The basic rules and combinatorics of poker are presented in Part 1.
So is the jargon.

Here, we continue our mathematical introduction to poker by
focusing on the rules and strategies for the most popular current poker variant:
No-Limit Texas Hold 'em (NLHE).
The second-most-popular modern form of poker is
Pot-Limit Omaha (PLO).

(2013-09-23) No-Limit Texas Hold 'em

The Texas State Legislature has recognized
Robstown
as the birthplace, sometime between 1900 and 1920 or so,
of what has become the most popular variant of poker.

Crandell Addington
(1938-) spotted the game in 1959 (it was just called hold 'em
at the time) and heralded its great
potential as a game of strategy, as it features 4 different betting opportunities
(draw poker features only 2).
Addington was instrumental in bringing the game to Las Vegas with a
group that included
Doyle Brunson (1933-) and
Amarillo Slim
(Thomas Austin Preston, Jr. 1928-2012).

For several years, only one casino in Las Vegas would offer
Texas Hold 'hem : The
Golden Nugget
of downtown Las Vegas (one of the oldest gambling hall in Las Vegas,
it hadn't yet been expanded into today's giant hotel-casino).

Rules :

Each player is dealt 2 down cards (hole cards
or pocket) for their exclusive use.
5 community cards are dealt.
In a showdown, the hand assigned to a player consists
of the best poker hand which can be formed with 5
of those 7 cards.

(2013-12-18) The Most Technical Rule in Poker
You're not allowed to check-behind the nuts on the river.

Checking behind
means checking (i.e., declining to start betting)
when every other active player has already done so.
When this happens before the river betting round,
the game proceeds with the next stage.
On the river, this terminates the game and a showdown takes place.

There's no obvious reason why a player who
"has the nuts"
(i.e., a hand that cannot be beaten by anything)
would want to check-behind on the river.
On the contrary, it may well be strong evidence of illegal softplay and
collusion between players in a tournament and it has been made against the
rules. The penalty is usually to sit out just one hand or,
possibly, one full orbit (until the dealer button
comes back to its current position).
The rule doesn't serve any purpose in a cash game and is
usually ignored beyond a mere reminder/warning.

During the 2010 WSOP "main event",
Darvin Moon
wasn't aware of the rule and committed the infraction
(he received the lightest possible penalty of sitting out a single hand).
He explained his play, before being pointed out that it was illegal,
by stating that he wanted to see what his opponent had and was
convinced that the opponent would have folded in response to any bet.
If you ask me, Moon's motivation was perfectly legitimate as
it could have helped him pinpoint the playing style of his opponent
for future hands. That's one good argument in favor of
repealing this overzealous rule...

Technically, this rule also implies careful scrutiny of the "odd chip" rule
for split pots, to avoid putting a fraction of
a chip at risk when not checking-behind the nuts on the
river (in case the nuts can be tied).

(2013-09-30) Rating the 169 possible pockets in Texas hold'em Pre-flop probability of winning a showdown in heads-up poker.

There's much more to a game of poker than its starting point.
A game may never go to a showdown but it's still very interesting to know
what your chances are to prevail if it ever does.

Before the flop in a game of Texas hold'em, the conditional
probabilities of winning, losing or splitting the pot
if the player takes part in a final showdown are
just functions of his 2-card starting hand.
Because all suits are on an equal footing in poker, there are only
169 types of starting hands and those functions can be expressed at a
glance in the fairly compact 13 by 13 table presented below
(valid for a heads-up match only).

Each entry is obtained
by tallying the outcomes of all possible ways of dealing 5 community cards and
2 hole cards for the opponent. The probabilities are equal to
the resulting integers divided into the total number of ways to compose,
with the remaining 50 cards, a 5-card community board
and a 2-card hand (for the opponent) namely:

C(50,5,2) = C(50,5) C(45,2) = 2097572400

For example, 1781508418 = 2 . 11119 . 80111 is the
numerator which gives the exact probability of winning outright by going all-in
preflop with pocket aces in an heads-up match, which is nearly 85%.
More precisely:

Let's illustrate the type of computations involved by enumerating the number
of ways (11402312) which lead to a tie when a player has pocket aces...
First are the cases where both players get the same flush or straight flush:

A straight flush on the board, between 6-high and Q-high, in either of the player's suits.
There are C(2,1) C(10-3,1) C(45,2) = 13860 possibilities.

