# Poker Probabilities

Working from the probabilities from the previous tables and equations, the probability P of making one of these runner-runner hands is a compound probability and odds of 8. If there is heavy action pre-flop, you have to assume you're either beat, or at best up against AK. The following are some general probabilities about what can occur on the board. Multiply the number of available cards by the number of available cards minus 1, then divide by two. Both the turn and river need to be the same suit, so both outs are coming from a common set of outs—the set of remaining cards of the desired suit.

## POKER PROBABILITIES

I consider that cheap entertainment, and Casino Pauma gives back more than that in player rewards and cashback. There, your only effective costs are your dealer tokes. I computed an outcome distribution over hands, starting out with a Ante bankroll i. The probability density function of the outcome of a single basic strategy UTH hand was calculated, and used to compute the session outcome distribution for a scenarios of bankrolls, goals, and session times.

In each scenario, the simulated 2. Subscribe to comments with RSS. Thank you for doing this. Math is great, but real world info is too.

How are you running these days and how big of swings have you experienced? Then, over five bad sessions, I can lose it all back. On the other hand, I see most all other players steadily lose money by not betting their hands. Hey — great graphs. Is there an error on the 2nd graph — the graph for 20 antes? What a fatalistic analysis on UTH. Making money in No-Limit Texas Hold'em starts with the hands you choose to play and when you choose to play them.

Since a definitive guide on every hand and how and when to play it in every situation would take more words than a novel, this article will touch on the major points of basic pre-flop hands with broad strokes. Although you can write volumes about detailed lines and theories on maximizing profit with this hand, other than folding there is rarely a scenario in which you can ever make a mistake with this hand pre-flop that is.

Even though this is the best starting hand, if the board doesn't improve your hand you only have one pair. Keep this in mind to avoid stacking off to random two pairs and sets. Pocket kings are almost identical to pocket aces pre-flop. Although players have folded KK pre-flop, it's rarely the correct thing to do.

If someone else is dealt AA when you have KK, chances are you're going to get it all in. Don't worry about this, just write it off as a cooler and move on. The same ideas about post-flop play with AA are applicable to KK. On top of the "one pair" concept, you also need to be on the lookout for an ace on the flop. Queens and jacks are right in the middle - below the big pairs and above the marginal pairs. These hands can be some of the trickiest to play. Unlike AA and KK, these hands are very foldable pre-flop in certain situations.

If you're playing at a tight table, where people are only raising with legitimate hands, many players would say that calling after one player raises and another re-raises pre-flop can be a mistake. If there is heavy action pre-flop, you have to assume you're either beat, or at best up against AK.

You only want to continue with these hands if the board improves your hand, or your opponents back off, showing signs of weakness. No set, no bet. The only goal with these hands is to flop a set and double up through the pre-flop raiser holding pocket aces. One Thing to Keep in Mind: The lower your pair, the greater the chance that you will find yourself in a set-over-set situation.

Anytime you flop the under set in a set-over-set situation, you will be lucky if you don't lose your entire stack. For this reason, many players will refuse to play pocket pairs below fives. At a loose table, these hands are great for raising when you have position and no one has raised ahead of you. The way to make money with these hands is to trap a loose opponent with the same top pair, weak kicker. The most important thing to keep in mind with hands such as K-Q or A-J is you almost never want to call a raise with these hands.

These hands are the most commonly dominated hands when faced with a raise, and as such will lose you significant money if you get into the habit of calling raises with them. Much like AA and KK, you need to remember that one pair is a hand easily beaten.

If your opponent is a very tight player there is little chance he will be putting in large bets against you if he can't beat top pair. Suited connectors can be some of the most valuable hands in No Limit Hold 'em cash games.

That being said, they aren't sure things and will miss everything far more often than they will hit it big. In middle to late position you want to play these hands with due diligence.

You don't want to be calling large raises to play these hands heads up. Your goal with these hands is to play the largest pots possible for the least amount of investment possible. You need great odds to make money on these. A dominated hand is a hand that is beaten by another hand the dominant hand and is extremely unlikely to win against it.

Often the dominated hand has only a single card rank that can improve the dominated hand to beat the dominant hand not counting straights and flushes. For example, KJ is dominated by KQ —both hands share the king, and the queen kicker is beating the jack kicker. Barring a straight or flush, the KJ will need a jack on the board to improve against the KQ and would still be losing if a queen appears on the board along with the jack. A pocket pair is dominated by a pocket pair of higher rank. Barring a straight or flush, a pocket pair needs to make three of a kind to beat a higher pocket pair.

See the section "After the flop" for the odds of a pocket pair improving to three of a kind. To calculate the probability that another player has a higher pocket pair, first consider the case against a single opponent. The probability that a single opponent has a higher pair can be stated as the probability that the first card dealt to the opponent is a higher rank than the pocket pair and the second card is the same rank as the first. Subtracting the two cards for the pocket pair leaves 50 cards in the deck.

After the first card is dealt to the player there are 49 cards left, 3 of which are the same rank as the first. So the probability P of a single opponent being dealt a higher pocket pair is.

The following approach extends this equation to calculate the probability that one or more other players has a higher pocket pair. Where n is the number of other players still in the hand and P m a is the adjusted probability that multiple opponents have higher pocket pairs, then the probability that at least one of them has a higher pocket pair is.

The calculation for P m a depends on the rank of the player's pocket pair, but can be generalized as. The following table shows the probability that before the flop another player has a larger pocket pair when there are one to nine other players in the hand. The following table gives the probability that a hand is facing two or more larger pairs before the flop.

From the previous equations, the probability P m is computed as. From a practical perspective, however, the odds of out drawing a single pocket pair or multiple pocket pairs are not much different. In both cases the large majority of winning hands require one of the remaining two cards needed to make three of a kind. The real difference against multiple overpairs becomes the increased probability that one of the overpairs will also make three of a kind.

When holding a single ace referred to as Ax , it is useful to know how likely it is that another player has a better ace —an ace with a higher second card. The weaker ace is dominated by the better ace. The probability that a single opponent has a better ace is the probability that he has either AA or Ax where x is a rank other than ace that is higher than the player's second card. When holding Ax , the probability that a chosen single player has AA is.

If the player is holding Ax against 9 opponents, there is a probability of approximately 0. The following table shows the probability that before the flop another player has an ace with a larger kicker in the hand. The value of a starting hand can change dramatically after the flop. Regardless of initial strength, any hand can flop the nuts—for example, if the flop comes with three 2 s, any hand holding the fourth 2 has the nuts though additional cards could still give another player a higher four of a kind or a straight flush.

By the turn the total number of combinations has increased to. The following are some general probabilities about what can occur on the board. These assume a " random " starting hand for the player.

It is also useful to look at the chances different starting hands have of either improving on the flop, or of weakening on the flop. One interesting circumstance concerns pocket pairs. When holding a pocket pair, overcards cards of higher rank than the pair weaken the hand because of the potential that an overcard has paired a card in an opponent's hand. The hand gets worse the more overcards there are on the board and the more opponents that are in the hand because the probability that one of the overcards has paired a hole card increases.

To calculate the probability of no overcard, take the total number of outcomes without an overcard divided by the total number of outcomes.

The number of outcomes without an overcard is the number of combinations that can be formed with the remaining cards, so the probability P of an overcard on the flop is. The following table gives the probability that no overcards will come on the flop, turn and river, for each of the pocket pairs from 3 to K. Notice, though, that those probabilities would be lower if we consider that at least one opponent happens to hold one of those overcards.

During play—that is, from the flop and onwards—drawing probabilities come down to a question of outs. All situations which have the same number of outs have the same probability of improving to a winning hand over any unimproved hand held by an opponent. For example, an inside straight draw e. Each can be satisfied by four cards—four 5 s in the first case, and the other two 6 s and other two kings in the second. The probabilities of drawing these outs are easily calculated.

The cumulative probability of making a hand on either the turn or river can be determined as the complement of the odds of not making the hand on the turn and not on the river. For reference, the probability and odds for some of the more common numbers of outs are given here.

With 20 or more outs, a hand is a better than 2: Many poker players do not have the mathematical ability to calculate odds in the middle of a poker hand.

One solution is to just memorize the odds of drawing outs at the river and turn since these odds are needed frequently for making decisions. Another solution some players use is an easily calculated approximation of the probability for drawing outs, commonly referred to as the "Rule of Four and Two". This approximation gives roughly accurate probabilities up to about 12 outs after the flop, with an absolute average error of 0.

This is easily done by first multiplying x by 2, then rounding the result to the nearest multiple of ten and adding the 10's digit to the first result. This approximation has a maximum absolute error of less than 0.