jjd323’sMathematics of PLO Ep. 2

Complex Starting Hand Combinatorics – Suitedness of AAXX

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 AA(KX)X - single-suited to the KAK(XX) - single-suited to the low cards(AX)(KX) or (AK)(XX)- double-suited

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. AsAhJdTcNon-nut single-suited pocket aces are of the form AaAbXcXc i.e.

AsAhJcTcNut single-suited pocket aces are of the form AaAbXaXc i.e. AsAhJsTcDouble-suited pocket aces are of the form AaAbXaXb i.e. AsAhJsTh

Recap : How many AA** ?

Counting AAXX: C(4,2) = 6 ways to get AA, with 48 non-A cards remainingC(48,2) = 1128 ways to get XX with each pair of AABeing 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

AA** vs. KK**

nut ss AAxx, 51%

nut ss AAxx (3-of-one-suit),

11%

nn ss AAxx, 11%

os AAxx, 12%

ss AAAx, 2%

AAAA & os AAAx, 1%

ds AAxx, 12%nn ds KKxx,

11%

ss KKxx, 56%

low ss KKxx, 10%

os KKxx, 15%

trips/worse, 3%

nut ss/ds AKKx, 5%

AA** Combinations

• How many AA** are offsuit ?– AAAA– AAAX– AAXX

• How many suited AAXX Combinations?– Single-suited– Double-suited– Nut vs. non-nut

How many AA** are offsuit ?

• 6768 AAXX hands• 192 AAAX hands• 1 AAAA hand

6961 total AAXX hands

QuadsAAAA cannot be suited∴ 1 os AAAA combo

TripsAAAX 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]

Offsuit AAXX - AaAbXcXd

PairsOffsuit 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 :

KsQsJsTs9s8s7s6s5s4s3s2sKdQdJdTd9d8d7d6d5d4d3d2dKhQhJhTh9h8h7h6h5h4h3h2hKcQcJcTc9c8c7c6c5c4c3c2c

Incorrect Counting MethodC(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 MethodWe 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

C(12,1) kickers of type Xc * C(12,1) kickers of type Xd =12 * 12 = 144 os kicker

combos

e.g. AsAdXhXc

Kh Qh Jh Th 9h 8h 7h 6h 5h 4h 3h 2h

Kc AsAdKhKc AsAdQhKc AsAdJhKc AsAdThKc AsAd9hKc AsAd8hKc AsAd7hKc AsAd6hKc AsAd5hKc AsAd4hKc AsAd3hKc AsAd2hKc

Qc AsAdKhQc AsAdQhQc AsAdJhQc AsAdThQc AsAd9hQc AsAd8hQc AsAd7hQc AsAd6hQc AsAd5hQc AsAd4hQc AsAd3hQc AsAd2hQc

Jc AsAdKhJc AsAdQhJc AsAdJhJc AsAdThJc AsAd9hJc AsAd8hJc AsAd7hJc AsAd6hJc AsAd5hJc AsAd4hJc AsAd3hJc AsAd2hJc

Tc AsAdKhTc AsAdQhTc AsAdJhTc AsAdThTc AsAd9hTc AsAd8hTc AsAd7hTc AsAd6hTc AsAd5hTc AsAd4hTc AsAd3hTc AsAd2hTc

9c AsAdKh9c AsAdQh9c AsAdJh9c AsAdTh9c AsAd9h9c AsAd8h9c AsAd7h9c AsAd6h9c AsAd5h9c AsAd4h9c AsAd3h9c AsAd2h9c

8c AsAdKh8c AsAdQh8c AsAdJh8c AsAdTh8c AsAd9h8c AsAd8h8c AsAd7h8c AsAd6h8c AsAd5h8c AsAd4h8c AsAd3h8c AsAd2h8c

7c AsAdKh7c AsAdQh7c AsAdJh7c AsAdTh7c AsAd9h7c AsAd8h7c AsAd7h7c AsAd6h7c AsAd5h7c AsAd4h7c AsAd3h7c AsAd2h7c

6c AsAdKh6c AsAdQh6c AsAdJh6c AsAdTh6c AsAd9h6c AsAd8h6c AsAd7h6c AsAd6h6c AsAd5h6c AsAd4h6c AsAd3h6c AsAd2h6c

5c AsAdKh5c AsAdQh5c AsAdJh5c AsAdTh5c AsAd9h5c AsAd8h5c AsAd7h5c AsAd6h5c AsAd5h5c AsAd4h5c AsAd3h5c AsAd2h5c

4c AsAdKh4c AsAdQh4c AsAdJh4c AsAdTh4c AsAd9h4c AsAd8h4c AsAd7h4c AsAd6h4c AsAd5h4c AsAd4h4c AsAd3h4c AsAd2h4c

3c AsAdKh3c AsAdQh3c AsAdJh3c AsAdTh3c AsAd9h3c AsAd8h3c AsAd7h3c AsAd6h3c AsAd5h3c AsAd4h3c AsAd3h3c AsAd2h3c

2c AsAdKh2c AsAdQh2c AsAdJh2c AsAdTh2c AsAd9h2c AsAd8h2c AsAd7h2c AsAd6h2c AsAd5h2c AsAd4h2c AsAd3h2c AsAd2h2c

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 AAAA 1 handos AAAX 48 handsos AAXX 864 hands

913 AA** hands are offsuit

913 of 6961 total AAXX hands; approx. 13% are offsuit.

