Poker is a card game with many variants.
This page mentions several of them but it focuses mainly
on the most popular one: NLHE, or no-limit Texas hold 'em
(which is the only kind of poker on television, nowadays).
All versions of poker are played with a single 52-card deck,
using the same basic ranking of 5-card poker hands, presented below.
.
A full-size poker table can accomodate up to 10 players
and typically measures 94'' by 45''
(a piece of felt fabric measuring
108'' by 60'' is wrapped around and stappled underneath).
Traditionally, a raised padded armrest surrounds the outer rim.
Cup-holders can be carved into a 4'' or 5'' wooden racetrack
or directly into the felt (or the armrest).
(2013-11-17) Five-card Draw Poker
This most elementary form of poker serves as the basis for video poker.
After posting antes and/or blinds, every player receives 5 cards face-down and a
live betting round takes place
(where players may fold).
Then, every active player is allowed to discard some or all of his initial cards
and receive replacements for them.
Once everybody has done this, a final betting round takes place.
If two or more active players remain, a showdown takes place between them and
the best hand wins the whole
pot.
If two or more hands have the same value
(the 4 suits are placed on the same footing in poker)
then the pot is split among the winners.
Video poker is essentially 5-card draw with antes only (no live betting)
and some conventional payout schedule which depends on the quality of the hand obtained.
So transformed, poker becomes a single-player game (gambler vs. machine)
which is now, by far, the most popular theme for slot machines in casinos.
(2002-07-17)
There are 2,598,960 possible 5-card poker hands.
They're divided into 10 classes whose values vary inversely as their sizes.
No cheating; a Smith & Wesson beats five aces.
In all variants of poker, only 5 cards are compared to determine the winner.
There are 10 types of such 5-card poker hands.
Within each type, ties are broken by comparing
lexicographically
the ranks of the cards as they appear in the standard designation of the hand
(based on the elementary order of lone cards AKQJT98765432).
As all four suits are placed on the same footing in poker, ties may subsist.
Actually, there are only 9 basic types, since the mythical royal flush
is just the highest-ranking straight flush according to
the above standard rules (it's an ace-high straight flush).
Some authors adopt this convention which may be technically better.
So do we, at times.
There are C(52,5) = 2598960 different poker hands [count them]
and each of them is dealt with the
same probability in the initial round of draw poker
The probability of a given type of hands is thus the number of such hands
divided by 2598960.
When the probability of something is x / (x + y),
its so-called odds are said to be either x to y in favor
(a player is said to be an "x to y favorite")
or y to x against (a player is then called a "y to x underdog" or a "y to x dog").
The former type of odds is shown in the table below,
where the bold numbers (10) within formulas
correspond to the number of straights under the standard single-wheel rules...
I'd prefer the full-wheel rule with 13
straights, if anybody actually played it...
I seem to recall that we used to play 5-card draw
under the no-wheel rule (with 9 straights) back in high-school.
That no-wheel rule is commonly used, among several other possibilities, in low poker
(lowball) where the object of the game is to make the lowest-ranking hand.
That form of lowball is called deuce
to seven (which is the description of the most desirable hand).
No form of lowball is played which exactly reverses the desirability of the standard rankings
corresponding to the following table (single-wheel; 10 straights).
Class
Number of Hands
Probability
Odds in Favor
Royal Flush
C(4,1) C(1,1)
4
1 / 649740
1 to 649739
Straight Flush
C(4,1) C(10-1,1)
36
3 / 216580
3 to 216577
4 of a Kind
C(13,1) C(48,1)
624
1 / 4165
1 to 4164
Full House
13 C(4,3) 12 C(4,2)
3744
6 / 4165
6 to 4159
Flush
C(4,1) [C(13,5) - 10]
5108
1277/649740
1277 to 648463
Straight
C(10,1) (45-4)
10200
5 / 1274
5 to 1269
3 of a Kind
13 C(4,3) C(12,2) 42
54912
88 / 4165
88 to 4077
Two Pairs
C(13,2) C(4,2)2 44
123552
198 / 4165
198 to 3967
Pair
13 C(4,2) C(12,3) 43
1098240
352 / 833
352 to 481
High Card
(C(13,5)-10) (45-4)
1302540
1277 / 2548
1277 to 1271
TOTAL
C(52,5)
2598960
1
1 to 0
"High Card" means five singletons not in sequence and not in the same suit.
Let's call p the pip count of the highest card in such a hand,
extended in the usual way to court cards
(p is 11,12,13,14 for J,Q,K,A respectively).
A p-high hand is obtained by selecting 4 different ranks between 2 and p-1
so that straights are avoided and then making any selection of five suits
besides the four that are all alike.
This yields the following number of p-high hands when p is between
5 and 13 included (in the case of ace-high hands, the bold
1 should be replaced by 2
because there are two disallowed collections of 4 kicker ranks, namely
KQJT and 5432).
Let's just factor out the second factor 45 - 4 = 1020
to list only small numbers to which the probabilities of the 8 possible named high-card hands
are proportional. (These add up to C(13,5)-10 = 1277.)
