.59090%	8.26802%	10.86923%	-
4-3-1-1-1	2.69494%	5.31726%	7.86891%	10.35180%	-
4-2-2-2	2.75924%	5.44236%	8.05144%	10.58853%	-
4-2-2-1-1	2.61818%	5.16781%	7.65069%	10.06857%	-
4-2-1-1-1-1	2.47664%	4.89195%	7.24744%	9.54460%	-
4-1-1-1-1-1-1	2.33464%	4.61477%	6.84167%	9.01659%	-
3-3-3-1	2.84000%	5.59935%	8.28034%	10.88519%	-
3-3-2-2	2.76348%	5.45059%	8.06344%	10.60409%	-
3-3-2-1-1	2.62217%	5.17559%	7.66206%	10.08333%	-
3-3-1-1-1-1	2.48040%	4.89929%	7.25817%	9.55855%	-
3-2-2-2-1	2.54506%	5.02535%	7.44251%	9.79816%	-
3-2-2-1-1-1	2.40294%	4.74813%	7.03698%	9.27083%	-
3-2-1-1-1-1-1	2.26034%	4.46959%	6.62891%	8.73942%	-
3-1-1-1-1-1-1-1	2.11728%	4.18973%	6.21830%	8.20393%	-
2-2-2-2-2	2.46771%	4.87452%	7.22195%	9.51145%	-
2-2-2-2-1-1	2.32523%	4.59640%	6.81476%	8.98154%	-
2-2-2-1-1-1-1	2.18228%	4.31695%	6.40503%	8.44755%	-
2-2-1-1-1-1-1-1	2.03887%	4.03617%	5.99275%	7.90944%	-
2-1-1-1-1-1-1-1-1	1.89498%	3.75406%	5.57790%	7.36719%	-
1-1-1-1-1-1-1-1-1-1	1.75062%	3.47060%	5.16047%	6.82076%	-

Example of using the table: Assume you are playing against three opponents, you have been dealt (77JJK), you hold (77JJ), discard (K) and replace it with (3).Your final hand is two pair: (77JJ3).
You want the probability for at least one opponent to hold three of a kind.

You have c = 6 seen cards, with the value distribution 2-2-1-1 (two J's, two 7's, one K and one 3).
By looking in the table at the intersection of row 2-2-1-1 with column n = 3, you find the probability 6.56288%.
Of course, this is not the probability that one or more opponents hold a higher formation than yours (there are also the straights, flushes, full houses, four of a kind and also the higher two pairs to count, but these can be also calculated).