Six-plus hold 'em (also known as Short-deck hold 'em) is a community card poker game variant of Texas hold 'em, where cards 2 through 5 are removed. Each player is dealt two cards face down and seeks make his or her best five card poker hand using from any combination of the seven cards (five community cards and their own two hole cards). From Wikipedia, the free encyclopedia Ultimate Texas Hold 'Em (also known as Ultimate Texas holdem and Ultimate Texas Hold'em) is registered trademark of Bally Gaming, Inc. And refers to a reinvented variant of the classic poker game Texas hold 'em. In this variation, the player does not compete against other players. For example, if after the flop in Texas hold 'em, a player has a backdoor flush draw (e.g., three spades), the probability of catching two outs on the turn and river is (10 ÷ 47) × (9 ÷ 46) = 4.16 percent. Backdoor draws are generally unlikely; with 43 unseen cards, it is equally likely to catch two out of seven outs as to catch one out of one.
The main underpinning of poker is math – it is essential. For every decision you make, while factors such as psychology have a part to play, math is the key element.
In this lesson we're going to give an overview of probability and how it relates to poker. This will include the probability of being dealt certain hands and how often they're likely to win. We'll also cover how to calculating your odds and outs, in addition to introducing you to the concept of pot odds. And finally we'll take a look at how an understanding of the math will help you to remain emotional stable at the poker table and why you should focus on decisions, not results.
What is Probability?
Probability is the branch of mathematics that deals with the likelihood that one outcome or another will occur. For instance, a coin flip has two possible outcomes: heads or tails. The probability that a flipped coin will land heads is 50% (one outcome out of the two); the same goes for tails.
Probability and Cards
When dealing with a deck of cards the number of possible outcomes is clearly much greater than the coin example. Each poker deck has fifty-two cards, each designated by one of four suits (clubs, diamonds, hearts and spades) and one of thirteen ranks (the numbers two through ten, Jack, Queen, King, and Ace). Therefore, the odds of getting any Ace as your first card are 1 in 13 (7.7%), while the odds of getting any spade as your first card are 1 in 4 (25%).
Unlike coins, cards are said to have 'memory': every card dealt changes the makeup of the deck. For example, if you receive an Ace as your first card, only three other Aces are left among the remaining fifty-one cards. Therefore, the odds of receiving another Ace are 3 in 51 (5.9%), much less than the odds were before you received the first Ace.
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Pre-flop Probabilities: Pocket Pairs
In order to find the odds of getting dealt a pair of Aces, we multiply the probabilities of receiving each card:
(4/52) x (3/51) = (12/2652) = (1/221) ≈ 0.45%.
To put this in perspective, if you're playing poker at your local casino and are dealt 30 hands per hour, you can expect to receive pocket Aces an average of once every 7.5 hours.
The odds of receiving any of the thirteen possible pocket pairs (twos up to Aces) is:
(13/221) = (1/17) ≈ 5.9%.
In contrast, you can expect to receive any pocket pair once every 35 minutes on average.
Pre-Flop Probabilities: Hand vs. Hand
Players don't play poker in a vacuum; each player's hand must measure up against his opponent's, especially if a player goes all-in before the flop.
Here are some sample probabilities for most pre-flop situations:
Post-Flop Probabilities: Improving Your Hand
Now let's look at the chances of certain events occurring when playing certain starting hands. The following table lists some interesting and valuable hold'em math:
Many beginners to poker overvalue certain starting hands, such as suited cards. As you can see, suited cards don't make flushes very often. Likewise, pairs only make a set on the flop 12% of the time, which is why small pairs are not always profitable.
PDF Chart
Texas Holdem Table Odds
We have created a poker math and probability PDF chart (link opens in a new window) which lists a variety of probabilities and odds for many of the common events in Texas hold ‘em. This chart includes the two tables above in addition to various starting hand probabilities and common pre-flop match-ups. You'll need to have Adobe Acrobat installed to be able to view the chart, but this is freely installed on most computers by default. We recommend you print the chart and use it as a source of reference.
Odds and Outs
If you do see a flop, you will also need to know what the odds are of either you or your opponent improving a hand. In poker terminology, an 'out' is any card that will improve a player's hand after the flop.
One common occurrence is when a player holds two suited cards and two cards of the same suit appear on the flop. The player has four cards to a flush and needs one of the remaining nine cards of that suit to complete the hand. In the case of a 'four-flush', the player has nine 'outs' to make his flush.
A useful shortcut to calculating the odds of completing a hand from a number of outs is the 'rule of four and two'. The player counts the number of cards that will improve his hand, and then multiplies that number by four to calculate his probability of catching that card on either the turn or the river. If the player misses his draw on the turn, he multiplies his outs by two to find his probability of filling his hand on the river.
