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Poker Probabilities
Mathematics

Title:

Probabilities in
Texas Hold’em

Rationale
A few months ago, my family and I moved to a new house
...
I have always been interested in
poker, both because of the randomness each round and the psychological aspect, e
...
poker face
...
So when the Mathematics Exploration came around, I knew I wanted to make it about
something I knew and was interested in
...

By making this exploration, I am also learning a great deal myself about the mathematical aspect of
Texas Hold’em, which will actually benefit me vastly in the future when I am playing poker
...
I am using this exploration to
mathematically prove that a well-known strategy in Texas Hold’em is actually very weak; betting
high in the first turn, just because you think your hand of two cards are of good value
...
It is played amongst groups of friends
as well as in huge casinos
...
However, there is a lot of probability
involved in Texas Hold’em, and putting aside the psychological aspects of poker, that is what I will
focus on in this exploration
...

A game of Texas Hold’em is played with a deck of 52 cards
...
Their respective value are in the same order
...

Figure 1 below contains the definition of some important terms:
Hand:

Community Cards:

Bet:

Suit:

The two cards a player
has, which are not
visible to the other
players

The cards that are face
up in the middle of the
poker table for all
players to see

To bet a certain
amount of money

The four symbols in poker:
spades, hearts, clubs, and diamonds

Fold:

Check:

Raise:

J–Q–K–A

Leave the current
round and wait for the
next one to begin

If no amount has been
bet yet, a player can
check and pass the turn
to the next player

If an amount has
already been bet, a
player can raise
that amount to a
new total

J: Jack
Q: Queen
K: King
A: Ace

Figure 1

1

There are four turns in a round of Texas Hold’em:
Turn 1: Each player receives two cards face down
...
Then it is the next player’s turn, and they can
either bet, fold, check, or raise, depending on the previous player’s actions
...
Each player now has a five
card combination from their own hand that no one but themselves can see, and the three
community cards
...

Turn 3: A fourth community card is dealt
...
Then the betting begins
...
Each player can now make a five card combination from
their own hand and the five community cards
...


Each player bases their bet
on how good their
combination of cards is,
where 1 is the best
combination and 10 is the
worst on the left
...
Figure 2 above contains a diagram listing the rankings of
the different combination of cards
...
The other rankings only depend on the number of the cards
...
If it can, you would bet a high amount, and
if cannot, you would fold
...
g
...
is much higher than it actually is, and so they blindly bet a high
amount and lose
...
I will then graph this on a bar chart to give
a realistic view of the probability of getting a high value five card combination and mathematically
show that you should not bet high in the first turn of Texas Hold’em
...
The suits of the
cards does not affect the probabilities of the first six columns, which is why the suits of the cards
are given right before the last three columns, whose probabilities are affected by the suits of the
card as explained before
...
D means that they
are of different suits, and S means that they are the same suit, e
...
two hearts
...
Therefore, the probability of getting a flush, straight flush
or royal flush is 0 before turn 2 if you have two of the same numbers in your hand
...
Therefore, I have
omitted the probability for a hand of different suits, as they would all equal zero, and this
exploration only focuses on turn 1 and the probabilities for turn 2
...

Hand

Pair

Two Pair

cards
2-2
2-3
2-4
2-5

%
100
33,09
33,09
33,09

%
16,90
1,010
1,010
1,010

Three of a
Straight Full House Four of a Kind
Kind
%
%
%
%
11,51
0
0,7347
0,2449
1,446
0,3265 0,09184
0,03061
1,446
0,3265 0,09184
0,03061
1,446
0,3265 0,09184
0,03061