A flush on the board using neither of the player's own suits when the
opponent doesn't hold a card of that suit which improves the hand
(either above the highest community card or plugging the hole to make a straight).

Other ties occur in the following mutually exclusive cases,
provided that any four community cards don't share the same suit as one hole card:

The opponent also has two aces.

The opponent has one ace and the board includes KQJT or 5432.

The opponent has no aces and the board includes AKQJT or 5432A.

The board forms a straight other than AKQJT or 5432A and the
opponent doesn't hold the card just above it.

Delicate as it may be, the above discussion is one of
the simplest of the 338 that would be needed to fill the following table
without a computer.

The convention adopted here (popularized by David Sklansky)
is to have entries above the main diagonal correspond to cards in the same suit
(suited pocket) and entries below the diagonal are for
off-suit pockets (consisting of two cards whose
suits are different). The diagonal is for cards of equal values
(which are necessarily off-suit).

In a nutshell, AK denotes Ace-King suited and KA denotes Ace-King off-suit.
The first number listed is the probability
of a win (expressed as a percentage) and the number below it
is the probability of a tie (split pot).

Heads-up Preflop Probabilities (percentages rounded to 3 decimal places)

win tie

A

K

Q

J

10

9

8

7

6

5

4

3

2

A

84.932 0.544

66.220 1.650

65.314 1.790

64.398 1.990

63.489 2.227

61.510 2.543

60.508 2.872

59.387 3.195

58.179 3.454

58.064 3.718

57.138 3.792

56.335 3.771

55.506 3.745

S u i t e d

h a n d s

a b o v e

d i a g o n a l

K

64.469 1.701

82.117 0.557

62.408 1.984

61.477 2.182

60.587 2.403

58.638 2.701

56.790 3.044

55.846 3.383

54.805 3.672

53.834 3.918

52.889 3.992

52.070 3.970

51.240 3.944

Q

63.509 1.846

60.432 2.047

79.632 0.586

59.071 2.377

58.171 2.594

56.223 2.883

54.417 3.201

52.523 3.558

51.679 3.867

50.713 4.112

49.763 4.185

48.938 4.162

48.102 4.134

J

62.535 2.056

59.441 2.255

56.906 2.457

77.153 0.633

56.155 2.746

54.112 3.101

52.312 3.408

50.454 3.741

48.574 4.063

47.821 4.332

46.869 4.404

46.042 4.380

45.202 4.351

T

61.568 2.307

58.494 2.489

55.947 2.687

53.826 2.843

74.660 0.703

52.377 3.301

50.509 3.650

48.651 3.976

46.800 4.281

44.939 4.554

44.204 4.653

43.378 4.628

42.540 4.599

9

59.450 2.646

56.408 2.809

53.862 2.997

51.639 3.224

49.816 3.432

71.666 0.783

48.856 3.889

46.990 4.255

45.151 4.554

43.313 4.818

41.407 4.910

40.807 4.915

39.973 4.884

8

58.374 2.997

54.432 3.177

51.931 3.339

49.714 3.553

47.818 3.806

46.068 4.057

68.717 0.891

45.684 4.504

43.819 4.850

41.990 5.109

40.103 5.198

38.283 5.182

37.679 5.185

7

57.170 3.343

53.417 3.540

49.904 3.723

47.726 3.912

45.830 4.157

44.072 4.451

42.693 4.715

65.725 1.021

42.829 5.085

40.979 5.393

39.109 5.481

37.304 5.465

35.440 5.432

6

55.870 3.624

52.297 3.852

48.997 4.055

45.714 4.261

43.848 4.488

42.103 4.776

40.697 5.089

39.654 5.338

62.700 1.169

40.348 5.570

38.481 5.705

36.685 5.696

34.838 5.663

5

55.742 3.908

51.254 4.120

47.959 4.322

44.905 4.552

41.857 4.787

40.137 5.065

38.741 5.373

37.675 5.675

37.013 5.863

59.640 1.370

38.533 5.842

36.759 5.869

34.930 5.840

4

54.733 3.994

50.225 4.205

46.925 4.406

43.869 4.635

41.055 4.897

38.086 5.171

36.709 5.475

35.662 5.776

35.003 6.015

35.075 6.161

56.257 1.533

35.727 5.829

33.918 5.822

3

53.855 3.979

