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As in every card game, in Hold'em Poker we deal with
a finite probability field, in which the events to be measured are
occurrences of certain card combinations (three-card combinations,
two-card combinations or one card) on the board, in your own hand and in
your opponents’ hands
Among all games of chance,
Texas Hold’em Poker
is highly predisposed to probability-based decisions because, by the
structure and dynamics of the game, the
poker odds change
significantly from one gaming moment to another.
We
have the so-called long-shot odds (probabilities of events
that are chronologically preceded by others, calculated by using
information available at the moment before the first event), which consist
of:
–
The own hand odds – dealing with odds for you to achieve
certain final card formations, calculated at two separate moments of the
game: the three community cards dealt on the table (after the flop) and
the four community cards dealt on the table (after the turn);
–
The opponents’ hands odds – dealing with odds for at least
one opponent to achieve a certain final card formation, calculated in
three separate moments of the game: three community cards dealt (after the
flop), four community cards dealt (after the turn) and all five community
cards dealt (after the river).
We
also have the immediate odds (probabilities of events that follow
immediately, calculated by using information of the moment just before
that event): preflop odds, turn odds, odds of improving expected
formations, and so on.
Although
the long-shot odds are the most important when making a probability-based
betting decision, because they give information about achieving specific
final card formations (which is the technical goal of the game), the
immediate odds could help in evaluating and creating an advantage during
each betting dialogue. This is something that distinguishes Hold’em Poker
from games like
blackjack, where odds are only
informative for the very next move.
The
immediate odds categories are:
– preflop odds (odds of being dealt certain 2-card combinations as
pocket cards before the flop)
– flop odds (odds for certain cards or certain 2- or 3-card combinations
to be contained in the flop, calculated after first card distribution and
right before the flop)
– turn odds (odds of a certain card being dealt as the turn card,
calculated right after the flop)
– river odds (odds of a certain card being dealt as the river card,
calculated right after the turn).
All
Hold'em odds are in very large number and may fill hundreds of pages. We
present here just a few examples of Texas Hold'em odds:
Preflop
odds
Odds of being dealt AA
The probability of
being dealt AA is 0.452%. The same
probability holds for any other pair.
Odds of being dealt AA or KK
The probability of
being dealt AA or KK is 0.904%. The same probability holds for any other
two pairs.
Odds of being dealt KK, QQ or JJ
The probability of being dealt KK, QQ or
JJ is 1.357%. The same probability holds for any other
three pairs.
Odds of being dealt a pair
The probability of being dealt a pair
(any) is 5.882%.
Odds of being dealt AK suited
The probability of being dealt AK
suited is 0.301%. The same probability holds for any other
two suited cards.
Odds of being dealt A and less than
J suited
The probability of being dealt A and less
than J suited is 2.714%.
Odds of being dealt A and less
than J offsuit
The probability of being dealt A and less
than J offsuit is 8.144%.
Odds of being dealt two suited
cards
The probability of being dealt two suited
cards is 23.529%.
Odds of being dealt A or pair
The probability of being dealt A or pair
is 20.361%.
Odds of being dealt two cards
higher than J
The probability of being dealt two cards
higher than J is 3.619%.
Odds of being dealt suited
connectors
The probability of being dealt suited
connectors (23, 34, ..., KQ) is 3.318%.
.....................................................................................................................................................
All previous probabilities also hold
for the hand of any of your opponents, from a neutral perspective (as
long as we do not take into account your own hole cards).
Pair
against higher pair
Assuming you are dealt a pair (except AA),
we can calculate the probability for at least one opponent being dealt a
pair higher than yours. We can find a unique formula that comprises all
possible situations. Such a formula holds as variables:
n = number of your opponents
p = number of pairs (as a value type)
higher than the one you hold.
For example, if you hold 88, there are six pairs higher than yours (99, 10
10, JJ, QQ, KK, AA), so p = 6; if you hold KK,
there is only one pair higher (AA) and p = 1. The formula
is:
P =
6np/1225 – n(n – 1)p(6p –
1)/460600 + n(n – 1)(n – 2)p(6p –
1)(6p – 2)/1430163000.