Suit

Flush Straight Flush Royal Flush

D/S
D
S
S
S

%
0
2,526
2,526
2,526

%
0
0,005102
0,005102
0,005102

%
0
0
0
0
3

2-6
2-7
2-8
2-9
2 - 10
2-J
2-Q
2-K
2-A

33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09

1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010

1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446

0,1633
0
0
0
0
0
0
0
0,1633

0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184

0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061

S
S
S
S
S
S
S
S
S

2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526

0,002551
0
0
0
0
0
0
0
0,002551

0
0
0
0
0
0
0
0
0

3-2
3-3
3-4
3-5
3-6
3-7
3-8
3-9
3 - 10
3-J
3-Q
3-K
3-A

33,09
100
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09

1,010
16,90
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010

1,446
11,51
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446

0,3265
0
0,3265
0,3265
0,3265
0,1633
0
0
0
0
0
0
0,1633

0,09184
0,7347
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184

0,03061
0,2449
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061

S
D
S
S
S
S
S
S
S
S
S
S
S

2,526
0
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526

0,002551
0
0,005102
0,005102
0,005102
0,002551
0
0
0
0
0
0
0,002551

0
0
0
0
0
0
0
0
0
0
0
0
0

4-2
4-3
4-4
4-5
4-6
4-7
4-8
4-9
4 - 10
4-J
4-Q
4-K
4-A

33,09
33,09
100
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09

1,010
1,010
16,90
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010

1,446
1,446
11,51
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446

0,3265
0,3265
0
0,3265
0,3265
0,3265
0,1633
0
0
0
0
0
0,1633

0,09184
0,09184
0,7347
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184

0,03061
0,03061
0,2449
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061

S
S
D
S
S
S
S
S
S
S
S
S
S

2,526
2,526
0
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526

0,005102
0,005102
0
0,005102
0,005102
0,005102
0,002551
0
0
0
0
0
0,002551

0
0
0
0
0
0
0
0
0
0
0
0
0

5-2
5-3
5-4
5-5
5-6
5-7
5-8
5-9
5 - 10
5-J
5-Q
5-K

33,09
33,09
33,09
100
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09

1,010
1,010
1,010
16,90
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010

1,446
1,446
1,446
11,51
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446

0,3265
0,3265
0,3265
0
0,3265
0,3265
0,3265
0,1633
0
0
0
0

0,09184
0,09184
0,09184
0,7347
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184

0,03061
0,03061
0,03061
0,2449
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061

S
S
S
D
S
S
S
S
S
S
S
S

2,526
2,526
2,526
0
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526

0,005102
0,005102
0,005102
0
0,005102
0,005102
0,005102
0,002551
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
4

5-A

33,09

1,010

1,446

0,1633

0,09184

0,03061

S

2,526

0,002551

0

6-2
6-3
6-4
6-5
6-6
6-7
6-8
6-9
6 - 10
6-J
6-Q
6-K
6-A

33,09
33,09
33,09
33,09
100
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09

1,010
1,010
1,010
1,010
16,90
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010

1,446
1,446
1,446
1,446
11,51
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446

0,1633
0,3265
0,3265
0,3265
0
0,3265
0,3265
0,3265
0,1633
0
0
0
0

0,09184
0,09184
0,09184
0,09184
0,7347
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184

0,03061
0,03061
0,03061
0,03061
0,2449
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061

S
S
S
S
D
S
S
S
S
S
S
S
S

2,526
2,526
2,526
2,526
0
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526

0,002551
0,005102
0,005102
0,005102
0
0,005102
0,005102
0,005102
0,002551
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0

7-2
7-3
7-4
7-5
7-6
7-7
7-8
7-9
7 - 10
7-J
7-Q
7-K
7-A

33,09
33,09
33,09
33,09
33,09
100
33,09
33,09
33,09
33,09
33,09
33,09
33,09

1,010
1,010
1,010
1,010
1,010
16,90
1,010
1,010
1,010
1,010
1,010
1,010
1,010

1,446
1,446
1,446
1,446
1,446
11,51
1,446
1,446
1,446
1,446
1,446
1,446
1,446

0
0,1633
0,3265
0,3265
0,3265
0
0,3265
0,3265
0,3265
0,1633
0
0
0

0,09184
0,09184
0,09184
0,09184
0,09184
0,7347
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184

0,03061
0,03061
0,03061
0,03061
0,03061
0,2449
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061

S
S
S
S
S
D
S
S
S
S
S
S
S

2,526
2,526
2,526
2,526
2,526
0
2,526
2,526
2,526
2,526
2,526
2,526
2,526

0
0,002551
0,005102
0,005102
0,005102
0
0,005102
0,005102
0,005102
0,002551
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0

8-2
8-3
8-4
8-5
8-6
8-7
8-8
8-9
8 - 10
8-J
8-Q
8-K
8-A

33,09
33,09
33,09
33,09
33,09
33,09
100
33,09
33,09
33,09
33,09
33,09
33,09

1,010
1,010
1,010
1,010
1,010
1,010
16,90
1,010
1,010
1,010
1,010
1,010
1,010

1,446
1,446
1,446
1,446
1,446
1,446
11,51
1,446
1,446
1,446
1,446
1,446
1,446

0
0
0,1633
0,3265
0,3265
0,3265
0
0,3265
0,3265
0,3265
0,1633
0
0

0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,7347
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184

0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,2449
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061

S
S
S
S
S
S
D
S
S
S
S
S
S

2,526
2,526
2,526
2,526
2,526
2,526
0
2,526
2,526
2,526
2,526
2,526
2,526

0
0
0,002551
0,005102
0,005102
0,005102
0
0,005102
0,005102
0,005102
0,002551
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0

9-2
9-3
9-4
9-5
9-6
9-7

33,09
33,09
33,09
33,09
33,09
33,09

1,010
1,010
1,010
1,010
1,010
1,010

1,446
1,446
1,446
1,446
1,446
1,446

0
0
0
0,1633
0,3265
0,3265

0,09184
0,09184
0,09184
0,09184
0,09184
0,09184

0,03061
0,03061
0,03061
0,03061
0,03061
0,03061

S
S
S
S
S
S

2,526
2,526
2,526
2,526
2,526
2,526

0
0
0
0,002551
0,005102
0,005102

0
0
0
0
0
0
5

9-8
9-9
9 - 10
9-J
9-Q
9-K
9-A

33,09
100
33,09
33,09
33,09
33,09
33,09

1,010
16,90
1,010
1,010
1,010
1,010
1,010

1,446
11,51
1,446
1,446
1,446
1,446
1,446

0,3265
0
0,3265
0,3265
0,3265
0,1633
0

0,09184
0,7347
0,09184
0,09184
0,09184
0,09184
0,09184

0,03061
0,2449
0,03061
0,03061
0,03061
0,03061
0,03061

S
D
S
S
S
S
S

2,526
0
2,526
2,526
2,526
2,526
2,526

0,005102
0
0,005102
0,005102
0,005102
0,002551
0

0
0
0
0
0
0
0

10 - 2
10 - 3
10 - 4
10 - 5
10 - 6
10 - 7
10 - 8
10 - 9
10 - 10
10 - J
10 - Q
10 - K
10 - A

33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
100
33,09
33,09
33,09
33,09

1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
16,90
1,010
1,010
1,010
1,010

1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
11,51
1,446
1,446
1,446
1,446

0
0
0
0
0,1633
0,3265
0,3265
0,3265
0
0,3265
0,3265
0,3265
0,1633

0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,7347
0,09184
0,09184
0,09184
0,09184

0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,2449
0,03061
0,03061
0,03061
0,03061

S
S
S
S
S
S
S
S
D
S
S
S
S

2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
0
2,526
2,526
2,526
2,526

0
0
0
0
0,002551
0,005102
0,005102
0,005102
0
0,005102
0,005102
0,005102
0,002551

0
0
0
0
0
0
0
0
0
0,002551
0,002551
0,002551
0,002551

J-2
J-3
J-4
J-5
J-6
J-7
J-8
J-9
J - 10
J-J
J-Q
J-K
J-A

33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
100
33,09
33,09
33,09

1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
16,90
1,010
1,010
1,010

1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
11,51
1,446
1,446
1,446

0
0
0
0
0
0,1633
0,3265
0,3265
0,3265
0
0,3265
0,3265
0,3265

0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,7347
0,09184
0,09184
0,09184