A-high
K-high
Q-high
J-high
10-high
9-high
8-high
7-high
493
329
209
125
69
34
14
4
In Texas hold 'em, the ultimate showdown between three
players would be when the winner has a royal flush,
the runner-up four aces and the third four kings
(everybody else would have aces and kings).
The probability of this is only 1 in 10,464,946,021,530.
Of course, it's never been reported!
Incredibly though, the two-player equivalent
(royal flush vs. four aces) was filmed in 2008
during the WSOP main event (see first link below).
The commentator didn't give the correct probability (1 in 790 million).
(2013-12-01) What's the kicker in a poker hand?
Straights and fulls don't have a kicker. Flushes do.
Within a poker hand (or, possibly, an incomplete poker hand)
the kicker is the highest card, if any, which could break a tie
between hands of the same name.
If two like hands have the same kicker, lower cards (lesser kickers)
may be used to break the tie (second kicker, third kicker and fourth kicker).
Straight flushes, full houses and straights don't have a kicker at all.
Among those hands, there is a tie between all hands that bear the same name.
(E.g., "Jacks full of aces").
In draw poker or stud poker,
the hands that are compared always consist of separate collections of cards.
In those games, kickers are never needed for
quads or trips.
In games with community cards however, including Texas Hold 'em,
the hands of two different players can share cards (in fact, they always do)
and kickers often play a rôle for those types of hands as well.
For example, if quads are on the board in Texas hold 'em and
the fitfth community card is weak, then the kicker will most probably come from the hole cards;
whoever has the highest will win...
(In Omaha hold 'em, kickers are never needed for quads
but they may play a rôle with trips.)
The kicker of a two-pair hand makes its description complete
(e.g., "Queens and Jacks with an Ace kicker").
For a high-card hand, the kicker is the second-highest card
(e.g., "Queen-high with a Jack kicker").
Some or all of the remaining three cards could be relevant in a showdown
(second, third and fourth kickers).
Likewise, the kicker in a flush is the second-highest card and the three lesser kickers
may help break a tie, if need be.
King-high flush with a Jack kicker :
(2013-10-14) Perfect Poker = Mathematician's Poker
A symmetry that would give "deuces or better" one to one odds exactly.
The above table is for standard rules which allow 10
possible "heights" for a straight;
The ace belongs to both the highest and lowest straights;
respectively AKQJT (broadway)
and 5432A (wheel, bicycle or bike).
Two lesser-known nonstandard rules exist:
Under the no-wheel rule, 5432A isn't a straight at all
and the above four bold occurences of 10 should be replaced by 9.
Conversely, under the full-wheel rule, 3 lower straights are
added (432AK, 32AKQ and 2AKQJ in that order) and the bold numbers 10
should be replaced by 13.
This changes the respective counts for (non-royal)
straight flushes, flushes, straights and "high cards" as follows:
32, 5112, 9180 and 1303540 (no-wheel; 9 different straights).
36, 5108, 10200 and 1302540 (standard; 10 different straights).
48, 5096, 13260 and 1299480 (full-wheel; 13 different straights).
Number of straight flushes which a given card belongs to
Rule for Straights
A
K
Q
J
T
9
8
7
6
5
4
3
2
No wheel (9 straights)
1
2
3
4
5
5
5
5
5
4
3
2
1
Single wheel (10 straights)
2
2
3
4
5
5
5
5
5
5
4
3
2
Full wheel (13 straights)
5
5
5
5
5
5
5
5
5
5
5
5
5
Any mathematician will find the full-wheel rule far more appealing,
as it respects a perfect symmetry between
the 13 heights within any of the 9 basic poker categories
(lumping royal flushes and straight flushes into the same category).
Heights are merely used as tiebreakers within each
category. Indeed, all probabilities become simpler fractions because of the regularity
of this nice symmetrical convention.
The ultimate reduction pertains to the probability of getting at least a pair, which becomes
exactly 1 to 1...
Perfect Poker = Mathematician's Poker
(adopting the full-wheel convention)
Class
Number of Hands
Probability
Odds in Favor
Royal Flush
C(4,1) C(1,1)
4
1 / 649740
1 to 649739
Straight Flush
C(4,1) C(13-1,1)
48
1 / 54145
1 to 54144
4 of a Kind
C(13,1) C(48,1)
624
1 / 4165
1 to 4164
Full House
13 C(4,3) 12 C(4,2)
3744
6 / 4165
6 to 4159
Flush
C(4,1) [C(13,5) - 13]
5096
1 / 510
1 to 509
Straight
C(13,1) (45-4)
13260
1 / 196
1 to 195
3 of a Kind
13 C(4,3) C(12,2) 42
54912
88 / 4165
88 to 4077
Two Pairs
C(13,2) C(4,2)2 44
123552
198 / 4165
198 to 3967
Pair
13 C(4,2) C(12,3) 43
1098240
352 / 833
352 to 481
High Card
(C(13,5)-13) (45-4)
1299480
1 / 2
1 to 1
TOTAL
C(52,5)
2598960
1
1 to 0
The three "new" type of straights are just ace-high
in standard poker. Thus, the probabilities of
the 8 possible high-card hands in perfect-poker are proportional
to the following numbers (adding up to
C(13,5)-13 = 1274).