49.331 4.189

46.025 4.389

42.967 4.618

40.155 4.879

37.428 5.183

34.750 5.468

33.718 5.769

33.070 6.016

33.165 6.200

32.066 6.159

52.839 1.708

33.092 5.785

2

52.947 3.963

48.423 4.172

45.110 4.371

42.049 4.598

39.239 4.858

36.517 5.162

34.087 5.482

31.710 5.747

31.079 5.993

31.194 6.182

30.117 6.165

29.239 6.128

49.385 1.898

Lower triangle
(below yellow diagonal) is for off-suit hole cards.

Highlighted in blue is what may be a puzzling anomaly at first:
A 5 is very slightly better
than a 6 when paired with a 2, a 3 or a 4 (either suited or off-suit).
The explanation is that the greater tiebreaking power of a 6 compared to a 5
is more than compensated by the greater numbers of ways a lower card can combine with a 5
instead of a 6 to form a straight (or a straight flush) because a 6 is more distant from those
low cards than a 5 is.

In the parlance implied by the above table, the Queen-Seven off-suit hand is denoted "7Q".
This particular pair of hole cards is known either as the "computer hand"
or the "average hand" because, as shown in the table,
it entails a probability of winning a showdown of nearly 50% (49.9%)
in the case of a heads-up game (poker jargon for a two-player game).

The table shows that the worst hand is 23 (3-2 offsuit)
in a heads-up match. This remains true with 3 or 4 players.
The reputation of 27 (7-2 offsuit) for being "the worst hand in poker" is
based on what the situation becomes when there are 5 or more players at the table.
It's the basis for the (optional) seven-deuce rule
which rewards players for winning with 7-2 (suited or not).

Suited hands have total probability C(13,2) C(4,1) / C(52,2) = 4/17
The probability of having a pair is C(13,1) C(4,2) / C(52,2) = 1/17
Other off-suit hands have probability C(13,2) C(4,2) 2! / C(52,2) = 12/17

4 / 17 +
1 / 17 +
12 / 17 = 1

From the above, we can derive the probabilities of each specific type
of suited hand (there are 78 of them) pair (there are 13)
or off-suit hands. Either that or we can figure directly that there are
C(52,2) = 1326 different hands, where:

Every suited hand is in an equivalence class of 4 (one in each suit).

Every pair is in a class of C(4,2) = 6 (choose 2 suits among 4).

Every off-suit hand is in a class of 12 (different top & bottom suits).

Simplify by 2 to obtain the respective probabilities: 2/663, 3/663, 6/663.

The Big Checksum :

Mathematics is a harsh mistress, but a loving one.
At the end of the day, she often provides a way to confirm your
results or detect the mistakes of whoever ignores her powers...
For example, 11 years ago
I could measure how trustworthy
several published tables of coefficients of elasticity really were,
as they failed to verify a simple mathematical relation between them
( 9/E = 3/G + 1/K ) at their claimed level of accuracy.

Instead of producing a precise version of the above poker table from scratch,
I decided to just validate the hard results of others
(discarding "simulations"
obtained by playing out a few million random hands,
which are only suitable for low-precision estimates).

A novelty calculator
(selling for less than $8, with a free deck of Queen "2000" cards)
gives as "winning odds" [sic] the expected value of every hand
(for 2 to 10 players). That's a single number equal to the probability
of a win plus half the probability of a tie.
The accuracy of the device is not documented
(who worked out the 1521 memorized results?)
but it gave correct results to the built-in precision
of 0.05% in all the cases I checked (limited to two players).