In the example of the four-flush, the player's probability of filling the flush is approximately 36% after the flop (9 outs x 4) and 18% after the turn (9 outs x 2).
Pot Odds
Another important concept in calculating odds and probabilities is pot odds. Pot odds are the proportion of the next bet in relation to the size of the pot.
For instance, if the pot is $90 and the player must call a $10 bet to continue playing the hand, he is getting 9 to 1 (90 to 10) pot odds. If he calls, the new pot is now $100 and his $10 call makes up 10% of the new pot.
Experienced players compare the pot odds to the odds of improving their hand. If the pot odds are higher than the odds of improving the hand, the expert player will call the bet; if not, the player will fold. This calculation ties into the concept of expected value, which we will explore in a later lesson.
Bad Beats
A 'bad beat' happens when a player completes a hand that started out with a very low probability of success. Experts in probability understand the idea that, just because an event is highly unlikely, the low likelihood does not make it completely impossible.
A measure of a player's experience and maturity is how he handles bad beats. In fact, many experienced poker players subscribe to the idea that bad beats are the reason that many inferior players stay in the game. Bad poker players often mistake their good fortune for skill and continue to make the same mistakes, which the more capable players use against them.
Decisions, Not Results
One of the most important reasons that novice players should understand how probability functions at the poker table is so that they can make the best decisions during a hand. While fluctuations in probability (luck) will happen from hand to hand, the best poker players understand that skill, discipline and patience are the keys to success at the tables.
A big part of strong decision making is understanding how often you should be betting, raising, and applying pressure.
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This 7+ hour course gives you applicable rules for continuation betting, barreling, raising, and easy ratios so that you ALWAYS have the right number of bluffing combos. Take the guesswork out of your strategy, and begin playing like the top-1%.
Conclusion
A strong knowledge of poker math and probabilities will help you adjust your strategies and tactics during the game, as well as giving you reasonable expectations of potential outcomes and the emotional stability to keep playing intelligent, aggressive poker.
Remember that the foundation upon which to build an imposing knowledge of hold'em starts and ends with the math. I'll end this lesson by simply saying…. the math is essential.
Related Lessons
By Gerald Hanks
Gerald Hanks is from Houston Texas, and has been playing poker since 2002. He has played cash games and no-limit hold'em tournaments at live venues all over the United States.
Related Lessons
Related Lessons
A poker player is drawing if they have a hand that is incomplete and needs further cards to become valuable. The hand itself is called a draw or drawing hand. For example, in seven-card stud, if four of a player's first five cards are all spades, but the hand is otherwise weak, they are drawing to a flush. In contrast, a made hand already has value and does not necessarily need to draw to win. A made starting hand with no help can lose to an inferior starting hand with a favorable draw. If an opponent has a made hand that will beat the player's draw, then the player is drawing dead; even if they make their desired hand, they will lose. Not only draws benefit from additional cards; many made hands can be improved by catching an out — and may have to in order to win.
Outs[edit]
An unseen card that would improve a drawing hand to a likely winner is an out. Playing a drawing hand has a positive expectation if the probability of catching an out is greater than the pot odds offered by the pot.