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

TripsSuited 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. AsAhJsTsDouble-suited pocket aces are of the form AaAbXaXb i.e. AsAhJsThNut single-suited pocket aces are of the form AaAbXaXc i.e. AsAhJsTc

Suited AAAX

TripsAAAX 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 handsAlternatively, we can count thirty-six suited kickers for each set of the four sets trips:

∴ 36 * 4 = 144 ss AAAX hands

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 :

KsQsJsTs9s8s7s6s5s4s3s2sKdQdJdTd9d8d7d6d5d4d3d2dKhQhJhTh9h8h7h6h5h4h3h2hKcQcJcTc9c8c7c6c5c4c3c2c

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

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

KsQsJsTs9s8s7s6s5s4s3s2sKdQdJdTd9d8d7d6d5d4d3d2dKhQhJhTh9h8h7h6h5h4h3h2hKcQcJcTc9c8c7c6c5c4c3c2c

3 * C(12,2) = 198 combos of AaAbXaXc per A ∴ 4 * 198 = 792 combos of AaAbXaXc hands

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 :KsQsJsTs9s8s7s6s5s4s3s2sKdQdJdTd9d8d7d6d5d4d3d2dKhQhJhTh9h8h7h6h5h4h3h2hKcQcJcTc9c8c7c6c5c4c3c2c

∴ 6 * 144 = 864 ds AaAbXaXb hands[Note that this is exactly the same number as for the offsuit AaAbXcXd hands]

Nut Single-Suited AAXX - AaAbXaXc

Nut Single-Suited AAXX

144 Trips792 Non-nut single-suited pocket aces are of the form AaAbXcXc i.e. AsAhJcTc864 Double-suited pocket aces are of the form AaAbXaXb i.e. AsAhJsTh792 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 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 AsKsQsJsTs9s8s7s6s5s4s3s2s with any one of 12 suited kickers AhAcAd there are 3 unchosen A, let’s choose AdKhQhJhTh9h8h7h6h5h4h3h2hKcQcJcTc9c8c7c6c5c4c3c2c which leaves 24 offsuit kickers.

AA** Combinations

6961 total AAXX hands

Suited/Offsuit AAXX combination breakdown :1 offsuit AAAA hand48 offsuit AAAX hands144 suited AAAX hands864 offsuit AaAbXcXd hands792 non-nut suited AaAbXcXc hands792 nut ss (3-of-one-suit) AaAbXaXa hands3456 nut ss AaAbXaXc hands864 ds AaAbXaXb hands

6961 total hands

nut ss AAxx, 51%

nut ss AAxx (3-of-one-suit),

11%

nn ss AAxx, 11%

os AAxx, 12%

ss AAAx, 2%

AAAA & os AAAx, 1%

ds AAxx, 12%

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 offsuitTo 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-suitedChoose one of four K, and assign it one of the 11 available kickers of the same suitAssign 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

KK**

6925 total KKXX hands (non-AA)

Suited/Offsuit AAXX combination breakdown := 1 offsuit KKKK hand

4 *12 = 48 offsuit KKKX hands12 *12 = 144 suited KKKX hands

(193)

4 *3 *11 = 132 offsuit AaKbKcXd hands4 *3 *22 = 264 nn ss (KX) AaKbKcXb hands4 *11 *3 = 132 nut ss (3-of-one-suit) AaKbKaXa hands4 *3 *22 = 264 nut ss (AK) AaKbKaXc hands4 *11 *3 = 132 nut ss (AX) AaKbKcXa hands4 *11 *3 = 132 nut ds AaKbKaXb hands

(1056)

6 *11^2 = 726 offsuit KaKbXcXd hands6 *2 *C(11,2) = 660 nn low ss KaKbXcXc hands4 *C(11,2) *3 = 660 nn ss (3-of-one-suit) KaKbXaXa hands4 *11 *3 *22 = 2904 nn K hi ss KaKbXaXc hands6 *11^2 = 726 nn ds KaKbXaXb hands

(5676)

6925 total hands

KK**

nut ss (Ax)KK, 2%

nn ss KK(xx), 10%

os KKxx, 10%

os AKKx, 2%

ss KKKx, 2%

os KKKx & KKKK, 1%

nn ss K(Kxx), 10%

nut ss (AKx)K, 2%

nut ss (AK)Kx, 4%

nut ds (AK)(Kx), 2%

nn ds (Kx)(Kx), 10%

nn ss K(Kx)x, 41%

nn ss AK(Kx), 4%

AA** vs. KK**

nut ss AAxx, 51%

nut ss Aaxx (3-of-one-suit),

11%

nn ss AAxx, 11%

os AAxx, 12%

ss AAAx, 2%

AAAA & os AAAx, 1%

ds AAxx, 12%nn ds KKxx,

11%

ss KKxx, 56%

low ss KKxx, 10%

os KKxx, 15%

trips/worse, 3%

nut ss/ds AKKx, 5%

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.

hand combos %total

AA** 6961 2.6%

KK** 6925 2.6%

QQ** 6889 2.5%

JJ** 6853 2.5%

TT** 6817 2.5%

99** 6781 2.5%

88** 6745 2.5%

77** 6709 2.5%

66** 6673 2.5%

55** 6637 2.5%

44** 6601 2.4%

33** 6565 2.4%

22** 6529 2.4%

32.4%