A few brave souls have published actual enumerations
(integers) obtained by splitting into wins, ties or losses the
C(50,5,2) = 2097572400 possible outcomes (per hand) of an heads-up showdown

There's a simple checksum which can be used to validate such sets of results,
since each time you win the opponent loses (obviously).
Consider now the grand totals W, T and L
obtained by summing up respectively the elementary tallies for wins, ties and losses
multiplied into the weights 2, 3 or 6 according to the nature of the hand
(suited, pair or off-suit). The sum W+T+L must equal 663 C(50,5,2).
By the above remark, L = W. So:

S = 1390690501200 - ( 2W + T) = 0

At the end of the day, you'll find a positive checksum S if you forgot to tally
something and a negative checksum if you counted some outcomes several times in different categories.
A zero checksum is no guarantee of perfection
(both types of errors could cancel each other out) but it's a strong indication thereof,
given the large magnitude of the integers involved...

As a result of this effort, I have
a spreadsheet giving the nontrivial result
T = 56565131908 (bottom-right)
which yields the exact probability of a tie in an heads-up match
with a forced showdown (as would happen if the stack of one player was equal
to the blind he has to post) namely:

(2013-11-26) How frequent are pocket aces when I have pocket kings?
The probability is a quadratic function of the number of players (m).

The well-known probability of having a given pair in a two-card hole
(two aces, two kings, two queens, etc.) is simply:

C(4,2) / C(52,2) = 1 / 221

The probability for an unspecified pair is 13 times as big, namely 1 / 17.

If you hold kings (or any two cards without an ace in them) the probability that the player
to your left (say) has aces is:

C(4,2) / C(50,2) = 6 / 1225 (that's 8.2% more than 1/221)

That fully covers heads-up situations (m=2, single opponent).
With three or more players, two of the m-1 opponents
could have aces at once and the probability that at least one
of them does is given by inclusion-exclusion:

(2014-01-08) The board is paired (single community pair).
How often does someone make trips with that pair? Quads?

In the following calculation, m is the number of initial players.
This isn't directly relevant to an actual gaming situation,
where some of those may well have folded before showdown...

The probability of quads is:

C(m,1) C(2,2) / C(47,2) = m / 1081

As two players could make trips together,
the probability that at least one does is a quadratic function of m
obtained from inclusion-exclusion:

m

[

_{ }92 - 2m

]

1081

That expression is tabulated below as a percentage:

m = 2

3

4

5

6

7

8

9

10

11

16.281

23.867

31.082

37.928

44.403

50.509

56.244

61.610

66.605

71.230

Note that with m = 23 initial players (the maximum possible)
the above formulas say that one player will have quads with probability 1/47
and at least one of them will have trips with probability 46/47.
Those two probabilities add up to 100% because 51 of the 52 cards have been
dealt, which makes it impossible to avoid both possibilities at once
(in all this, we do include the mucked hands, as advertised).

(2013-12-17) You have pocket clubs. Three clubs are on the board.
How often will someone else have a flush (or straight flush) ?

In the following calculation, m is the number of initial players.
This isn't directly relevant to an actual gaming situation,
where some of those may well have folded before showdown...

Eight clubs are unaccounted for.
In an m-handed game, up to four of the m-1 opponents
could have two clubs which would give them a flush.
The probability that at least one opponent does is thus given by an
inclusion-exclusion enumeration with four terms
involving multichoice numbers, namely:

Probability (%) that several of m initial players have exactly 5 clubs if one does :

m = 2

3

4

5

6

7

8

9

10

11

2.828

5.610

8.344

11.033

13.675

16.272

18.823

21.330

23.792

26.210

Again, the hands of some of those m initial players may be in the muck.

(2013-10-23) Nontransitivity of pairwise showdown matches:
If A beats B and B beats C in a showdown, then C may beat A...