0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,2449
0,03061
0,03061
0,03061

S
S
S
S
S
S
S
S
S
D
S
S
S

2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
0
2,526
2,526
2,526

0
0
0
0
0
0,002551
0,005102
0,005102
0,005102
0
0,005102
0,005102
0,005102

0
0
0
0
0
0
0
0
0,002551
0
0,002551
0,002551
0,002551

Q-2
Q-3
Q-4
Q-5
Q-6
Q-7
Q-8
Q-9
Q - 10
Q-J
Q-Q
Q-K
Q-A

33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
100
33,09
33,09

1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
16,90
1,010
1,010

1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
11,51
1,446
1,446

0
0
0
0
0
0
0,1633
0,3265
0,3265
0,3265
0
0,3265
0,3265

0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,7347
0,09184
0,09184

0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,2449
0,03061
0,03061

S
S
S
S
S
S
S
S
S
S
D
S
S

2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
0
2,526
2,526

0
0
0
0
0
0
0,002551
0,005102
0,005102
0,005102
0
0,005102
0,005102

0
0
0
0
0
0
0
0
0,002551
0,002551
0
0,002551
0,002551

6

K-2
K-3
K-4
K-5
K-6
K-7
K-8
K-9
K - 10
K-J
K-Q
K-K
K-A

33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
100
33,09

1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
16,90
1,010

1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
11,51
1,446

0
0
0
0
0
0
0
0,1633
0,3265
0,3265
0,3265
0
0,3265

0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,7347
0,09184

0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,2449
0,03061

S
S
S
S
S
S
S
S
S
S
S
D
S

2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
0
2,526

0
0
0
0
0
0
0
0,002551
0,005102
0,005102
0,005102
0
0,005102

0
0
0
0
0
0
0
0
0,002551
0,002551
0,002551
0
0,002551

A-2
A-3
A-4
A-5
A-6
A-7
A-8
A-9
A - 10
A-J
A-Q
A-K
A-A

33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
33,09
100

1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
1,010
16,90

1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
1,446
11,51

0
0
0
0
0
0
0
0
0,1633
0,3265
0,3265
0,3265
0

0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,09184
0,7347

0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,03061
0,2449

S
S
S
S
S
S
S
S
S
S
S
S
D

2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
2,526
0

0
0
0
0
0
0
0
0
0,002551
0,005102
0,005102
0,005102
0

0
0
0
0
0
0
0
0
0,002551
0,002551
0,002551
0,002551
0

The values above have all been rounded to 4 significant figures, as any more figures would not
change the percentage enough to be relevant for this exploration
...
However, there are fifteen unique
calculations, which I have explained below by giving sample calculations of the actual probabilities
above
...


𝑷𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚(𝟐 − 𝟑, 𝐏𝐚𝐢𝐫)
3
47
46
= (( ) × ( ) × ( )) × 3 × 2
50
49
48
× 100% = 33
...
09%

To get a pair with a hand of a 2 and a 3, one of the three community
cards need to be a 2 or a 3
...
Then a 47/49
chance of getting anything but a 2, since if you get another 2, you
would have a two pair
...

This is then multiplied by three as the second 2 can show up in any
order, and then it is multiplied by two as you could also form a pair
with the 3 in your hand
...
8980% = 16
...
01020% = 1
...
5102% = 11
...
44612% = 1
...
326531% = 0
...
To get another pair, there is a
4/50 chance of getting a specific number, then a 3/50 chance of
getting that number again, and then a 46/48 chance of getting any
number but that specific number
...

3/50 chance of getting another 2, then a 3/50 chance of getting
another 3, then a 44/48 chance of getting anything but a 2 or 3
...


You already have two 2’s
...

These two probabilities are then added and result in the final
probability
...
This is
then multiplied by three as the cards could show up in any order,
and then multiplied by two, as 3 could also be used to form a three
of a kind instead of 2
...
This is then multiplied by twelve,
as any of the thirteen numbers in the deck, except two, can act as
the three of a kind
...