A-high
K-high
Q-high
J-high
10-high
9-high
8-high
7-high
490
329
209
125
69
34
14
4
(2013-09-20) Poker Chips: User's Guide & Buyer's Guide
This is a topic related to, but distinct from, casino gaming chips.
Gambling with actual coins is possible but presents difficulties.
Pieces of ivory begat the chip. Printed checks begat the plate.
... / ...
When only three or four chip denominations are used,
their values are often in a geometric progression of ratio 4
or 5 (e.g., 1-5-25 or 1-5-25-100).
The fourth denomination is sometimes used as a "unit of account",
much larger than the other three, for high-limit games.
One example might be 1, 5, 25, 250.
In the televised Million-Dollar Cash Game they use
1,5,10,250 (actual denominations are 100, 500, 1000 and 25000).
Large tournaments start with low denominations in the early elimination
stages and work up their way to huge denominations in the final table(s).
For visual effects in successive video, they use different denominations
which translate in a rainbow of colors from start to finish but,
usually, only 3 or 4 of those are used simultaneusly on each table.
The colors are consistent within a tournament, a casino or a set of casinos
at a given time but there's little or no overall coordination as the following
table will serve to demonstrate.
Some Color-Codes for Poker Chips (starting with home-use tradition)
Pastel high denominations are rare.
A true mint-colored 25000 chip appears in
Sun-Fly's Valentino ceramic collection.
Unfortunately, their Lucky Dragon 25000 chip is beige...
At right, is the most affordable ($0.10) 39 mm chip of that denomination;
the brown-rimmed Ace Casino chip with laser-graphic.
For home use, denominations beyond 25000 are nowhere to be found.
Commercial poker chips are produced in the following grades:
Composite: Mixture of clay and resin (d = 1.85). Typically 8 g.
Full Clay: Compressed clay with a binding agent. About 9 grams.
Ceramic: Synthetic ceramics, with direct imprint. About 10 grams.
Slugged: Composite clay with a metal insert (between 11 g and 14 g).
Lightweight plastic chips
are only recommended for travelling.
Most experts favor the traditional 9-gram clay chips ("full clay").
However, the newer composites (with or without metal inserts) are cheaper and
can withstand more abuse while retaining "nearly" the same feel and sound.
Finally, ceramics are the most durable kind
(the graphics are imprinted on the chip itself, not on a sticker)
at a cost midway between top-notch clay chips and ordinary composites.
In the following table, the "colors" column tells how many of the 12 basic
denominations (0.25,0.5,1,5,10,25,50,100,500,1000,5000,10000)
are available in the colors listed above.
The "+" sign indicates that other colors are offered in addition to those (or instead of them).
Prices are listed for a single chip to allow comparisons, although you may have to buy a
whole pack (usually, 25 chips) to get that price.
Gaming chips (including, at least, $1, $5, $25 and $100). Prices were current in 2013.
The lightweight interlocking radial poker chips were once virtually the only
ones available for home use. They came in only three colors, without denominations,
universally interpreted as white for 1, red for 5 and blue for 10
(mirroring the lowest denominations in US coins; pennies, nickels and dimes).
Due to the interlocking features, a stack of N of these has a height of about
(1.9 N + 0.2) milimeters (for the Bicycle brand) they weigh
only 1.6 g each for a diameter of 38.1 mm (compared to a nominal
diameter of 39.4 mm for standard casino chips).
The effective volume (2.1 cc) is about half the volume (4.2 cc)
of typical casino chips. Having a density of 0.76,
ABS chips float in water.
Chips of that type are now
available
in the full "standard" spectrum made popular by other types of poker chips:
brown (0.25), dark grey (0.5), white (1), red (5), blue(10), green (25), black (100), purple (500),
yellow (1000), orange (10000) and pink (either 2.50 or 5000). At this writing,
this leaves out only the need for a darker shade of pink for an unambiguous 5000, and light-blue (or cyan) for 50.
Beyond that are the various grades of full-sized poker chips, inspired by standard 39 mm
casino chips (the nominal diameter is sometimes quoted as 39.4 mm,
the thickness is 3.4 mm, the volume is 4.1 cc or 4.2 cc).
The first mass-produced full-sized poker chips for home-use were the premium bicycle chips
(7.8 g)
the suited design marketed by Wal-Mart
and the Da Vinci striped dice chips.
The latter feature the 6 possible dice pips close to the rim between
squarish white stripes (blue stripes for the white chip)
which wrap around to the other side (it was the first time a chip
intended for home use had any kind of edge stripe mimicking those found on casino chips).
None of the above have imprinted denominations.
In the Fall of 2003, an amateur (Chris Moneymaker) grabbed the most prestigious title
in the world of poker (the WSOP
"main event" in Texas hold 'hem).
This grabbed the imagination of the public who suddenly became interested in home poker tournaments
and serious poker chips, ressembling casino chips,
to make them more glamorous.
Traditional makers of casino chips could only fill the upper-end of that new market with overpriced pieces.
New companies jumped in and began to provide quality chips at a more reasonnable price level
(see article by
Jeffrey Smith,
who helped create Claysmith in 2003 and
Brybelly in 2004).
Here are reviews of some lines of poker chips currently available for home use, as tabulated above:
The 2-Stripe Twist line is one of the best deals around at $0.08 per chip.
Those chips have exactly the same weight as the original premium ("tournament") Bicycle chips
(7.8 g, which is about 20% less than casino clay chips). You get a small stack of these
for what you'd pay for a single Paulson clay chip.
The graphic design has no wording besides the denomination in large black numbers
(without a dollar symbol, which is an annoying habit in most other denominated chip lines).
The color scheme is nice, except for the 10 and 50 denominations which come in the same shade of
blue with the same yellow stripes, which makes is risky to use both denominations on the
table (which is of little concern to me, as don't care for either
of those marginal denominations, anyway).
The loaded (13.2 g) Yin Yang line is also an excellent deal.
I am partial to the nifty design which includes no wording and no currency symbol.
This is the type I have selected for my own use.
The only thing I don't
like about those chips is the lack of contrast between the 100 and 500 denominations
(the former is black, the latter is a very dark shade of purple).
I have chosen to discard the 10 and 50 denominations
(the less common 2, 20, 250 and 2000 denomination aren't available at all in this collection)
and to supplement my basic set
with a dozen of the above
25000 Ace Casino chips which are made by the same manufacturer
(with identical rims).
The Scroll Ceramic line is missing only 4 non-essential denominations
($0.25, $10, $50 and $10000). The color coding of the rest is flawless.
This makes it a perfect candidate to be supplemented by just two off-brand plaques:
$25,000 (mint) &100,000 (lavender).
Those could be issued for re-buys by obtaining a color-cange from the chip leaders...
The Venerati line is apparently
no longer
live up to the hype.
What they now sell under that name are mere replicas of the stunning originals
at the same hefty price ($0.39 "on sale")
with less vibrant colors, poorer print and less weight (9.3g instead of 10g).
(2013-10-21) Breakout of a set of poker chips
What's the best composition of a set of chips with several denominations?
A typical set of 1000 chips for home tournaments might include:
300 white chips (1),
200 reds (5),
200 greens (25),
200 blacks (100),
50 purples (500), and 50 yellows (1000).
What's the pattern here?
What would be the most satisfying breakout?
I own a set of 324 poker chips with 9 different denominations, from 1 to 25000.
The number of chips of every denomination is a whole number of dozens.
Thus, the chips can be distributed evenly among 1,2,3,4,6 or 12 players.
For other numbers of players, the simplest number to distribute the chips at the
beginning of the game is to leave a few of them in the bank so that the rest is evenly divisible
by the number of players.
The following table shows how many chips of each denomination this method assigns to every player,
as a function of the total number of players:
Color
Value
1
2
3
4
5
6
7
8
9
10
White
1
96
48
32
24
19
16
13
12
10
9
Red
5
48
24
16
12
9
8
6
6
5
4
Green
25
48
24
16
12
9
8
6
6
5
4
Black
100
48
24
16
12
9
8
6
6
5
4
Purple
500
24
12
8
6
4
4
3
3
2
2
Yellow
1000
24
12
8
6
4
4
3
3
2
2
Melon
5000
12
6
4
3
2
2
1
1
1
1
Pumpkin
10000
12
6
4
3
2
2
1
1
1
1
Brown
25000
12
6
4
3
2
2
1
1
1
1
The total face values of the chips is 522336.
In the case of 5 players, the naive scheme illustrated by the above table will assign 60 chips
to every player for a total value of:
The value of the 300 chips in play is five times that (435945).
The 24 leftover chips have a value of 86381.
Can we waste less than that and distribute the leftover among the players?
Yes, we can. We must leave $1 in the bank
(since 522336 isn't divisible by 5) but we can distribute the rest evenly
among the 5 players in several ways.
None of them is trivial. The following table illustrates one of them:
Splitting the above set of chips (minus $1) evenly among 5 players : A,B,C,D,E
Color
Value
Pieces
A
B
C
D
E
Bank
White
1
96
17
17
17
22
22
1
Red
5
48
10
10
10
9
9
0
Green
25
48
12
12
8
8
8
0
Black
100
48
11
11
12
7
7
0
Purple
500
24
6
6
2
5
5
0
Yellow
1000
24
5
5
2
6
6
0
Melon
5000
12
3
3
2
2
2
0
Pumpkin
10000
12
3
3
4
1
1
0
Brown
25000
12
2
2
2
3
3
0
Every player gets 104467
(2015-05-15)
2013 NBC National Heads Up Poker Championship
Accelerated structure sheet of NBC Heads Up at Caesar's Palace.
This invitational tournament had a $25000 buy-in.
Each player started round one with 25000, in chips of 3 denominations
(25, 100, 500).
The field of 64 is divided into 4 brackets (Spades, Hearts, Clubs, Diamonds)
of 16 players, with 8 matches taking place at once.
Only the winner of a match advances to the next round.
The winner is decided among the last two players in "best of three" matches
(the starting stack being reset to 800k at the beginning of each of those matches).
Only three or four different chip denominations are in play at a time,
according to the following detailed structure sheet :
Round of ...
64
32
16
8
4
2
Blinds
150 300
300 600
600 1200
1k 2k
2k 4k
5k 10k
Starting Stack
25k
50k
100k
200k
400k
800k
Green & Red
25
Grey & Pink
100
50
White & Purple
500
30
Yellow & Green
1 k
20
Orange & Purple
5 k
12
Yellow & Black
25 k
Money (750k to the winner)
25k
50k
100k
300k
The total prize money of 1650k exceeds by 50k the buy-in fees
collected from the 64 participants (25k each).
NBC paid for that difference.
(2013-12-30) Handling Poker Chips
Counting chips. Betting with chips.
Chips are counted by heaps of five. Four such heaps form a proper
stack (20 chips of the same denomination).
Avoid dirty stacks (including at least one chip of
another denomination) or barber poles
(several stacks of different denominations intentionally put on top
of each other). Both of these can be illegal in some clubs.
It's acceptable to put incomplete stacks on top of two stacks of equal
height (preferably of 20 or 40 chips of the same denomination).
(2013-09-28) Poker Chip Tricks
Fidgeting like a pro is part of the intimidation...
In almost any video of high-stakes poker, there are silent moments
where you hear only one noise, produced continuously by several players
around the table: The characteristic clicking produced by
chip shuffling (also called riffling).
It's almost as if you can't enter a poker tournament
unless your fingers are trained to do this move,
which is the most popular and possibly the easiest of all chip tricks.
The butterfly
is called the "coin star" in magic circles
(tutorial by Rich Ferguson).
(2013-11-17) Seven-card Stud
"Down the River" was the most popular form of poker
before the rise of NLHE and PLO.
In seven-card stud poker, there are neither blind bets nor community cards.
The cards dealt to each player are for his own use only.
However, all but two of those (the hole cards)
are dealt face-up for everyone to see.
Initially. everyone posts an ante bet and
every player is dealt two hole cards and one face-up card.
The player with the lowest face-up card is required to start
a standard round of betting
by wagering at least the minimum agreed upon ahead of time.
This forced wager is called a bring-in bet.
Players who fold put all their cards into the muck.
It up to the remaining player to remember what up-cards were thus taken out of play.
As the rest of the game unfolds, that information can be critical.
(2013-09-22) Seven-Card Combos
A combo has the same value as the best 5-card poker hand it contains.
In several variants of poker, including
7-card stud and
Texas hold 'em,
your final hand is the best 5-card poker hand that can be extracted from
7 randomly-dealt cards. In that case, the probability of finally obtaining
each type of poker hand is given by the following table:
Best (5-card) poker hands
extracted from 7 random cards
Class
Combos
Probability
Odds
Royal Flush
4324
1 in 30940
1 to 30939
Straight Flush
37260
81 / 290836
81 to 290755
4 of a Kind
224848
1 / 595
1 to 594
Full House
3473184
726 / 27965
726 to 27239
Flush
4047644
1011911 / 33446140
1011911 to 32434229
Straight
6180020
44143 / 955604
44143 to 911461
3 of a Kind
6461620
14047 / 290836
14047 to 276789
Two Pairs
31433400
785835 / 3344614
785835 to 2558779
Pair
58627800
4455 / 10166
4455 to 5711
High Card
23294460
166389 / 955604
166389 to 789215
Total: C(52,7)
133784560
1
1 to 0
A royal flush is thus 21 times more frequent if you
have 7 cards to choose from, instead of just 5.
You get less than a pair with probability 17.41%.
Enumerating Flushes :
The enumeration of the various kinds of flushes demonstrates the delicate computations
involved in building the above table, using choice numbers:
We first remark that there are four times as many
flushes as there are spade flushes (that wouldn't be true if we
had at least 10 cards to choose from, as we could have then have several flushes in
different suits and would need to use inclusion-exclusion
enumeration.) So, let's just count the spade flushes...
The number of ways to have at least 5 spades among 7 cards is:
The last two results, multiplied by 4, give us directly the tabulated numbers of
royal flushes and other straight flushes, respectively:
4 (1081) = 4324
and
4 (10396 - 1081) = 37260
Now, there are no fulls or quads among the 1022307 combos of 7 cards
which include at least 5 spades. Therefore,
they are all tallied as spade flushes unless they contain a straight flush
and the total number of 7-card combos that make a flush is, as tabulated:
4 (1022307 - 10396) = 4047644
Enumerating Straights :
A 7-card combo is tallied as a straight when it contains a sequence of at least
five consecutive ranks, provided the most common suit occurs less than five times.
This can be achieved in the four distinct ways listed below.
Each of those leads to a different way to count the allowed repartitions of suits:
3372180 combos of 7 singletons containing a straight but no flush: 217 [ 47 - 4 C(7,5) 32 - 4 C(7,6) 3 - 4 C(7,7) ]
= 3372180
2530440 combos of a pair and 5 singletons, likewise restricted: 71 C(6,1) C(4,2) [ 45 - C(2,1) {C(5,4)3 + C(5,5)} - C(4-2,1) ]
Two pairs and 3 singletons, containing a straight but no flush: 10 C(5,2) [ C(4,2)2 43 - 4 32 ]
= 226800
Three-of-a-kind and 4 singletons, containing a straight but no flush: 10 C(5,1) C(4,3) [ 44 - C(3,1) ]
= 50600
This adds up to 3372180 + 2530440 + 226800 + 50600 = 6180020
The Remaining Enumerations :
The number of 7-cards combos with quads in them is obtained
by multiplying the 13 ways of picking four like cards into
the number of ways to pick three other cards from the rest of the deck:
C(13,1) C(52-4,7-4) = 224848
A full house is obtained in one of three ways:
A set, a pair, two singletons:
C(13,1) C(4,3) C(13-1,1) C(4,2) C(13-2,2) C(4,1)2 = 3294720
A set, two pairs:
C(13,1) C(4,3) C(13-1,2) C(4,2)2 = 123552
Two sets, a singleton:
C(13,2) C(4,3)2 C(52-8,7-6) = 54912
This adds up to 3294720 + 123552 + 54912 = 3473184
Three of a kind can only be extracted from a combo consisting of a set and four singletons
forming neither a straight nor a flush. A proper choice of the suits is
obtained by first picking 3 suits for the set and, then, any suits for the singletons
except a single suit present in the set. The tally is thus:
Two-pair hands come from only two types of 7-card combos, namely:
Three pairs and a singleton:
C(13,3) C(4,2)3 C(52-12,7-6) = 2471040
Two pairs and three singletons forming neither a straight nor a flush:
(C(13,5)-10) C(5,2) [ C(4,2)2 43 - 4 C(4-1,2)2 ]
= 28962360
This adds up to 2471040 + 28962360 = 31433400
To form a 7-card hand which amounts to a pair, you first choose a collection of 6 ranks
besides the 71 in which there's a straight.
Then, you choose one of those six as the rank of the pair and pick 2 suits for the pair.
Last, you choose 5 suits for the singletons by avoiding having either at least 4 singleton
matching a suit from the pair or 5 singletons of the same suit not represented in the pair:
As advertised in the above table, this boils down to 58627800
Finally, the other 7-card combos
(which contain neither several cards of the same rank nor straights nor flushes) are obtained
by selecting 7 distinct ranks besides the 217 which
form a straight and then assigning suits to those in such a way that five cards never belong to the same suit,
The choices of the latter kind correspond to the square bracket below
(where the three negative terms tally combos with 5, 6 or 7 cards of the dominant suit):
(2013-10-12) Betting Rules
Folding, checking, calling and raising. Limit, pot-limit and no-limit.
There are two type of forced bets:
The ante
(from the Latin word for "before") is a wager that all players are
required to make, before any cards are dealt, for the priviledge to play a hand.
The amount of the ante is set by the organizer of the game
(the host, the house, the casino) and it can be zero
(in poker, it often is).
The blinds which must be posted by the player to
the left of the dealer button (big blind) and the next one
(small blind) "blindly", regardless of their cards (hence the name).
The small blind is usually at most half of the big blind (BB) but it's up to the
organizer to determine the values of the blinds ahead of time...
For example, in a low-stake game played with whole denominations chips,
the big blind could be $5 and the small blind could be $2.
On the first round of betting, the minimum bet for the player in the small blind
position is the difference between the big blind and the small blind.
Thereafter, the basic choice is always to fold or to bet.
To fold is to give up entirely without putting any more money at risk but without
the possibility of winning anything.
In certain circumstances, the minimum bet is zero and taking that opportunity of a zero bet
is called checking. You stay in the game for free, so to speak.
It usually males no sense to fold if you can check, unless you absolutely want to avoid
revealing the hand you were dealt
(presumably a weak one which almost certainly can't win).
If you fold
(or if you win because everybody else has folded)
you don't have to reveal you hand.
Showing your hand is a valuable piece of information for skilled opponents
and it can be a good idea not to do that in certain cases,
even if it means "foolishly" giving up a tiny chance at winning the pot.
A betting round always proceeds clockwise around the table,
giving every player an opportuniy to call, raise or fold until everybody who didn't
fold has bet the same amount.
The first betting starts in different ways depending on the poker variant.
In seven-card studs, it starts with whoever has the lowest up-card.
