The usual rationale for checking on the end is that in many circumstances, it risks money for no gain. If you believe that your opponents are on a draw against you, they will fold if they miss, or raise if they hit. In high-only poker variants, players check on the end for this reason. But, in high-low games, a hit draw might be for only half the pot. In these cases a raise does not necessary mean you lose everything. This is why having a good read on the end is so important. It is always dangerous to bet with a high-only hand into a made low-hand because your opponent has nothing to loose by raising and could have you beat. However, not every hand that shows low cards on the board represents a qualified low-hand. Many times these are high-only hands that will still call your bet when you have them beat. Here are some cues for reading hands.
Suited cards: Obviously, an opponent showing four, or even three suited cards on the board is a threat to make a flush on the end. However, two suited cards can be threat if one of them includes a high door card, especially if that high door card was either dead, or an under-card after the deal. For example, a person playing a Queen, when other Queens are exposed and/or higher cards such as Aces or Kings are exposed, is almost always playing three suited cards. Even though, in most instances this is a bad play, three suited cards is an irresistible starting hand for most players. If a second card with a suit matching the door card appears, that person will stay to the end looking to complete the flush. Pay attention to how live your opponent's flush draw remains when deciding if you want to bet into this kind of hand at the end.
Sequential cards: Opponents showing sequential mid-range cards, such as 8, 9, 10, or even cards with a gap, such as 7, 9, 10, are usually on a straight draw because that is one of the principal reasons for continuing with mid-range cards that have little value for a low-hand and are often dominated for high hand. Often the straight-draw is backed into from a hand that started as a potential low-hand. Many times these straight-draws can be live and open-ended, so proceed with caution on the end if you do not finish with a qualified low-hand, and cannot beat a mid-range straight.
The exposure of 4s and 5s: As pointed out in a previous post, the completion of a straight that simultaneously qualifies as a low-hand requires 4s and 5s. If you do not see any of the 4s or 5s, any two exposed low-cards in an opponent's hand can be a threat to make a low straight on the end, because most likely the straight-draw is live.
Paired door cards that prompt raising: A hand such as (X, X) 5, 7, K, 5 (X) in which the player suddenly started raising on Sixth Street when the second 5 appeared, has usually made trip 5s. The hand cannot qualify for low on Sixth Street and if there are trips 5s it will not qualify for low. Unless you can beat 5s-full you should probably check to this hand on the river.
Paired door cards that are high in a hand that did not raise on prior streets: Consider a hand such as (X, X) K, K, 7, 5 (X). When a high door card such as this is paired, the possibility of trips must be considered, but usually the player will begin raising immediately. If a raise does not occur when the card is paired, that is often sign of a wired pair, and the player now has two pair. Players with two pair are often aggressive when there are no potential low-hands, but if there are one or more potential low-hands, two pair is a vulnerable holding and many players check and call while hoping to fill-up. What this means is that if you finish with two pair, you need to be able to beat a hand with two pair that includes the pair on the board.
Paired door cards that are low in a hand that did not raise on prior streets: A hand such as (X, X) 4, 4, J, 10 (X) that checked and called all the way, most likely went to the river as a pair 4s with hopes of making two-pair or trips on the end. Most likely this hand started as three small cards but never improved for low. If you can beat two small pair you should bet because this player will call with any two pair, no matter how small, and might even call with the pair of 4s if the 4s beat the board.
Paired cards for Fourth Street and later: For hands such as (X, X) 3, 4, J, J (X) or (X, X) 3, 4, Q, 4 (X) it is unlikely that the player had anything better than the single pair going to the river. The player is hoping to make two pair, or trips on the end, or complete a low-hand. If you have a two pair that can beat the pair on the board it is usually worth it to bet.
Low door cards that remain in the hand after picking up high cards: A player with a hand such as (X, X) 5, J, Q, K, almost always started with a low pair, either split or wired, because if the starting hand had been 4, 5, 6, it would have been abandoned. This player is hoping make two pair, or trips on the end. You should not have to worry about the straight. If you are heads-up against this kind of hand and have two small pair, you should check. This is the kind of hand that will often fold to a bet unless it makes Jacks-up or better, in which case you are beat.
Any two exposed wheel cards, especially on early streets: A player with (X, X), 5, 4, J, Q (X) is going to the river looking to complete a low-hand at the minimum, and possibly a low straight. You should check to this hand unless you have a high hand that can beat a low straight because you have nothing to gain from a bet. The player will fold to any bet unless he completed a low-hand.