For
p = 1, the formula returns the following results:
Probabilities
that at least one opponent holds AA, in case you are dealt KK
|
Opponents
(n)
|
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
|
P
(%)
|
0.489
|
0.979
|
1.462
|
1.946
|
2.427
|
2.906
|
3.383
|
3.912
|
4.330
|
A
table of values for the probability (P) of one opponent (a specific one,
not at least one) holding a pair higher than yours follows:
Probabilities
that one opponent holds a higher pair
|
Your
own
pair
|
|
33
|
44
|
55
|
66
|
77
|
88
|
99
|
T
T
|
JJ
|
QQ
|
KK
|
|
P
(%)
|
5.877
|
5.387
|
4.897
|
4.408
|
3.918
|
3.428
|
2.938
|
2.448
|
1.959
|
1.469
|
0.979
|
0.489
|
.............................................................................................................................................................
Flop
odds
We
have two cards showing (your own pocket cards), 50 unknown cards and C(50, 3) = 19600 possible combinations for the flop cards.
Odds of improving to one pair, three of a kind, or four of a kind
on the flop
Let C be a card (as a repeated value) from the expected
formation and let c be the number of your own hole C-cards. The
next table notes the probabilities for your hand to improve to one pair,
three of a kind, or four of a kind, on the flop.
Let S be a symbol (hearts,
diamond, clubs, or spades)
and let s be the number of your own hole
S-cards. The next table notes the probabilities for your hand to improve
to a flush partially or totally on the flop.
For a specific opponent and a symbol S, let us denote by
A' the event: "That opponent will achieve a flush (SSSSS) as his/her
final hand" and by B' the event: "At least one opponent will achieve a
flush (SSSSS) as his/her final hand". We calculated the probabilities
P(A') and P(B') at the moment when 3 community cards were dealt. The
variables the probabilities depend on are:
- s'' = number
of viewed community S-cards
- s = number of
viewed S-cards
- n = number of
your opponent
The next table notes the probabilities P(B') for
all possible values of these variables (the probabilities
P(A') are on the row n=1).
Table of values for the probability of at least
one opponent achieving a flush (SSSSS)
|
|
s''=0 |
s''=1 |
s''=2 |
s''=3 |
|
s |
|
1 |
2 |
3
|
2 |
3 |
4 |
3 |
4 |
5 |
|
n |
|
1 |
0 |
0.277% |
0.184% |
0.117% |
3.515% |
2.606% |
1.860% |
19.409% |
16.048% |
12.895% |
|
2 |
0 |
0.546% |
0.365% |
0.233% |
6.359% |
4.777% |
3.454% |
30.402% |
23.533% |
20.998% |
|
3 |
0 |
0.806% |
0.542% |
0.294% |
8.595% |
6.894% |
4.439% |
37.557% |
32.861% |
26.517% |
|
4 |
0 |
1.059% |
0.714% |
0.459% |
10.373% |
8.053% |
6.890% |
41.889% |
37.139% |
32.376% |
|
5 |
0 |
1.303% |
0.882% |
0.569% |
12.041% |
9.350% |
7.057% |
45.721% |
40.553% |
35.631% |
|
6 |
0 |
1.539% |
1.046% |
0.677% |
13.610% |
10.579% |
7.945% |
49.129% |
43.599% |
37.987% |
|
7 |
0 |
1.767% |
1.206% |
0.783% |
15.081% |
11.738% |
8.819% |
52.117% |
46.276% |
40.314% |
|
8 |
0 |
1.987% |
1.362% |
0.887% |
16.455% |
12.871% |
9.678% |
54.685% |
48.903% |
42.610% |
|
9 |
0 |
2.200% |
1.493% |
0.990% |
17.832% |
13.794% |
10.524% |
57.514% |
50.894% |
44.874% |
|
- Example:
- Community
cards: (♣♣♥)
- Your own hole
cards: (♥♣)
- 6 opponents
- Let us see
what is the probability for at least one opponent to achieve a flush
(♣♣♣♣♣).
- We have S=♣,
s''=2, s=3, and n=6.
- Searching in
the table, we find that P(B') = 10.579%.
The opponents’
hands odds – Three of a kind odds
in turn stage
For a specific opponent, let us
denote by A'' the event: "That opponent will achieve three of a kind
(TTTxy) as his/her final hand" and by B'' the event: "At least one
opponent will achieve three of a kind (TTTxy) as his/her final hand". We
calculated the probabilities P(A'') and P(B'') at the moment when 4
community cards were dealt. The variables the probabilities depend on
are:
- t'' = number
of viewed community T-cards
- t = number of
viewed T-cards
- n = number of
your opponent
The next table notes the probabilities P(B'') for all possible values of
these variables (the probabilities P(A'') are on the row n=1).