In televised games of hold 'em poker,
the hole cards revealed by "pocket cams" are often shown
with computer graphics which display the player's name and
the corresponding percentage of winning.

That percentage is computed using all the information available
to the show's producers. They know not only the board card revealed so
far but the hole cards of every player.
The displayed percentage is obtained by trying out every possibility for
the community cards yet to come (there are at most C(48,5) or 1712304 such
possibilities, which is mincemeat for modern computers).

The only rational way to compare the strength of two hands is to compare
the values of their respective entries in the above table.
However, one can be tempted to match them up against each other
in a fictitious heads-up game using the
numbers normally displayed by video producers
(the proper software is available for free
online).

Such pairwise comparisons have no reason to be transitive
and indeed they're not.
(Transitivity is a key property of a consistent ordering relation which says that
if the first thing is better than the second and the second is better than the third,
then the first is better than the third.)
The worst violation of transitivity is for the following three hands:

(2013-09-30) How Much to Bet?
It's a weakness to have a betting scheme be a predictable function of the pocket hand.

How Much to Bet?

Phil Hellmuth (1964-)
is a professional American poker player who cultivates a temperamental personnality
(at times, he proudly sports a football jersey bearing his nickname: "The Poker Brat").
Countless videos of his frequent obnoxious outbursts (during or after poker games)
are floating around YouTube.
It seems fairly clear that at least some of those are calculated to induce
a specific impression on his peers which may influence their decision-making process
during competitions (the skilled Hellmuth can then cash in on that).

In one instance
[ 1 ],
the 21-year old Tom Dwan with pocket tens eliminates
Hellmuth with pocket aces as both push all-in and Dwan
luckily draws a third 10. Hellmut goes ballistic, patronizing and insulting Dwan.

Although visibly shaken by the verbal agression, Dwan catches on to the
poker aspect of what Hellmuth is trying to do and plays along with
the type of follow-up questions one would be asking to narrow down the
risk tolerance of Hellmuth (at least
in the context of a tournament of the same type).
This is a gem if you know what to look for.
Let me assure you that neither participant is feeble-minded or faint-hearted,
but the superior reaction of Dwan floored me:
"Would you have gone for 3100, Phil?"

(2013-09-23) Omaha Hold'em
Pot-Limit Omaha (PLO) is the second most popular form of poker.

The Omaha game has the same structure as Texas hold'em concerning
the 5 community cards (flop, turn and river) but every player
receives 4 hole cards (instead of 2) and must form the best possible
poker hand using exactly 2 of his hole cards and 3 community cards.

In Pot Limit Omaha wagers beyond the blinds can't be larger than the pot.

An Omaha combo consists of 9 cards in two disjoint collections;
the hole (4 cards) and the board (5 cards).
The total number of those is the
multichoice number
C(52,4,5) = C(52,5,4) which is nearly 464 billion:

Odds of forming various poker hands out of random Omaha combos

Class

Combos

Probability

Odds

Royal Flush

42807600

1 in 10829

1 to 10828

Straight Flush

368486160

/

to

4 of a Kind

2225270496

/

to

Full House

29424798576

/

to

Flush

31216782384

/

to

Straight

52289648688

/

to

3 of a Kind

40712657408

/

to

Two Pairs

170775844104

/

to

Pair

122655542152

/

to

High Card

13851662832

/

to

Total: C(52,5,4)

463563500400

1

1 to 0

Thus, royal flushes are 60 times more common in Omaha combos
than in 5-card combos. They're 20/7 = 2.857 more frequent than
in Texas hold'em.

Note that a straight beats three-of-a-kind in all forms of poker.
Yet, in Omaha poker, a straight is more likely than three-of-a-kind...

To enumerate Omaha royal flushes, we first choose the suit (there can't be
two different royal flushes in the same combo, since we've fewer than 10 cards).
Then, we choose 2 cards from that hand for the hole and three for the board,
in one of C(5,2) = 10 ways. Finally, we complete the combo
with two cards for the hole and two cards for the board. All told: