Poker Odds Probabilities
Some people may argue that poker is a game of chance, instead of a skills game. And you know what? These people are actually right, since most of the time you will not be able to predict the outcome of the next card which will determine if you’ll win – or lose a hand.
- Poker Odds Probability Calculator
- Poker Odds And Probabilities
- Texas Holdem Poker Odds And Probabilities
- Poker And Probability
Each of the 2,598,960 possible hands of poker is equally likely when dealt 5 cards from a standard poker deck. Because of this, one can use probability by outcomes to compute the probabilities of each classification of poker hand. The binomial coefficient can be used to. While you don't have to be a mathematical genius to play poker (although it can certainly help), it is worth knowing some of the basic odds, probabilities and percentages behind the various hands. A poker odds calculator shows you the exact odds of your hand winning in any scenario.
Poker odds give you the probability of winning any given hand. Higher odds mean a lower chance of winning, meaning that when the odds are large against you it’ll be a long time until you succeed. Probabilities in poker Probability means the degree of certainty that a possible event will occur. The classic definition is: 'The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which.
But knowing how to calculate odds and probabilities in poker will help you ‘guessing’ the next card based on a few simple formulas which are described below. If you play more than just a few hands of poker you will learn to understand that skill is actually a huge part of the game, and luck becomes a lot less significant.
Knowing this, poker indeed is a game of skill. If poker is played seriously, skill simply dominates luck.
Here are some related pages and posts:
Probabilities
The following calculates the frequency of each poker hand for all combinations that can be made out of 5 cards drawn at random from a full deck of 52 playing cards, without the use of wild cards.
The probability in poker is determined based on the number 2,598,960, which represents the total number of five card combinations that can be created. In this case, the probability is the frequency of a hand divided by the total number of five card hands, and the odds are defined by the formula ((1 / p) – 1 : 1), where p corresponds to the probability.
Frequency of Five Card Poker hands
In the following table, the frequencies are precise, while the probabilities and the odds given are approximate.
Also, the royal flush is included in the table as a straight flush, since the royal flush can be made in four different ways (hearts, spades, clubs and diamonds). This calculates a probability of 0.000001539077169 and odds of 649,739 : 1.
| Hand | Frequency | Probability | Odds against |
|---|---|---|---|
| Straight flush | 40 | 0.00154 % | 64,973 : 1 |
| Four of a kind | 624 | 0.0240 % | 4,164 : 1 |
| Full house | 3,744 | 0.144 % | 693 : 1 |
| Flush | 5,108 | 0.197 % | 508 : 1 |
| Straight | 10,200 | 0.392 % | 254 : 1 |
| Three of a kind | 54,912 | 2.11 % | 46.3 : 1 |
| Two pair | 123,552 | 4.75 % | 20.0 : 1 |
| One pair | 1,098,240 | 42.3 % | 1.37 : 1 |
| No pair | 1,302,540 | 50.1 % | 0.995 : 1 |
| Total | 2,598,960 | 100 % | 0 : 1 |
The probability of a straight flush or an ace low straight will be reduced in case they won’t be counted. They will become 9/10 as common as they usually would be.
This pages will tell you more about card outs:
Pot odds
In poker, pot odds are used to determine the expected value of a play. Poker players use pot odds to calculate the profitability over the long term. Generally, odds are expressed as a win:loss ratio. The following formula can be used to calculate the percentage probability:
win:loss odds = win / (win + loss) % probability
For instance, this means that 1:4 odds equals 1 / (1 + 4) = 20% probability.
Instead, odds are also expressed as a loss:win ratio, or also known as odds against (see table above). To make it totally clear, we have used win:loss odds in remainder of this explanation.
The correct poker playing strategy is influenced by pot odds for every possible table action (check, fold, call or raise) at every single moment during a poker game.
Most likely, calling is the correct play when facing a call with higher pot odds, just like folding is the correct play when facing a call with lower pot odds.
For instance, if you can win a $1000 pot by making a call for just $1, essentially no hand should be folded, since you only have to win this call one time every thousand hands for it to be profitable.
The probability of winning is equal to the chance that a player will win with either the best hand at showdown or simply because the opponents will fold out of the hand before opening up.
Be sure to also read:
Pot odds in Texas Hold’em Poker
The approximate percentage probability that a player hits one of his outs on the first coming card in Texas Hold’em is calculated as follows:
(number of outs) x 2 + 1
In case you have the outs for a potential flush, i.e. 9 cards could improve your hand, then you’ll have about 19% chance to complete your flush with the next card (9 x 2 + 1 = 19).
Poker Odds Probability Calculator
With two more cards to come (the turn and the river), the approximate percentage probability is calculated as follows:
(number of outs) x 4 – 1
This means that you’ll have 35% roughly to hit a flush with two more cards to come (9 x 4 – 1 = 35).
For a correctly classified table action, the odds for winning a hand should never be lower than the applicable pot odds.
One of the games that have seen a flurry of interest over the last few months is Six Plus Hold’em, also referred to as Short Deck Poker.
Six Plus Hold’em is an exciting and fun poker variant based on Texas Hold’em where the game is played with a deck of 36 cards as opposed to the usual 52 cards in traditional hold’em. Deuces through fives are removed from the deck giving the game its name Six Plus Hold’em/6+ or Short Deck Poker.
Aces are played both low and high, making both a low-end straight A6789 and the high JQKTA. Also, with a shortened deck, the game changes a bit in terms of hand rankings and rules. A Flush beats a Full House and in most places where Six Plus is offered, a Set or a Three-of-a-Kind beats a Straight.
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Because the low cards are removed, there are more playable hands compared with traditional Hold’em, and so it is more of an action-orientated game. Not only are the hand rankings modified but so are the mathematics and odds/probabilities of the majority of hands.