Four exposed low-cards in a hand that did not raise on prior streets: Hand such as (X, X) 2, 5, 6, 7 (X) that did not raise earlier have not qualified for the low-pot before the river. Clearly the hand could be a qualified low-hand after the river, but frequently the last card is a brick, especially if many of the low cards are dead. If the hand is not already a qualified low-hand before the river, it is because low hole cards paired, resulting in two small pair, or there was a wired pair to begin with that did not make trips. Often in these situations the wired pair is high, which meant the player never expected to get a low-hand. If you have a hand with a reasonably high two pair or better, you should bet because often you will scoop in this situation, even with all the scary looking low-cards.
Monday, February 28, 2011
Sunday, January 30, 2011
The High-Only Four-Flush
One of the most misunderstood hands in Seven-Card Stud High-Low is the high-only four-flush. These are hands such as 2, 9, Q, K suited that can never qualify for the low-pot, but need just one additional suited card to complete a flush for a powerful high-hand. These hands occur frequently on Fourth Street when players who automatically play any three suited cards catch their suit. These players almost always stay to the end no matter what action follows.
In high-only Seven-Card Stud, a four-flush is usually a positive expectation holding. With a live draw and three cards to come, the probability of completing the flush by the end is about 50%. That means as long as the pot is paying better than even money--which it always is because of the antes--you will win money over the long run. It is the equivalent of betting on coin-flips that pay back more than you wager.
But the math is very different if you are in a Stud High-Low game and completing the flush will only win half the pot. If you win half the pot, half the time, your pot equity is only 25%. The pot must be paying at 3 to 1 for you to just break-even on a wager. That means heads-up play against a player with a better high hand who is also drawing to the low-pot is usually not worth the risk, unless you have some other way of backing into the high-hand.
To get the proper pot odds to play a high-only four-flush usually requires a multi-way pot. With three or more players your bets are being multiplied, not just matched. But, even then it is difficult to get a positive expectation. For example, in a three-way pot your bets are essentially being matched 2 for 1, which is not the 3 for 1 payoff needed to break even. Imagine a betting game in which three players put up equal amounts of money and 50% of the time you win 50% of the amount wagered. If the bets were $10 it would cost you $100 to play this game 10 times. Your expected winning of 5 x $15, or $75, doesn't cover your cost.
Playing a high-only four-flush against two other players with possible low-hands is essentially this kind of a betting game. The difference is that in poker, there is usually some dead money in the pot from antes and bets on prior streets. But, the dead money has to be large and your cost small to profitably compete for it.
To see how this works lets consider a specific example. Consider a Fifth Street scenario with the following conditions.
Stakes: $1-2
Hands Dealt: Seven
Current pot size: $10
Remaining hands:
You (2-Diamonds, 3-Diamonds) K-Diamonds, 10-Diamonds, 9-Spades
Alice (X, X) A-Clubs, 6-Spades, 7-Clubs
Bob (X, X) 2-Spades, A-Hearts, A-Spades
The number of exposed cards is 11 plus the 4 door cards that were dealt and mucked, a total of 15. If none of the mucked door cards were diamonds, you have 9 outs with 2 cards to come and 37 unseen cards. Under these circumstances, the probability of completing the flush by the end is 43%. But, this board appears to give someone a low-hand and you are currently behind for high. You are attempting a draw for half of a $10 pot. The expected values for some possible betting scenarios can be calculated.
Scenario 1: You believe Bob will lead the betting for the next three streets and Alice will call, in which case you will call (likely to happen if Alice is drawing for a low-hand). If this happens the final pot will be $28 ($10 + $2 x 3 players x 3 streets). You will have to invest $6 for a 43% chance of winning $14. Because 43% of $14 is $6.02 this is essentially a break-even proposition.
Scenario 2: You believe Bob will lead the betting for the next three streets and Alice will raise, and if you call, Bob will re-raise and Alice will cap (likely if Alice already has a lock on the low-pot and is free-rolling. If this happens the final pot will be $82 ($10 + $2 x 3 players x 4 bets x 3 streets). You will have to invest $24 for a 43% chance to win $41. Because 43% of $41 is $17.63 this is a losing proposition.
