The Mathematics Of Limit Holdem

Before understanding the mathematics of Limit Texas Holdem it is imperative that you understand how to play the game. Limit Texas Holdem is a game generally played with two to ten people. There are four betting rounds: pre-flop, the flop, the turn, and the river. In Texas Holdem each player is dealt two cards face down. The play begins with a pre-flop betting round in which the player to the left of the big blind is first to act. Then, in the next betting round the dealer deals the community cards or the first three shared cards. On fourth street or the turn the players are allowed to bet again and on the river or fifth street is the last betting round. The player with the best 5 card combination with any combination of the hole cards and community cards wins. The objective of the game is to win the maximum when you got a winning hand and lose the minimum when you have a losing hand. That is easier said than done for most people. The blinds are forced bets to encourage action. In a $5/$10 limit game the small blind must put in $5 and the big blind must put in $10. In the pre-flop and flop the betting is usually only the amount of the small blind. On the turn and river you may bet the big blind. The rank of the hands is high card (no pair), pair (two of the same card), 2-pair (two of the same card 2 times), 3 of a kind (3 of the same card), straight (5 consecutive cards), flush (5 cards of the same suit), full-house (3 of a kind + pair), four of a kind (4 of same card), straight flush(5 consecutive cards of same suit), royal flush(5 consecutive cards of same suit from 10 to Ace). The rank of the cards are from {2,3,4,5,6,7,8,9,10,Jack,Queen,King,Ace} from worst to best.

In limit holdem poker, understanding mathematics is essential for survival. The difference between a limit game and a no-limit game is that in a limit your raises are capped and in no-limit you may bet any amount. In no-limit poker psychology is much more important than mathematics. Most of your mathematically defined decisions will be based on weight your hand odds against your pot odds. Hand odds simply mean your odds of having or making the best hand. Pot odds simple mean the amount of money in the pot compared to the amount of money it costs to contest it. Some important concepts I will discuss are expected value, pot odds, implied odds, odds, variance, bankroll management, and observing betting patterns,

It is important when you sit down at an online or brick and mortar table that you pick your opponents. You want to sit down next to less skillful and looser opponents. The long term average outcome or expected value of playing against loose and less skillful players is positive. You make money in poker at tables which offer a lot of "action" or betting and raising.

Poker is a game of long term skill and short term luck or variance. A simple definition of variance without going into mathematical jargon is an event that departs from expectation. If you play millions of hands of the course of playing poker the luck will even out and if you capitalized on positive expected value situations you should come out ahead. What do I mean by a positive expected value situation? It's a situation in poker where you will most likely get the best of it. These situations don't come up often so it is important to bet aggressively when you have the best of it.

Understanding the mathematics of pot odds is important for any poker player. Pot odds, as I defined earlier are the amount of money in the pot compared to the amount of money to contest it. For example, in a $5/10 limit holdem game I am dealt {Ac,Kc} and the board comes {Qc,2d,8c}. The pot already contains $100 and your opponent bets $10. It costs $10 to play and you are getting 11-1 pot odds. I calculated that by: ($100+10)/10 which equal 11-1 pot odds. Your hand odds are 4-1 against which means you will improve to the best hand one time in five. We will lose $10 on four attempts and win $110 on one attempt. That means that you will profit $70 dollars over five hands. This comes out to $12 per hand. With these pot odds you only need to win the hand one time in twelve to break even. You will make this call every time in a limit poker game.

The mathematics of counting outs is very important for any poker player. The definition of "outs" is the number of unseen cards that will give you the best hand. If you count the number of cards that will give you the best hand and multiply them by 2 and add one you will get a percentage that you will win the hand. For example when chasing a flush draw, you will have nine unseen cards. So that is, (9 x 2) +1 which means you will make the draw about 19% of the time on the next card.

Suppose in a $5-$10 limit holdem game there is 60 dollars in the pot and a number of callers after the pre-flop betting. You have {6c,6d} and the flop comes out {Ad,5h,Tc}. Everyone checks to the dealer who bets out $5. What is the mathematically correct play here? The pot is offering you $65 to $5 which is 13-1 pot odds. You will win the hand around one chance in twenty. If you convert that to odds that means 19-1 against and 19-1 against means you will break even. This is a fold because even with the chance of hitting the six, there is a chance that any of the players will re-raise you making the odds not in your favor.

Knowing odds (percentage certain plays will win or lose) is also important to a poker player. Four important draws and percentage of hitting that every poker player should know are the flush draw, open ended straight draw, inside straight draw, and pair that needs to hit set. With a flush draw you are about 19% chance to hit on the next card with 9 outs. With an open ended straight draw you have 18% chance to hit on the next card with 8 outs. To hit an inside straight draw or "bellybuster" straight you have 9% chance to hit on the next card and 4 outs. With a pair that needs to hit a set to win you have about a 5% chance to win which is 2 outs.

In poker you have two kinds of hands: drawing hands and made hands. With made hands you have the best hand at the moment but it could get beaten by a drawing hand. A drawing hand is a hand which is not best at the moment but could win by the last card. For example in a limit poker game mathematically it is imperative to play your top two pair aggressively on a board with 2 cards drawing to a flush draw. If the flush does come on the last two cards you can just call it down if you believe your opponent is on that draw.

Another concept that is mathematics is starting hand match-ups. For example in a limit holdem hand the larger pair vs. smaller pair is a 4-1 favorite to win the hand. In the match up of higher pair vs. two lower cards the higher pair is a 4.5-1 favorite. The most classic confrontation