Before we talk about the odds and probabilities of some of the hands, let’s have a look at the hand rankings offered in Six Plus Hold’em (ranked from the highest hand to the lowest):
Six Plus Hold’em Hand Rankings Comparison
| Traditional Hold’em | 6+ Plus Hold’em (Trips beat Straight) | 6+ Plus Hold’em (Straight beat Trips) |
|---|---|---|
| Royal Flush | Royal Flush | Royal Flush |
| Straight Flush | Straight Flush | Straight Flush |
| Four of a Kind | Four of a Kind | Four of a Kind |
| Full House | Flush | Flush |
| Flush | Full House | Full House |
| Straight | Three-of-a-Kind | Straight |
| Three-of-a-Kind | Straight | Three-of-a-Kind |
| Two Pair | Two Pair | Two Pair |
| One Pair | One Pair | One Pair |
| High Card | High Card | High Card |
Poker Odds And Probabilities
One may wonder why a Flush is ranked higher than a Full House or why Three-of-a-Kind is ranked above a Straight. That’s because in Six Plus Hold’em, a Flush is harder to make since there are only nine cards in each suit instead of thirteen. Similarly, the stripped-deck also means that the remaining 36 cards are much closer in rank and so there will be smaller gaps between the cards in the hand and those on the board. This increases the probability of a hand becoming a Straight and hence Straights are ranked higher than a Three-of-a-Kind.
Texas Holdem Poker Odds And Probabilities
However, it is worth noting that the rules vary from game to game. For example, in the Short Deck variant offered in the Triton Poker Series, a Straight is ranked higher than a Three-of-a-Kind like in traditional hold’em even though mathematically a player would hit a Straight more.
One of the reasons why an operator would rank a Straight higher than Three-of-a-Kind is because it would generate more action. If Trips were ranked higher, a player with a Straight draw would have no reason to continue the hand as he or she would be drawing dead.
Let’s take a look at the odds/probabilities of hitting some of the hands:
Six Plus Hold’em vs Traditional Hold’em (Odds and Probabilities comparison)
| Traditional Hold’em | Six Plus Hold’em/Short Deck Poker | |
|---|---|---|
| Getting Dealt Aces | 1 in 221 (0.45%) | 1 in 105 (0.95%) |
| Aces Win % vs a Random Hand | 85% | 77% |
| Getting Dealt any Pocket Pair | 5.90% | 8.60% |
| Hitting a Set with a Pocket Pair | 11.80% | 18% |
| Hitting an Open-Ended Straight by the River | 31.50% | 48% |
| Possible Starting Hands | 1326 | 630 |
As you can see in the table above, the odds of being dealt pocket Aces are doubled as you now get the powerful starting hand dealt once in every 105 hands, as opposed to once in every 221 hands with a full 52-card deck. However, the probability of winning a hand with aces vs a random hand decreases from 85% in traditional hold’em to 77% in Six Plus Hold’em.
The probability of hitting a Set with pocket pairs increases to 18% from 11.8%, and the probability of hitting an open-ended Straight by the River also increases to 48% in 6+ Hold’em compared with 31.5% in traditional Hold’em.
Let’s now have a look at some of the pre-flop all-in hand situations:
Six Plus Hold’em vs Traditional Hold’em (Hands Comparison)
| Hand All-in Pre-Flop | Traditional Hold’em | 6+ Hold’em (Trips beat Straight) | 6+ Hold’em (Straight beat Trips) |
|---|---|---|---|
| Ac Ks vs Th Td | 43% vs 57% | 47% vs 53% | 49% vs 51% |
| Ac Ks vs Jc Th | 63% vs 37% | 53% vs 47% | 52% vs 48% |
| As Ah vs 6s 6h | 81% vs 19% | 76% vs 24% | 76% vs 24% |
As mentioned earlier, the equities run very close to each other with the shortened deck and so a hand like Ace-King versus Jack-Ten is almost a coin-flip, whereas the former is a favorite in Texas Hold’em. Again, a hand like Ace-King versus a pocket pair like Tens is a coin-flip in 6+, whereas a pocket pair is a slight favorite in normal Hold’em.
Now, let’s take a look at the probabilities when a connected or wet Flop is dealt:
Player 1: Ac Ks
Player 2: Td 9h
Flop: Kh 8c 7d
| Traditional Hold’em | 6+ Hold’em (Trips beat Straight) | 6+ Hold’em (Straight beat Trips) | |
|---|---|---|---|
| Player 1 vs Player 2 | 66% vs 34% | 52% vs 48% | 48% vs 52% |
In traditional Hold’em, Ace-King is a favorite with 66% and Player 2 is chasing the Straight draw with a close to 34% chance of hitting it. However, the probability significantly changes in both variants of 6+ Hold’em. In a variant where Trips beat a Straight, Player 1 is only a slight favorite with just 52% (more like a coin-flip). However, in a Short Deck game where a Straight beat Trips, Player 2 is now slightly favorite with 52% chance of hitting a Straight by the river.
Another hand:
Player 1: As Ah
Player 2: Qd Jh
Poker And Probability
Flop: Ad Th 9s
| Traditional Hold’em | 6+ Hold’em (Trips Beat a Straight) | 6+ Hold’em (Straight beat Trips) | |
|---|---|---|---|
| Player 1 vs Player 2 | 74% vs 26% | 100% vs 0% | 68% vs 32% |
It’s pretty clear when it comes to normal Hold’em, but in a Short Deck variant where Trips beat a Straight, Player 2 is drawing dead as opposed to the other variant where Player 2 still has a 32% of chance of completing a Straight by the River.