These scenarios are all thinking ahead after seeing the fifth card, but prior to any Fifth Street action. At that moment your expected return from further investment in the hand is between break-even and negative. It will require more than $10 of dead money to shift Scenario 1 to a positive expectation and much more to shift Scenario 2 to your favor.
The lesson from this analysis is that you should not automatically play any three suited cards if one or more is high. If Alice and Bob are showing high door cards go ahead. A low-hand is unlikely to develop and if you catch your suit on Fourth Street, you have an advantage. Plus if the King is the high card on the board, you can represent it as a pair of Kings and put pressure on Bob and Alice to fold without completing the flush. But, if Alice and Bob have low door cards and/or Aces, even catching your suit might leave you with a negative expectation if Alice and Bob develop viable low-hands. You cannot pressure anyone with a lock on the low-pot, so your fold equity is gone.
All this re-enforces the fundamental concept for playing Stud High-Low that is: if you can't foresee a scenario in which you have a high probability of scooping, don't play the hand.
In high-only Seven-Card Stud, a four-flush is usually a positive expectation holding. With a live draw and three cards to come, the probability of completing the flush by the end is about 50%. That means as long as the pot is paying better than even money--which it always is because of the antes--you will win money over the long run. It is the equivalent of betting on coin-flips that pay back more than you wager.
But the math is very different if you are in a Stud High-Low game and completing the flush will only win half the pot. If you win half the pot, half the time, your pot equity is only 25%. The pot must be paying at 3 to 1 for you to just break-even on a wager. That means heads-up play against a player with a better high hand who is also drawing to the low-pot is usually not worth the risk, unless you have some other way of backing into the high-hand.
To get the proper pot odds to play a high-only four-flush usually requires a multi-way pot. With three or more players your bets are being multiplied, not just matched. But, even then it is difficult to get a positive expectation. For example, in a three-way pot your bets are essentially being matched 2 for 1, which is not the 3 for 1 payoff needed to break even. Imagine a betting game in which three players put up equal amounts of money and 50% of the time you win 50% of the amount wagered. If the bets were $10 it would cost you $100 to play this game 10 times. Your expected winning of 5 x $15, or $75, doesn't cover your cost.
Playing a high-only four-flush against two other players with possible low-hands is essentially this kind of a betting game. The difference is that in poker, there is usually some dead money in the pot from antes and bets on prior streets. But, the dead money has to be large and your cost small to profitably compete for it.
To see how this works lets consider a specific example. Consider a Fifth Street scenario with the following conditions.
Stakes: $1-2
Hands Dealt: Seven
Current pot size: $10
Remaining hands:
You (2-Diamonds, 3-Diamonds) K-Diamonds, 10-Diamonds, 9-Spades
Alice (X, X) A-Clubs, 6-Spades, 7-Clubs
Bob (X, X) 2-Spades, A-Hearts, A-Spades
The number of exposed cards is 11 plus the 4 door cards that were dealt and mucked, a total of 15. If none of the mucked door cards were diamonds, you have 9 outs with 2 cards to come and 37 unseen cards. Under these circumstances, the probability of completing the flush by the end is 43%. But, this board appears to give someone a low-hand and you are currently behind for high. You are attempting a draw for half of a $10 pot. The expected values for some possible betting scenarios can be calculated.
Scenario 1: You believe Bob will lead the betting for the next three streets and Alice will call, in which case you will call (likely to happen if Alice is drawing for a low-hand). If this happens the final pot will be $28 ($10 + $2 x 3 players x 3 streets). You will have to invest $6 for a 43% chance of winning $14. Because 43% of $14 is $6.02 this is essentially a break-even proposition.
Scenario 2: You believe Bob will lead the betting for the next three streets and Alice will raise, and if you call, Bob will re-raise and Alice will cap (likely if Alice already has a lock on the low-pot and is free-rolling. If this happens the final pot will be $82 ($10 + $2 x 3 players x 4 bets x 3 streets). You will have to invest $24 for a 43% chance to win $41. Because 43% of $41 is $17.63 this is a losing proposition.
These scenarios are all thinking ahead after seeing the fifth card, but prior to any Fifth Street action. At that moment your expected return from further investment in the hand is between break-even and negative. It will require more than $10 of dead money to shift Scenario 1 to a positive expectation and much more to shift Scenario 2 to your favor.
The lesson from this analysis is that you should not automatically play any three suited cards if one or more is high. If Alice and Bob are showing high door cards go ahead. A low-hand is unlikely to develop and if you catch your suit on Fourth Street, you have an advantage. Plus if the King is the high card on the board, you can represent it as a pair of Kings and put pressure on Bob and Alice to fold without completing the flush. But, if Alice and Bob have low door cards and/or Aces, even catching your suit might leave you with a negative expectation if Alice and Bob develop viable low-hands. You cannot pressure anyone with a lock on the low-pot, so your fold equity is gone.
All this re-enforces the fundamental concept for playing Stud High-Low that is: if you can't foresee a scenario in which you have a high probability of scooping, don't play the hand.
Monday, December 27, 2010
Fifth Street Tactics: Keeping opponents in versus driving them out
If you have a made low on Fifth Street and no one else does, it is often advantageous to keep players in the hand contributing to the pot. If the high hand is willing to bet, calling might be a more profitable option because you do not want to just get your money back if it becomes a heads-up contest with the high hand. However, there are situations where raising with the intent of driving others out of hand is preferable. Some examples:
-- Your low is vulnerable to another player with a low draw. Suppose you have (2, 4) A, 7, 8 and another player has an exposed 6, 3, 3. The pair means a low has not been made, but she could have a draw to a better low and she holds three cards that would improve your low. Your low hand is very vulnerable to becoming second best later in the hand. If the high bets you should raise and attempt to force her out.
-- You can freeroll your low for a powerful high hand and do not want other hands to compete against it for high. Suppose you have (2, 5) 3, 4, 7 in a multi-way pot in which one of the players appears to be on a flush draw. Another player with an exposed pair of 8s bets. You should raise to force out the flush draw. If you make your straight, you want it to hold up for high.
-- The high hand appears weak and possibly the result of a busted low. Suppose you have (6, 3) A, 5, 7 against a player with an exposed 2, 2, J. The pair of 2s might be the only thing he has going for him and pairing any one of your cards could lead to a better high. Aggressive raising might win the pot outright. He could judge that it is too risky to pursue the hand for high.
-- You have a lock on low in a multi-way pot in which raising will not drive the others out. There could be several powerful high hands that have developed or are developing in such a way that your opponents will pay any price to stay in the hand. Obviously in that situation, raising will maximize the amount in the pot and comes with no risk, even if you are drawing dead to high.
-- Your low is vulnerable to another player with a low draw. Suppose you have (2, 4) A, 7, 8 and another player has an exposed 6, 3, 3. The pair means a low has not been made, but she could have a draw to a better low and she holds three cards that would improve your low. Your low hand is very vulnerable to becoming second best later in the hand. If the high bets you should raise and attempt to force her out.
-- You can freeroll your low for a powerful high hand and do not want other hands to compete against it for high. Suppose you have (2, 5) 3, 4, 7 in a multi-way pot in which one of the players appears to be on a flush draw. Another player with an exposed pair of 8s bets. You should raise to force out the flush draw. If you make your straight, you want it to hold up for high.
-- The high hand appears weak and possibly the result of a busted low. Suppose you have (6, 3) A, 5, 7 against a player with an exposed 2, 2, J. The pair of 2s might be the only thing he has going for him and pairing any one of your cards could lead to a better high. Aggressive raising might win the pot outright. He could judge that it is too risky to pursue the hand for high.
-- You have a lock on low in a multi-way pot in which raising will not drive the others out. There could be several powerful high hands that have developed or are developing in such a way that your opponents will pay any price to stay in the hand. Obviously in that situation, raising will maximize the amount in the pot and comes with no risk, even if you are drawing dead to high.
Monday, November 29, 2010
Missed Opportunities on the River
Extracting maximum value from winning hands is as important as limiting financial damage from losing hands. Here are some examples of mistakes I've made on the end.
Missing an extra bet
Opponent: (X, X) A-Clubs, A-Spades, 10-Hearts, 10-Spades, (X)
Me: (3-Clubs, 5-Hearts) 4-Clubs, 6-Spades, 5-Diamonds, 5-Clubs, (J-Clubs)
Action: Obviously I was unhappy to see the brick on the end that denied me a qualifying low-hand. My opponent had led all the way in this hand with the exposed pair of Aces, and I had serious doubts that my trip 5s would hold up for high. But, on the river my opponent checked and after I thoughtlessly checked back, I won the entire pot.
