1. jjd323’s Mathematics of PLO Ep. 2 Complex Starting Hand Combinatorics – Suitedness of AAXX

2. Abbreviations & Notation os- offsuit(e.g. AsAd7c2h) ss- single-suited(e.g. AsKd7d2h) ds- double-suited(e.g. AsJdJsTd) nn- non-nut(e.g. AsAd7c2c not AsAc7d2c) cf. nut (e.g. AsAc7d2c not AsAd7c2c) AKXX- offsuit (AK)XX or (AX)KX- single-suited to the A A(KX)X- single-suited to the K AK(XX)- single-suited to the low cards (AX)(KX) or (AK)(XX)- double-suited

3. Suited AAXX Combinations Let the suits, normally given as s, h, d, and c be given algebraic notation as a, b, c, and d. Offsuit pocket aces are of the form AaAbXcXdi.e. AsAhJdTc Non-nut single-suited pocket aces are of the form AaAbXcXci.e. AsAhJcTc Nut single-suited pocket aces are of the form AaAbXaXci.e. AsAhJsTc Double-suited pocket aces are of the form AaAbXaXbi.e. AsAhJsTh

4. Recap : How many AA** ? Counting AAXX: C(4,2) = 6 ways to get AA, with 48 non-A cards remaining C(48,2) = 1128 ways to get XX with each pair of AA Being careful to not double-count AAAX and AAAA hands. C(4,2) * C(48,2) 6* 1128 = 6768 AAXX hands Counting AAAX hands: C(4,3) = 4 ways to get AAA, leaving 48 non-A cards C(48,1) = 48 ways to deal X with each set of AAA 4 * 48 = 192 AAAX hands Counting AAAA: C(4,4) = 1 AAAA hand 6768 AAXX hands 192 AAAX hands 1 AAAA hand 6961total AAXX hands

5. AA** vs. KK**

6. AA** Combinations How many AA** are offsuit ? –AAAA –AAAX –AAXX How many suited AAXX Combinations? –Single-suited –Double-suited –Nut vs. non-nut

7. How many AA** are offsuit ? 6768 AAXX hands 192 AAAX hands 1 AAAA hand 6961total AAXX hands Quads AAAA cannot be suited ∴ 1 os AAAA combo Trips AAAX can be suited to any one of the three aces; count the twelve possible offsuit kickers for each of the four sets of AAA : i.e. AsAhAdXc can have any one of twelve kickers : 2c, 3c, 4c, 5c, 6c, 7c, 8c, 9c, Tc, Jc, Qc, Kc. 12 * 4 = 48 os AAAX hands We also count the single-suited trips : ∴ 192 - 48 = 144 ss AAAX hands [We’ll use this result later]

8. Offsuit AAXX - AaAbXcXd Pairs Offsuit pocket aces are of the form AaAbXcXd. We can calculate the total combinations of hands by consider all the choices separately : 1) We already know that there are C(4,2) = 6 ways to pick AaAb from the deck. 2) This leaves 48 non-A cards to choose as kickers; of these, 24 will be off-suited to both the aces chosen. i.e. for AsAdXhXc : KsQsJsTs9s8s7s6s5s4s3s2sKdQdJdTd9d8d7d6d5d4d3d2d KhQhJhTh9h8h7h6h5h4h3h2hKcQcJcTc9c8c7c6c5c4c3c2c Incorrect Counting Method C(24,2) = 276 possible kicker combos, including the nn ss AAXX (e.g. AsAd7c2c) (!!) Some combinations of these 24 kickers give us suited kickers in the 4-card hand. Correct Counting Method We must consider only the possible combinations of XcXd where c and d are different suits. C(12,1) Xc * C(12,1) Xd = ∴ 12 * 12 = 144 os kicker combos

9. C(12,1) kickers of type Xc * C(12,1) kickers of type Xd = 12* 12 = 144 os kicker combos e.g. AsAdXhXc KhQhJhTh9h8h7h6h5h4h3h2h KcAsAdKhKcAsAdQhKcAsAdJhKcAsAdThKcAsAd9hKcAsAd8hKcAsAd7hKcAsAd6hKcAsAd5hKcAsAd4hKcAsAd3hKcAsAd2hKc QcAsAdKhQcAsAdQhQcAsAdJhQcAsAdThQcAsAd9hQcAsAd8hQcAsAd7hQcAsAd6hQcAsAd5hQcAsAd4hQcAsAd3hQcAsAd2hQc JcAsAdKhJcAsAdQhJcAsAdJhJcAsAdThJcAsAd9hJcAsAd8hJcAsAd7hJcAsAd6hJcAsAd5hJcAsAd4hJcAsAd3hJcAsAd2hJc TcAsAdKhTcAsAdQhTcAsAdJhTcAsAdThTcAsAd9hTcAsAd8hTcAsAd7hTcAsAd6hTcAsAd5hTcAsAd4hTcAsAd3hTcAsAd2hTc 9cAsAdKh9cAsAdQh9cAsAdJh9cAsAdTh9cAsAd9h9cAsAd8h9cAsAd7h9cAsAd6h9cAsAd5h9cAsAd4h9cAsAd3h9cAsAd2h9c 8cAsAdKh8cAsAdQh8cAsAdJh8cAsAdTh8cAsAd9h8cAsAd8h8cAsAd7h8cAsAd6h8cAsAd5h8cAsAd4h8cAsAd3h8cAsAd2h8c 7cAsAdKh7cAsAdQh7cAsAdJh7cAsAdTh7cAsAd9h7cAsAd8h7cAsAd7h7cAsAd6h7cAsAd5h7cAsAd4h7cAsAd3h7cAsAd2h7c 6cAsAdKh6cAsAdQh6cAsAdJh6cAsAdTh6cAsAd9h6cAsAd8h6cAsAd7h6cAsAd6h6cAsAd5h6cAsAd4h6cAsAd3h6cAsAd2h6c 5cAsAdKh5cAsAdQh5cAsAdJh5cAsAdTh5cAsAd9h5cAsAd8h5cAsAd7h5cAsAd6h5cAsAd5h5cAsAd4h5cAsAd3h5cAsAd2h5c 4cAsAdKh4cAsAdQh4cAsAdJh4cAsAdTh4cAsAd9h4cAsAd8h4cAsAd7h4cAsAd6h4cAsAd5h4cAsAd4h4cAsAd3h4cAsAd2h4c 3cAsAdKh3cAsAdQh3cAsAdJh3cAsAdTh3cAsAd9h3cAsAd8h3cAsAd7h3cAsAd6h3cAsAd5h3cAsAd4h3cAsAd3h3cAsAd2h3c 2cAsAdKh2cAsAdQh2cAsAdJh2cAsAdTh2cAsAd9h2cAsAd8h2cAsAd7h2cAsAd6h2cAsAd5h2cAsAd4h2cAsAd3h2cAsAd2h2c

10. Offsuit AA** Offsuit AAXX There are 6 ways of being dealt AA, with each having 144 off-suited kicker combinations : ∴ 6 * 144 = 864 os AaAbXcXd hands So far we have calculated the offsuit AAAA, AAAX and AAXX combinations: os AAAA1 hand os AAAX48 hands os AAXX864 hands 913 AA** hands are offsuit 913 of 6961 total AAXX hands; approx. 13% are offsuit.

11. How many AA** are suited ? 6768 AAXX hands 192 AAAX hands 1 AAAA hand 6961total AAXX hands We have already counted 913 offsuit hands containing AA** : 6961 – 913 = 6048 suited AA** hands Trips Suited Pairs Non-nut single-suited pocket aces are of the form AaAbXcXci.e. AsAhJcTc Three-suited pocket aces are of the form AaAbXaXai.e. AsAhJsTs Double-suited pocket aces are of the form AaAbXaXbi.e. AsAhJsTh Nut single-suited pocket aces are of the form AaAbXaXci.e. AsAhJsTc

12. Suited AAAX Trips AAAX can be suited to any one of the three aces; count the twelve possible offsuit kickers for each of the four sets of AAA : i.e. AsAhAdXc can have any one of twelve kickers : 2c, 3c, 4c, 5c, 6c, 7c, 8c, 9c, Tc, Jc, Qc, Kc. 12 * 4 = 48 os AAAX hands ∴ 192 - 48 = 144 ss AAAX hands Alternatively, we can count thirty-six suited kickers for each set of the four sets trips: ∴ 36 * 4 = 144 ss AAAX hands

13. Non-nut Suited AAXX - AaAbXcXc Non-nut suited AAXX Remember from choosing kickers that are offsuit to both AA gave us 24 possible kickers: i.e. for AsAdXhXc : KsQsJsTs9s8s7s6s5s4s3s2sKdQdJdTd9d8d7d6d5d4d3d2d KhQhJhTh9h8h7h6h5h4h3h2hKcQcJcTc9c8c7c6c5c4c3c2c C(24,2) = 276 kicker combos, including the nn ss AAXX (e.g. AsAd7c2c) 12 * 12 = 144 os kicker combos 276 - 144 = 132 nn ss kicker combos per AA [cf. “2 *C(12,2)”] ∴ 6 * 132 = 792 nn ss AaAbXcXc hands Alternatively we can take both groups of twelve kickers, and choose two : 2 * C(12,2) = 2 * 66 = 132 nn ss kicker combos per AA

14. Three-of-one-suit AAXX - AaAbXaXa Three-of-one-suit AAXX For the AaAbXaXa we are heavily restricted in combinations; we must choose two from twelve kickers for each of the four Aa, then multiply by three for each of the offsuit Ab remaining : i.e. for AsAbXsXs : Ab = Ad, Ac and Ah KsQsJsTs9s8s7s6s5s4s3s2sKdQdJdTd9d8d7d6d5d4d3d2d KhQhJhTh9h8h7h6h5h4h3h2hKcQcJcTc9c8c7c6c5c4c3c2c 3 * C(12,2) = 198 combos of AaAbXaXc per A ∴ 4 * 198 = 792 combos of AaAbXaXc hands

15. Double-Suited AAXX - AaAbXaXb Double-Suited AAXX For each Aa and Ab there are C(12,1) Xa and Xb kickers of the same suit : 12 * 12 = 144 kicker combos i.e. for AsAdXsXd : KsQsJsTs9s8s7s6s5s4s3s2sKdQdJdTd9d8d7d6d5d4d3d2d KhQhJhTh9h8h7h6h5h4h3h2hKcQcJcTc9c8c7c6c5c4c3c2c ∴ 6 * 144 = 864 ds AaAbXaXb hands [Note that this is exactly the same number as for the offsuit AaAbXcXd hands]

16. Nut Single-Suited AAXX - AaAbXaXc Nut Single-Suited AAXX 144 Trips 792 Non-nut single-suited pocket aces are of the form AaAbXcXci.e. AsAhJcTc 864 Double-suited pocket aces are of the form AaAbXaXbi.e. AsAhJsTh 792 Three-suited pocket aces are of the form AaAbXaXai.e. AsAhJsTs 2592 of 6048 leaves 6048 – 2592 =3456 3456 Nut single-suited pocket aces are of the form AaAbXaXci.e. AsAhJsTc 4248 - 792 = 3456 combos of nut ss AaAbXaXc hands Alternatively, we can count this as : Aa * Xa * Ab * Ac = C(4,1)* C(12,1)* C(3,1)* C(24,1)= 4* 12* 3* 24= 3456 i.e. AsAhAcAd one of four A, let’s consider As KsQsJsTs9s8s7s6s5s4s3s2swith any one of 12 suited kickers AhAcAdthere are 3 unchosen A, let’s choose Ad KhQhJhTh9h8h7h6h5h4h3h2hKcQcJcTc9c8c7c6c5c4c3c2cwhich leaves 24 offsuit kickers.

17. AA** Combinations 6961total AAXX hands Suited/Offsuit AAXX combination breakdown : 1 offsuit AAAA hand 48 offsuit AAAX hands 144 suited AAAX hands 864 offsuit AaAbXcXd hands 792 non-nut suitedAaAbXcXc hands 792 nut ss (3-of-one-suit)AaAbXaXa hands 3456 nut ssAaAbXaXc hands 864 ds AaAbXaXb hands 6961 total hands

18. Shortcuts – KK** example We can count combinations by taking some shortcuts; with time, familiarity with these calculations will allow you to actually compare combinations of different hand-types at the table. KKXX offsuit To count the total combinations of a particular hand-type quickly we must break the hand into distinguishable groups. i.e. in offsuit KKXX : 6 pairs of KK, each using up two suits Then there are eleven kickers from each of the remaining two suits to choose from : 6 * 11 * 11 = 726 offsuit KKXX hands In this case the kickers are distinguishable. cf. this next example where the kickers don’t get to choose what suit they are, (KX)KX single-suited Choose one of four K, and assign it one of the 11 available kickers of the same suit Assign one of the three remaining K of a different suit, and any one of the twenty-two kickers from either of the two remaining suits : 4* 11 * 3 * 22 = 2904 ss (KX)KX hands

19. KK** 6925total KKXX hands (non-AA) Suited/Offsuit AAXX combination breakdown : = 1 offsuit KKKK hand 4 *12 = 48 offsuit KKKX hands 12 *12 = 144 suited KKKX hands (193) 4 *3 *11 = 132 offsuit AaKbKcXd hands 4 *3 *22 = 264 nn ss (KX)AaKbKcXb hands 4 *11 *3 = 132 nut ss (3-of-one-suit)AaKbKaXa hands 4 *3 *22 = 264nut ss (AK) AaKbKaXc hands 4 *11 *3 = 132nut ss (AX)AaKbKcXa hands 4 *11 *3 = 132 nut ds AaKbKaXb hands (1056) 6 *11^2 = 726 offsuit KaKbXcXd hands 6 *2 *C(11,2) = 660 nn low ss KaKbXcXc hands 4 *C(11,2) *3 = 660nn ss (3-of-one-suit)KaKbXaXa hands 4 *11 *3 *22 = 2904 nn K hi ssKaKbXaXc hands 6 *11^2 = 726 nn ds KaKbXaXb hands (5676) 6925 total hands

20. KK**

21. AA** vs. KK**

22. PLO Pairs+ Starting Hands None of this leads to many useful conclusions until we apply it to real poker situations. We can see that each pair contributes to approx. 2.5% of total starting hands; ~32% of all starting hands are paired. What about the rest? Next episode Rundowns Gappers “Junk” Note : in the table on the right; AA** includes AA22, but 22** does not include 22AA. There are 6961 starting hands that contain at least a pair of 2s, just as there are 6961 starting hands that contain at least a pair of As. handcombos%total AA**69612.6% KK**69252.6% QQ**68892.5% JJ**68532.5% TT**68172.5% 99**67812.5% 88**67452.5% 77**67092.5% 66**66732.5% 55**66372.5% 44**66012.4% 33**65652.4% 22**65292.4% 32.4%