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jjd323’s Mathematics of PLO Ep. 2

jjd323’s Mathematics of PLO Ep. 2. Complex Starting Hand Combinatorics – Suitedness of AAXX. Abbreviations & Notation. os - offsuit (e.g. As Ad 7c 2h ) ss - single-suited (e.g. As Kd7d 2h ) ds - double-suited (e.g. As Jd Js Td ) nn - non-nut (e.g. As Ad 7c2c not As Ac 7d 2c )

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jjd323’s Mathematics of PLO Ep. 2

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  1. jjd323’sMathematics 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 AaAbXcXd i.e. AsAhJdTc Non-nut single-suited pocket aces are of the form AaAbXcXc i.e. AsAhJcTc Nut single-suited pocket aces are of the form AaAbXaXc i.e. AsAhJsTc Double-suited pocket aces are of the form AaAbXaXb i.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 6961 total 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 6961 total 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

  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 AaAbXcXdhands So far we have calculated the offsuit AAAA, AAAX and AAXX combinations: os AAAA 1 hand os AAAX 48 hands os AAXX 864 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 6961 total 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 AaAbXcXc i.e. AsAhJcTc Three-suited pocket aces are of the form AaAbXaXa i.e. AsAhJsTs Double-suited pocket aces are of the form AaAbXaXb i.e. AsAhJsTh Nut single-suited pocket aces are of the form AaAbXaXc i.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 AaAbXcXchands 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 AaAbXaXchands

  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 AaAbXaXbhands [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 AaAbXcXc i.e. AsAhJcTc 864 Double-suited pocket aces are of the form AaAbXaXb i.e. AsAhJsTh 792 Three-suited pocket aces are of the form AaAbXaXa i.e. AsAhJsTs 2592 of 6048 leaves 6048 – 2592 = 3456 3456 Nut single-suited pocket aces are of the form AaAbXaXc i.e. AsAhJsTc 4248 - 792 = 3456 combos of nut ss AaAbXaXchands 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 KsQsJsTs9s8s7s6s5s4s3s2s with any one of 12 suited kickers AhAcAd there are 3 unchosen A, let’s choose Ad KhQhJhTh9h8h7h6h5h4h3h2hKcQcJcTc9c8c7c6c5c4c3c2c which leaves 24 offsuit kickers.

  17. AA** Combinations 6961 total 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 suited AaAbXcXc hands 792 nut ss (3-of-one-suit) AaAbXaXa hands 3456 nut ss AaAbXaXc 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** 6925 total 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 = 264 nut ss (AK) AaKbKaXc hands 4 *11 *3 = 132 nut 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 = 660 nn ss (3-of-one-suit) KaKbXaXa hands 4 *11 *3 *22 = 2904 nn K hi ss KaKbXaXc 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.

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