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Poker Probability Texas Holdem Wikipedia Rules
The probability P1{displaystyle P_{1}} of catching an out with one card to come is:
- P1=outsunseencards{displaystyle P_{1}={frac {mathrm {outs} }{mathrm {unseen} ,mathrm {cards} }}}
The probability P2{displaystyle P_{2}} of catching at least one out with two cards to come is:
- P2=1−nonoutsunseencards×nonouts−1unseencards−1{displaystyle P_{2}=1-{frac {mathrm {non} ,mathrm {outs} }{mathrm {unseen} ,mathrm {cards} }}times {frac {mathrm {non} ,mathrm {outs} -1}{mathrm {unseen} ,mathrm {cards} -1}}}
- nonouts=unseencards−outs{displaystyle mathrm {non} ,mathrm {outs} ={mathrm {unseen} ,mathrm {cards} }-mathrm {outs} }
| Outs | One Card % | Two Card % | One Card Odds | Two Card Odds | Draw Type |
|---|---|---|---|---|---|
| 1 | 2% | 4% | 46 | 23 | Backdoor Straight or Flush (Requires two cards) |
| 2 | 4% | 8% | 22 | 12 | Pocket Pair to Set |
| 3 | 7% | 13% | 14 | 7 | One Overcard |
| 4 | 9% | 17% | 10 | 5 | Inside Straight / Two Pair to Full House |
| 5 | 11% | 20% | 8 | 4 | One Pair to Two Pair or Set |
| 6 | 13% | 24% | 6.7 | 3.2 | No Pair to Pair / Two Overcards |
| 7 | 15% | 28% | 5.6 | 2.6 | Set to Full House or Quads |
| 8 | 17% | 32% | 4.7 | 2.2 | Open Straight |
| 9 | 19% | 35% | 4.1 | 1.9 | Flush |
| 10 | 22% | 38% | 3.6 | 1.6 | Inside Straight & Two Overcards |
| 11 | 24% | 42% | 3.2 | 1.4 | Open Straight & One Overcard |
| 12 | 26% | 45% | 2.8 | 1.2 | Flush & Inside Straight / Flush & One Overcard |
| 13 | 28% | 48% | 2.5 | 1.1 | |
| 14 | 30% | 51% | 2.3 | 0.95 | |
| 15 | 33% | 54% | 2.1 | 0.85 | Flush & Open Straight / Flush & Two Overcards |
| 16 | 34% | 57% | 1.9 | 0.75 | |
| 17 | 37% | 60% | 1.7 | 0.66 |
A dead out is a card that would normally be considered an out for a particular drawing hand, but should be excluded when calculating the probability of catching an out. Outs can be dead for two reasons:
- A dead out may work to improve an opponent's hand to a superior hand. For example, if Ted has a spade flush draw and Alice has an outside straight draw, any spades that complete Alice's straight are dead outs because they would also give Ted a flush.
- A dead out may have already been seen. In some game variations such as stud poker, some of the cards held by each player are seen by all players.
Types of draws[edit]
Flush draw[edit]
A flush draw, or four flush, is a hand with four cards of the same suit that may improve to a flush. For example, K♣ 9♣ 8♣ 5♣ x. A flush draw has nine outs (thirteen cards of the suit less the four already in the hand). If a player has a flush draw in Hold'em, the probability to flush the hand in the end is 34.97 percent if there are two more cards to come, and 19.56 percent (9 live cards divided by 46 unseen cards) if there is only one more card to come.
Outside straight draw[edit]
An outside straight draw, also called up and down, double-ended straight draw or open-ended straight draw, is a hand with four of the five needed cards in sequence (and could be completed on either end) that may improve to a straight. For example, x-9-8-7-6-x. An outside straight draw has eight outs (four cards to complete the top of the straight and four cards to complete the bottom of the straight). Straight draws including an ace are not outside straight draws, because the straight can only be completed on one end (has four outs).
Inside straight draw[edit]
An inside straight draw, or gutshot draw or belly buster draw, is a hand with four of the five cards needed for a straight, but missing one in the middle. For example, 9-x-7-6-5. An inside straight draw has four outs (four cards to fill the missing internal rank). Because straight draws including an ace only have four outs, they are also considered inside straight draws. For example, A-K-Q-J-x or A-2-3-4-x. The probability of catching an out for an inside straight draw is half that of catching an out for an outside straight draw.
Double inside straight draw[edit]
A double inside straight draw, or double gutshot draw or double belly buster draw can occur when either of two ranks will make a straight, but both are 'inside' draws. For example in 11-card games, 9-x-7-6-5-x-3, or 9-8-x-6-5-x-3-2, or in Texas Hold'em when holding 9-J hole cards on a 7-10-K flop. The probability of catching an out for a double inside straight draw is the same as for an outside straight draw.
Other draws[edit]
Sometimes a made hand needs to draw to a better hand. For example, if a player has two pair or three of a kind, but an opponent has a straight or flush, to win the player must draw an out to improve to a full house (or four of a kind). There are a multitude of potential situations where one hand needs to improve to beat another, but the expected value of most drawing plays can be calculated by counting outs, computing the probability of winning, and comparing the probability of winning to the pot odds.
Backdoor draw[edit]
A backdoor draw, or runner-runner draw, is a drawing hand that needs to catch two outs to win. For example, a hand with three cards of the same suit has a backdoor flush draw because it needs two more cards of the suit. The probability Prr{displaystyle P_{rr}} of catching two outs with two cards to come is:
- Prr=outsunseencards×outs−1unseencards−1{displaystyle P_{rr}={frac {mathrm {outs} }{mathrm {unseen} ,mathrm {cards} }}times {frac {mathrm {outs} -1}{mathrm {unseen} ,mathrm {cards} -1}}}
For example, if after the flop in Texas hold 'em, a player has a backdoor flush draw (e.g., three spades), the probability of catching two outs on the turn and river is (10 ÷ 47) × (9 ÷ 46) = 4.16 percent. Backdoor draws are generally unlikely; with 43 unseen cards, it is equally likely to catch two out of seven outs as to catch one out of one. A backdoor outside straight draw (such as J-10-9) is equally likely as a backdoor flush, but any other 3-card straight combination is not worth even one out.