In hold'em poker (Texas or Omaha variants)
every round starts with the player to the left of the dealer button
(which indicates the nominal dealer,
who doesn't do any actual dealing if a professional dealer is on duty).
However, in the first round only, that player is required
to post one "big blind" before receiving any cards and the next player must
likewise post one "small blind" (usually equal to half of the big blind).
After everyone has spoken, those two players get a chance to call or raise
the previous player by deducted from the amount the owe
the amount that they've already payed "blindly".
In subsequent rounds, "checking" is allowed (which is effectively betting or
calling a zero amount) until somebody actually bets something.
In a betting round, wagers can only increase.
After the first nonzero bet (see checking, as discussed above)
betting is called either calling (when waging an amount equal to
the previous bet) or raising
(when the wager is greater than the previous bet).
The usual rules impose that a raise must be at least twice the amount of the previous bet.
When a betting round ends with a call, the game proceeds with the next stage
(either more cards are dealt or the final showdown occurs where actual
hands are compared).
Otherwise, another round of betting occurs, starting with a minimum bet of zero
(which is to say that players are allowed to "check" until one of them "bets").
Examples of Uniform Starting Stack Sizes and Final Tables
The poker subculture has its own vocabulary.
Some of it is technical jargon for key aspects of the game.
The rest is simply colorful lingo used for its entertainment value.
Below is most of the former and some of the latter.
This is still a work in progress...
Naming Single Cards :
Bullet : Ace (A).
Cowboy : King (K).
Lady : Queen (Q).
Fishhook : Jack (J). Because of the J-shape and its appeal to fish.
Paint : King, Queen or Jack.
Dime : 10, ten (T).
Niner, neener : 9.
Snowman : 8.
Cane, walking stick : 7.
Sax : 6.
Nickel : 5.
Sailboat : 4.
Trey, crab : 3.
Deuce, duck : 2.
Rag : An unspecified card lower than 9 (as in "Ace-rag pocket").
Start hands in Texas Hold 'em (2 hole cards):
Pocket : The two face-down cards dealt to a player (Texas hold 'em).
Pocket Pair : Two matching hole cards.
Suited : Said of two cards belonging to the same suit.
Off-suit, unsuited :
Qualifier for two cards in different suits.
Player : Someone who has been dealt cards in a hand.
Calling the clock :
Upon request of any player, the organizers must force a player
to make a decision within one minute (sometimes, 30 s or 90 s).
This is usually automatic (without the need for a player request) in online games.
Railbird : A spectator (not a player) behind an actual rail or not.
Heads-up :
A game of poker played between only two players.
Satellite : Low-fee competition for free admission into another tournament.
Buy-in : The initial purchase of chips to participate in a tournament.
Re-buy : Purchase of chips to re-enter a multiple-table tournament after elimination
at an early stage (or to replenish one's stack in a cash game).
Compulsory re-buys take the form of lammers.
Bubble : The set of finalists who are not awarded any prize money.
In the money (ITM) : Tournament winners sharing the prize money.
Big Blind : The amount that the person left of the dealer must wager before cards are dealt.
Often used as a unit (BB) to measure stacks.
Splashing the Pot : The impolite practice of throwing chips instead of pushing them neatly in stacks.
Small Blind : The amount that the second person to the dealer's left must wager
"blindly", before cards are dealt.
It's normally about half of the big blind
(the organizer of the game must specify that).
Ante : The wager which everybody must put into the pot before cards are dealt.
The organizer may set the ante to zero (and often does).
Rake : The house's take, off every pot (e.g., 10% up to $4).
Pot bonus :
Reverse rake, put by the house into the pot to encourage play.
Dead chips : Statutory contributions to the pot (bonus, blinds, ante).
Straddle : By posting twice the big blind before cards are dealt,
the player under the gun gets last preflop action (Nevada, Atlantic City).
No-wheel : Nonstandard house-rule disallowing
5432A as a straight.
Full-wheel : Rule whereby
432AK, 32AKQ & 2AKQJ are straights.
One-orbit penalty :
Sitting out until the button returns to the same place.
One-chip rule :
A single-chip bet isn't a raise unless otherwise stated.
Color up :
Exchange chips for higher denominations (e.g., for the next stage of a tournament).
Positions (clockwise) around the table :
Button : The marker put in front of the player reputed to deal the cards.
Position : A player's location, with respect to the dealer button.
On the Button : The position of the dealer (the strongest spot).
Dead Button : The button marks the spot of someone who's just left.
Under the Gun : The third spot left of the dealer (after both blinds).
Under the Gun Plus One : The fourth spot (UTG+1).
Cutoff: The position to the right of the dealer.
Hijack position : The position to the right of the cutoff.
In position :
Out of position :
Psychology & Superstition :
Card-dead : Qualifies a player who believe he can't get good cards.
Cold deck, cooler :
Tells : Involuntary reactions to the cards a player is dealt.
Reads : Interpretations of the tells to put a player on a hand.