One way to form a straight, is if the community cards are 4, 5 and 6
...

Another way is if the community cards are A, 4 and 5
...

These two probabilities are then added
...
163265% = 0
...

There is a 4/50 chance of getting a 3, then a 4/49 chance of getting
a 4, then a 4/48 chance of getting a 5
...
734694% = 0
...
There is a 4/50 chance to get a specific number, then a 3/49
to get that number again, then a 2/48 chance to get that number
again
...


𝑷𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚(𝟕 − 𝟖, 𝐅𝐮𝐥𝐥 𝐇𝐨𝐮𝐬𝐞)
3
3
2
= (( ) × ( ) × ( )) × 3 × 2
50
49
48
× 100% = 0
...
09184%

To get a full house, you need another 7 and another two 8’s
...
Multiplied by three due
to the order, then by two as one 8 and two 7’s also works
...
244898% = 0
...
Multiplied by three due to the order
...
0306122% = 0
...
Multiplied by 3 due to the order,
then by 2, as three 9’s can also form a four of a kind
...
52551% = 2
...
00510204% = 0
...
Since you already have two
cards of the same suit, there are eleven other cards with that suit
...

One way to get a straight flush is to get a 4, 5 and 6 of the same suit
...
Multiplied by three due to the order
...


𝑷𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚(𝟐 − 𝟔, 𝐒𝐭𝐫 𝐅𝐥𝐮𝐬𝐡, 𝐒 𝐒𝐮𝐢𝐭)
1
1
1
= (( ) × ( ) × ( )) × 3 × 100%
50
49
48
= 0
...
002551%

Only way to form a straight flush, is if the community cards are 3, 4
and 5 of the same suit
...


𝑷𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚(𝟏𝟎 − 𝐉, 𝐑𝐲𝐥 𝐅𝐥𝐮𝐬𝐡, 𝐒 𝐒𝐮𝐢𝐭)
1
1
1
= (( ) × ( ) × ( )) × 3 × 100%
50
49
48
= 0
...
002551%

Only way to form a royal flush, is if the community cards are Q, K
and A of the same suit
...


9

Charts

Figure 3 on the left contains a bar chart that I
have created, showing the probabilities of
getting all the different poker combinations
with a hand of 10 – J
...


Figure 3

Figure 4 below contains a bar chart of the probabilities of a hand of a 10 and every other number,
all with the same suit
...
As you can see, they vary from very small to very large, so I have limited the y-axis
to only show up to 3% so the smaller probabilities are visible, and then written the values for the
probabilities that exceed this limit
...
The only differ if you have
the same numbers, 10 – 10 in this case
...
To put that into real life perspective, if you had a 10 – J hand, mathematically you would have
to play 39,200 rounds of Texas Hold’em before you have a royal flush in turn 2 with that hand
...
On the other hand,
having one of the biggest value hands in Texas Hold’em, e
...
10 – A, has the same chance of getting
a pair, two pair, three of a kind, full house, four of a kind, flush as every other hand except 10 – 10
...
002551% chance of being extremely lucky
...
However, as I have
shown by listing out the probabilities for every single possible combination of cards in Texas Hold’em
by turn 2, the probability to get a very good card combination, is extremely small, which can
practically be seen as nothing
...
This shows that betting high beyond turn 2, e
...
in
turn 3 or turn 4, is mathematically more beneficial, as there are more cards to make a combination
from
...
The strategy of betting high on turn 1 could be used to make others think you have good
cards and cause them to fold, making you win whatever money was on the table
...
This is where Texas Hold’em
gets really interesting, but combining the mathematical aspect of poker with the psychological
aspect is way beyond the focus of this exploration
...
" Gaming Nites
...
6 Jan
...

...
com/poker-hand-rankings/>

---

Title: Poker Mathematics
Description: A complete exploration of mathematical probabilities in Texas Hold’em.

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