Drawing dead[edit]
A player is said to be drawing dead when the hand he hopes to complete will nonetheless lose to a player who already has a better one. For example, drawing to a straight or flush when the opponent already has a full house. In games with community cards, the term can also refer to a situation where no possible additional community card draws results in a win for a player. (This may be because another player has folded the cards that would complete his hand, his opponent's hand is already stronger than any hand he can possibly draw to or that the card that completes his hand also augments his opponent's.)
See also[edit]
- Poker strategy
Texas Holdem Probability Equation
References[edit]
Poker Probability Texas Holdem Wikipedia Codes
- ^Odds Chart. 'How to play texas holdem poker'. Howtoplaytexasholdempoker.org. Archived from the original on 13 January 2010. Retrieved 22 February 2010.
Texas Holdem Statistics
By Gerald Hanks
Gerald Hanks is from Houston Texas, and has been playing poker since 2002. He has played cash games and no-limit hold'em tournaments at live venues all over the United States.
Related Lessons
Related Lessons
A poker player is drawing if they have a hand that is incomplete and needs further cards to become valuable. The hand itself is called a draw or drawing hand. For example, in seven-card stud, if four of a player's first five cards are all spades, but the hand is otherwise weak, they are drawing to a flush. In contrast, a made hand already has value and does not necessarily need to draw to win. A made starting hand with no help can lose to an inferior starting hand with a favorable draw. If an opponent has a made hand that will beat the player's draw, then the player is drawing dead; even if they make their desired hand, they will lose. Not only draws benefit from additional cards; many made hands can be improved by catching an out — and may have to in order to win.
Outs[edit]
An unseen card that would improve a drawing hand to a likely winner is an out. Playing a drawing hand has a positive expectation if the probability of catching an out is greater than the pot odds offered by the pot.
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Poker Probability Texas Holdem Wikipedia Rules
The probability P1{displaystyle P_{1}} of catching an out with one card to come is:
- P1=outsunseencards{displaystyle P_{1}={frac {mathrm {outs} }{mathrm {unseen} ,mathrm {cards} }}}
The probability P2{displaystyle P_{2}} of catching at least one out with two cards to come is:
- P2=1−nonoutsunseencards×nonouts−1unseencards−1{displaystyle P_{2}=1-{frac {mathrm {non} ,mathrm {outs} }{mathrm {unseen} ,mathrm {cards} }}times {frac {mathrm {non} ,mathrm {outs} -1}{mathrm {unseen} ,mathrm {cards} -1}}}
- nonouts=unseencards−outs{displaystyle mathrm {non} ,mathrm {outs} ={mathrm {unseen} ,mathrm {cards} }-mathrm {outs} }
| Outs | One Card % | Two Card % | One Card Odds | Two Card Odds | Draw Type |
|---|---|---|---|---|---|
| 1 | 2% | 4% | 46 | 23 | Backdoor Straight or Flush (Requires two cards) |
| 2 | 4% | 8% | 22 | 12 | Pocket Pair to Set |
| 3 | 7% | 13% | 14 | 7 | One Overcard |
| 4 | 9% | 17% | 10 | 5 | Inside Straight / Two Pair to Full House |
| 5 | 11% | 20% | 8 | 4 | One Pair to Two Pair or Set |
| 6 | 13% | 24% | 6.7 | 3.2 | No Pair to Pair / Two Overcards |
| 7 | 15% | 28% | 5.6 | 2.6 | Set to Full House or Quads |
| 8 | 17% | 32% | 4.7 | 2.2 | Open Straight |
| 9 | 19% | 35% | 4.1 | 1.9 | Flush |
| 10 | 22% | 38% | 3.6 | 1.6 | Inside Straight & Two Overcards |
| 11 | 24% | 42% | 3.2 | 1.4 | Open Straight & One Overcard |
| 12 | 26% | 45% | 2.8 | 1.2 | Flush & Inside Straight / Flush & One Overcard |
| 13 | 28% | 48% | 2.5 | 1.1 | |
| 14 | 30% | 51% | 2.3 | 0.95 | |
| 15 | 33% | 54% | 2.1 | 0.85 | Flush & Open Straight / Flush & Two Overcards |
| 16 | 34% | 57% | 1.9 | 0.75 | |
| 17 | 37% | 60% | 1.7 | 0.66 |
A dead out is a card that would normally be considered an out for a particular drawing hand, but should be excluded when calculating the probability of catching an out. Outs can be dead for two reasons:
- A dead out may work to improve an opponent's hand to a superior hand. For example, if Ted has a spade flush draw and Alice has an outside straight draw, any spades that complete Alice's straight are dead outs because they would also give Ted a flush.