Table
of values for the probability of at least one opponent achieving three
of a kind (TTTxy)
|
|
t''=0 |
t''=1 |
t''=2 |
t''=3
t''=4 |
|
t |
0 |
1 |
2 |
1 |
2 |
3
|
2 |
3 |
4 |
|
|
n |
|
1 |
0.026% |
0.006% |
0 |
0.660% |
0.193% |
0 |
12.657% |
6.521% |
0 |
1 |
|
2 |
0.052% |
0.013% |
0 |
1.295% |
0.386% |
0 |
20.579% |
10.869% |
0 |
1 |
|
3 |
0.079% |
0.019% |
0 |
1.903% |
0.579% |
0 |
28.115% |
15.217% |
0 |
1 |
|
4 |
0.105% |
0.026% |
0 |
2.485% |
0.772% |
0 |
35.265% |
19.565% |
0 |
1 |
|
5 |
0.131% |
0.032% |
0 |
3.041% |
0.966% |
0 |
42.028% |
23.913% |
0 |
1 |
|
6 |
0.158% |
0.039% |
0 |
3.570% |
1.159% |
0 |
48.405% |
28.260% |
0 |
1 |
|
7 |
0.184% |
0.046% |
0 |
4.073% |
1.352% |
0 |
54.396% |
32.608% |
0 |
1 |
|
8 |
0.210% |
0.052% |
0 |
4.549% |
1.545% |
0 |
60% |
36.956% |
0 |
1 |
|
9 |
0.237% |
0.059% |
0 |
5% |
1.739% |
0 |
65.217% |
41.304% |
0 |
1 |
|
-
Example:
- Community
cards: (37QQ)
- Your own
hole cards: (5Q)
- 3
opponents
- Let us see
what is the probability for at least one opponent to achieve three of a
kind (QQQxy).
- We have
T=Q, t''=2, t=3, and n=3. Searching in the table, we find that P(B'') =
15.217%.
.............................................................................................................................................
The above probabilities belong to
specific card formations, what we call simple events. However, a
player also needs the probabilities of complex events (like events of
type higher than or one or the other). The chapter Operating with
and Weighting the Odds of the book
Texas Hold’em
Poker Odds
for
Your Strategy, with Probability-Based Hand Analyses
deals with the rules of estimating and evaluating the probabilities of
complex gaming events in order to use them in a strategy, by operating
with the
odds of the simple events. When we are allowed to add together the
partial probabilities of a union of events for an overall probability
and when we are not, methods of approximation and tips of avoiding hard
calculations, are all
explained in that chapter.
- The strength of a Hold'em hand
and the probability-based hand analysis
Any poker strategy consists of
rules and if-then-else algorithms, which are based, among other
criteria, on what players call the strength of a hand at
various moments of the game. This strength, even though quantified
in an intermediate moment of the game, is directly related to the
final moment of the game, which is in the future. That is because we
take the strength of a hand as an indicator of how good that hand is
now in order to win at the end. Therefore we can refer
to the strength of a hand only in terms of mathematical probability,
the only scientific tool we have for measuring the possibility of a
future event occurring, since the strength-of-a-hand concept
involves measurability and predictability. The strength matrix of
a hand is the main object of the
mathematical model of the
strength of a Hold’em hand and an important tool for weighting
the odds within a probability-based strategy.
Every poker strategy should take into account probabilities as
objective information, however this information will be used
through subjective criteria based on the personal threshold of
afforded risk. Thus,
any hand analysis should be based on probabilities.
Click the left button below
for an overview on the strength matrix of a hand and the strength
indicator of a hand. Click the right
button below for some probability-based analyses of concrete
Hold'em hands.
-
-
Hold'em Poker
Mathematics
online course
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Author |
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The author of this page is Catalin Barboianu
(PhD). Catalin is a games mathematician and problem gambling researcher,
science writer and consultant for the mathematical aspects of gambling
for the gaming industry and problem-gambling institutions.
Profiles:
Linkedin
Google Scholar
Researchgate |
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