Probabilities In Texas Hold’em Poker Research Paper

Introduction

This paper is intended to explain the math of probability in the most popular at present varieties of poker - Texas Hold'em. Mathematics in poker is fairly simple and does not require deep knowledge in this field for successful application in practice. However, the basic knowledge of statistics and probability theory is necessary.
In Texas Hold'em, it is very important to know a type and a number of possible combinations of starting hands with whom or against whom you have to play. Playing poker, you have to anticipate, which possible hands could go against you opponent and from your guesses will depend the decision. The understanding and representation of the overall picture of possible starting hands will help to avoid unpleasant situations in preflop.

Body

The first thing to remember, starting to play poker, is the rules of the game and combinations. The rules define the game for just such a deck of 52 cards:

Probability Distribution of Starting Hands in Texas Hold'em

The probability of passing a given hand can be calculated directly. For Texas Hold'em, which uses a 2 pocket cards, the first card can be one of 52 in the deck, the second - one of the remaining 51. The number of possible combinations is 52 * 51/2 = 1326. Another way to calculate the number of combinations is to use the binomial coefficient: C (52,2) = 1326. It is calculated in several ways, but the most convenient to use the following formula:
C(n,k) = n! / (k! * (n - k)!)
where n! is factorial. Factorial is the product of a series of numbers, each of which is smaller than subsequent to the previous one (e.g. 2! = 2 * 1 = 2, 5! = 5 * 4 * 3 * 2 * 1 = 120, etc.)

In our case, we get:

C (52, 2) = 52! / (2! * (52 - 2)!) = 52! / (2! * 50!)
The expression can be reduced by 50!, because 52! = 52 * 51 * 50!, then eventually we get:
Although the total number of possible starting hands is still 1326, in preflop (but not in postflop) it does not matter what color the card is in his hand. Therefore, the amount of net combinations is 169 (13 pocket pairs, 78 suited and 78 assorted cards). The probability distribution of a pocket pair is:
13 * C (4,2) / 1326 = 13 * 6/1326 = 78/1326 = 0.0588 or 17:1,
where C (4,2) is a number of ways in which you can collect a couple of four suits, for example as for kings К♠К♦, К♠К♣, К♠К♥, K♦K♣, K♦K♥ and K♣K♥.

Probability distribution of suited cards is:

C * 78 (4.1) / 1326 = 78 * 4/1326 = 0.2353 or 4.25:1

The probability distribution of the unpaired suited cards is:

78 * C (4,1) * C (3.1) / 1326 = 936/1326 = 0.7059 or 0.417:1
Perform the further calculations similar to the given above and create a table of probabilities of different hands in Texas Hold'em. The probabilities of different starting combinations are given below:

Analysis of situations when you play one-on-one in hold'em (heads-up)

In poker, the most important part of the game is when you are alone with your opponent. Therefore, it is useful to represent how likely your opponent will have a particular starting hand.
When you already have on hand is a combination of the two cards, the opponent may have one of the 50 * 49/2 = 1225 remaining possible hands. From this we can get the total number of possible comparisons 1 to 1:
C (52,2) * C (50,2) / 2 = 812 175
Division by 2 in the previous terms is due to the fact that we consider the comparison, for example, when you hold AA and your opponent holds KK, and vice versa. But since the number of "pure" hand in Hold'em only 169, the overall number of "pure" comparison equals 169 * 1225 = 207 025. The number of unique boards of five cards is:
С(48,5) = 1 712 304

Knowing this number can get the total number of possible outcomes in a heads-up:

207 025 * 1 712 304 = 354 489 735 600

The probabilities of different comparisons in heads-up

Here are some of the most interesting games for the probability of 1 vs 1 in Texas Hold'em. In the following table we consider situations such as favorite against underdog for the most important types of comparisons, where the columns of probability and chance is given the probability of winning favorite in the hand (it should be noted that this average value as the probability of winning will depend on various factors, such as suited, connectivity, rank cards, etc.):

Starting hands in multiway

The number of possible starting hands in 2 or more players is high and equal to:
C (50,2) * C (48,2) = 1381800
However, this is the total number of possible hand, from which should be excluded a situation where one player holds, for example, AA, and the other holds AK and vice versa. To do this, divide the number of possible starting hands by the factorial of the number of opponents (for example, 2 players for 2!, 3 - 3!, Etc.). We obtain:
Two opponens - C (50,2) * C (48,2) * 2! = 690,900
Three: C (50,2) * C (48,2) * C (46,2) * 3! = 238360500
As we can see, even when playing against 3, the number of possible situations is enormous. For clarity, summarizes the results for different number of opponents in the following table, the number of possible combinations for different number of opponents:

The total number of possible situations on the flop when playing against 9 opponents is huge:

169 * C(50,18) * 17!! * C(32,5) = 2,117 x 1028

The probability of Hands Dominance in Texas Hold'em

For poker players will be helpful to think of what the probability that the opponent has a hand that dominates his pocket pair. In this article, we consider only two situations - the likelihood that your pocket pair to dominate, and the likelihood that your hand with an ace has a worse kicker than your opponent. These probabilities we reduce for the simplicity and clarity in the table.

The probability of dominance hand with an ace:

Conclusion
The paper discusses some of the mathematical aspects of starting hands in Texas Hold'em. As we can see, the understanding of the basic concepts of probability theory is very useful to estimate the changes of the winning. Hence, this knowledge helps anyone to become a good poker player. Texas Hold'em is the perfect game for strategic and mathematical analysis; for the application of the theory of probability and mathematical statistics in practice. The basic principles that should be enjoyed by all, playing Texas Hold'em is the adoption of a mathematically correct (plus) solutions for obtaining a positive result in the long run. Do not try to win every hand, taking minus solutions and remaining thus, in the end, in the red.

Works Cited

Sklansky, David (2005). The Theory of Poker (Fourth ed.). Las Vegas: Two plus two.
Brunson, Doyle (1978). Super/System: A course in power poker. B&G Publishing Company., emphasis in original
"Texas Hold'em Rules". WorldSeriesOfPoker.com. Retrieved August 16, 2009.
Harrington, Dan and Bill Robertie (2004). Harrington on Hold'em: Expert Strategy For No-Limit Tournaments; Volume I: Strategic Play. Two Plus Two Publications. ISBN 1-880685-33-7.