- A dead out may have already been seen. In some game variations such as stud poker, some of the cards held by each player are seen by all players.
Types of draws[edit]
Flush draw[edit]
A flush draw, or four flush, is a hand with four cards of the same suit that may improve to a flush. For example, K♣ 9♣ 8♣ 5♣ x. A flush draw has nine outs (thirteen cards of the suit less the four already in the hand). If a player has a flush draw in Hold'em, the probability to flush the hand in the end is 34.97 percent if there are two more cards to come, and 19.56 percent (9 live cards divided by 46 unseen cards) if there is only one more card to come.
Outside straight draw[edit]
An outside straight draw, also called up and down, double-ended straight draw or open-ended straight draw, is a hand with four of the five needed cards in sequence (and could be completed on either end) that may improve to a straight. For example, x-9-8-7-6-x. An outside straight draw has eight outs (four cards to complete the top of the straight and four cards to complete the bottom of the straight). Straight draws including an ace are not outside straight draws, because the straight can only be completed on one end (has four outs).
Inside straight draw[edit]
An inside straight draw, or gutshot draw or belly buster draw, is a hand with four of the five cards needed for a straight, but missing one in the middle. For example, 9-x-7-6-5. An inside straight draw has four outs (four cards to fill the missing internal rank). Because straight draws including an ace only have four outs, they are also considered inside straight draws. For example, A-K-Q-J-x or A-2-3-4-x. The probability of catching an out for an inside straight draw is half that of catching an out for an outside straight draw.
Double inside straight draw[edit]
A double inside straight draw, or double gutshot draw or double belly buster draw can occur when either of two ranks will make a straight, but both are 'inside' draws. For example in 11-card games, 9-x-7-6-5-x-3, or 9-8-x-6-5-x-3-2, or in Texas Hold'em when holding 9-J hole cards on a 7-10-K flop. The probability of catching an out for a double inside straight draw is the same as for an outside straight draw.
Other draws[edit]
Sometimes a made hand needs to draw to a better hand. For example, if a player has two pair or three of a kind, but an opponent has a straight or flush, to win the player must draw an out to improve to a full house (or four of a kind). There are a multitude of potential situations where one hand needs to improve to beat another, but the expected value of most drawing plays can be calculated by counting outs, computing the probability of winning, and comparing the probability of winning to the pot odds.
Backdoor draw[edit]
A backdoor draw, or runner-runner draw, is a drawing hand that needs to catch two outs to win. For example, a hand with three cards of the same suit has a backdoor flush draw because it needs two more cards of the suit. The probability Prr{displaystyle P_{rr}} of catching two outs with two cards to come is:
- Prr=outsunseencards×outs−1unseencards−1{displaystyle P_{rr}={frac {mathrm {outs} }{mathrm {unseen} ,mathrm {cards} }}times {frac {mathrm {outs} -1}{mathrm {unseen} ,mathrm {cards} -1}}}
For example, if after the flop in Texas hold 'em, a player has a backdoor flush draw (e.g., three spades), the probability of catching two outs on the turn and river is (10 ÷ 47) × (9 ÷ 46) = 4.16 percent. Backdoor draws are generally unlikely; with 43 unseen cards, it is equally likely to catch two out of seven outs as to catch one out of one. A backdoor outside straight draw (such as J-10-9) is equally likely as a backdoor flush, but any other 3-card straight combination is not worth even one out.
Drawing dead[edit]
A player is said to be drawing dead when the hand he hopes to complete will nonetheless lose to a player who already has a better one. For example, drawing to a straight or flush when the opponent already has a full house. In games with community cards, the term can also refer to a situation where no possible additional community card draws results in a win for a player. (This may be because another player has folded the cards that would complete his hand, his opponent's hand is already stronger than any hand he can possibly draw to or that the card that completes his hand also augments his opponent's.)
See also[edit]
- Poker strategy
Texas Holdem Probability Equation
References[edit]
Poker Probability Texas Holdem Wikipedia Codes
- ^Odds Chart. 'How to play texas holdem poker'. Howtoplaytexasholdempoker.org. Archived from the original on 13 January 2010. Retrieved 22 February 2010.
Texas Holdem Statistics
External links[edit]
Texas Holdem Mathematics