Level : (Noun) A ruse. (Verb) To outsmart.
Leveling War : "He knows that I know that he knows, etc."
White Magic : The black art of reading opponents.
[Phil Hellmuth]
Tilt (emotional state) : Reckless play by someone who is upset.
Hollywooding : Adopting a fake composure to mislead opponents.
Slowroll :
Delayed disclosure of the winning hand (rude & useless).
Showboating : Revealing hole cards after a successfull bluff.
Putting a player on a hand : Guessing the hand of a player.
One-Time :
The overused joke of asking the gods of poker to grant you luck just for once ("let me use my one-time now").
Fubar, foobar : F...ed up beyond all recognition.
Fugazi : F...ed up, got ambushed, zipped in [a body bag].
Money Management & Betting Strategy :
Stack : The value of all the chips in front of a player,
who can wager nothing beyond that on the next hand (in a "cash game" he may be allowed to "buy in" more chips
from the organizer before the start of the next play). Chips can't be taken out of play from the stack
unless the player leaves the table early (as is only allowed in cash games).
Chip leader : Player with the most chips (in tournament or table).
Short stack : The player with the fewest chips.
Lammer :
A special token indicating a specific player priviledge.
In some
tournaments, one or two lammers are given along the starting stack.
They can be used to reload
(for a predetermined equivalent in regular tournament chips)
at any time between hands, during the initial stages.
A lammer cannot be wagered directly and is not at risk during a
hand. Two lammers would thus enable a player to all-in twice and
still be alive (all lammers must be redeemed before the later
stages of the tournament).
The prizes for a satellite may be
awarded in the form of lammers redeemable
for the buy-in of the main tournament.
Stages of the Game & Outcomes:
Muck : The pile of discarded cards (to muck = to fold).
Preflop : The betting round before any community cards are dealt.
Board : The community cards dealt so far (flop, turn and river in Hold'em).
Flop : The first three cards of the community hand (Hold'em).
To flop : To obtain after the first three community cards are revealed.
Turn : The fourth community card. Fourth street. (Hold'em)
River ("torrent") :
The last community card. Fifth street. (Hold'em)
Rainbow Flop : Flop consisting of three cards from different suits.
Flush draw : Four cards of the same suit, before the river is drawn.
Straight draw : Incomplete straight (either gut shot or open-ender).
Up-and-down straight draw : Four consecutive cards, without an ace.
Gut shot : Inside straight draw (also used for AKQJ or 432A).
Broadway draw : An ace and three of the four cards K, Q, J, T.
Wheel draw : An ace and three of the four cards 5, 4, 3, 2.
Air : Community or hole cards which have no tangible value.
Brick, blank : A community card which doesn't help any player
or a hole card which doesn't help its owner.
To brick, to brick out : To miss, to fail for lack of hits.
Gin Card : A community card which completes the winning hand.
Runner-runner, backdoor : Hand made with bothturn and river.
Running spades, running sevens, etc. :
Turn and river completing a hand.
Suck-Out : A lucky draw which turns a weak hand into a winner.
Bad Beat :
A loss occuring despite sound play (cf. suck-out).
Showdown : Final comparison of hands which haven't been folded.
Flop the big hand : Show the winning hand at showdown time.
Side Pot : Settles a showdown when some players are all-in.
Odd chip :
In a split pot,
a chip is awarded if it can't be divided.
Running it Twice
(or more) :
A gentleman's agreement whereby the remaining community cards
are dealt twice or more, after all betting has ceased.
The winners of every deal get equal portions of the pot.
Betting :
Chip : Gaming token, often standing for a definite amount of money.
Action : Either the act of betting or the amount so wagered.
Pot : The aggregate of all chips put in play at some point in time.
Squeeze Play :
Covering : Betting at least as much as a previous bettor in the round.
Refund : The uncovered part of a bet is refunded.
To fold, to pass, to lay down : To give up the hand entirely.
Check : Bet no additional amount on one's turn (without folding).
Check-raise :
Raise after having checked in the same betting round.
Check-fold : In online play, a preset decision to fold if anybody bets.
Value bet :
Thin value bet :
Bluffing : Pretending to have a stronger hand to make opponents fold.
Representing a hand:Bluffing with that
(strong) hand in mind.
Float :
A bluffing bet meant to steal the pot on a later round.
Semi-Bluff : Bluffing with a hand having an outside chance to win.
To Call : To wager the same as the previous player in the round.
Flat call : An emphatic way to describe a call, as opposed to a raise.
Instacall : Call immediately (with little or no time delay).
Raise :
To "raise it", one must post at least twice the last bet wagered.
Instaraise : Raise immediately (with little or no time delay).
Reraising, going over the top [of someone] : Raising after someone's raise.
Three-betting : Raising after someone else's reraise.
Four-betting : Raising after someone else's three-bet.
Overbet :
5p Gunning (UK) : Overbetting to secure a very meager pot.
To push all in, to be all-in :
To wager one's entire stack, either agressively or to stay "in the pot" till showdown
[standing up].