Analysis: I missed picking up an extra bet on the end because it is not possible for my opponent to have trip Aces or trip 10s. If he has three of either rank his hand would be a full house, which is a holding that he would certainly bet. In fact he would bet quads, a full house, a flush, or an Ace-high straight. The only reason for a check is that he has none of these holdings, and fears losing to a possible small straight. Therefore my trip 5s has to be the nuts and I should bet. With Aces-up, he has to call because the pot was large and I could be betting with only a low-hand.
Missing a chance at half the pot
Opponent: (X, X) 10-Clubs, 9-Hearts, 8-Spades, 7-Clubs, (X)
Me: (2-Spades, Q-Clubs) Q-Spades, 5-Spades, 6-Spades, A-Clubs, (7-Diamonds)
Action: This was a heads-up hand, that because of the high door-cards, I bet out thinking that no qualified low-hand would result. I checked on Sixth Street and when my opponent responded by betting into my Queens, I read him for a straight and stayed because of my flush draw. I missed the flush-draw on the river but backed into a 7-high nut-low. My opponent bet on the end and I made the mistake of calling. He won the high-pot with two pair 10s and 7s.
Analysis: There was no reason for me not to raise in this situation. I have no risk of being scooped by a straight and a raise would force him to make a difficult decision if he missed his draw, which in this case he did. Do you call someone raising on the end with two overcards, when all you have is two small pair? He's not expecting a low-hand on my side anymore than I did. Most likely he would fold because his river bet amounts to a semi-bluff.
Part of the reason for my errors in both these cases was backing into a different kind of hand than what I had sought. In the latter case I had too much mental focus on playing a high-hand without thinking about the backdoor low possibilities. The call was an afterthought because I had not been looking for a low-hand. In the former case I was looking for a low straight because I believed trip 5s and even 5s-full would lose. But, backdoor low-hands and backdoor high-hands occur frequently in Stud-Eight. You need to quickly switch your thought processes when they occur and think about the new tactical possibilities that they present.
Missing an extra bet
Opponent: (X, X) A-Clubs, A-Spades, 10-Hearts, 10-Spades, (X)
Me: (3-Clubs, 5-Hearts) 4-Clubs, 6-Spades, 5-Diamonds, 5-Clubs, (J-Clubs)
Action: Obviously I was unhappy to see the brick on the end that denied me a qualifying low-hand. My opponent had led all the way in this hand with the exposed pair of Aces, and I had serious doubts that my trip 5s would hold up for high. But, on the river my opponent checked and after I thoughtlessly checked back, I won the entire pot.
Analysis: I missed picking up an extra bet on the end because it is not possible for my opponent to have trip Aces or trip 10s. If he has three of either rank his hand would be a full house, which is a holding that he would certainly bet. In fact he would bet quads, a full house, a flush, or an Ace-high straight. The only reason for a check is that he has none of these holdings, and fears losing to a possible small straight. Therefore my trip 5s has to be the nuts and I should bet. With Aces-up, he has to call because the pot was large and I could be betting with only a low-hand.
Missing a chance at half the pot
Opponent: (X, X) 10-Clubs, 9-Hearts, 8-Spades, 7-Clubs, (X)
Me: (2-Spades, Q-Clubs) Q-Spades, 5-Spades, 6-Spades, A-Clubs, (7-Diamonds)
Action: This was a heads-up hand, that because of the high door-cards, I bet out thinking that no qualified low-hand would result. I checked on Sixth Street and when my opponent responded by betting into my Queens, I read him for a straight and stayed because of my flush draw. I missed the flush-draw on the river but backed into a 7-high nut-low. My opponent bet on the end and I made the mistake of calling. He won the high-pot with two pair 10s and 7s.
Analysis: There was no reason for me not to raise in this situation. I have no risk of being scooped by a straight and a raise would force him to make a difficult decision if he missed his draw, which in this case he did. Do you call someone raising on the end with two overcards, when all you have is two small pair? He's not expecting a low-hand on my side anymore than I did. Most likely he would fold because his river bet amounts to a semi-bluff.
Part of the reason for my errors in both these cases was backing into a different kind of hand than what I had sought. In the latter case I had too much mental focus on playing a high-hand without thinking about the backdoor low possibilities. The call was an afterthought because I had not been looking for a low-hand. In the former case I was looking for a low straight because I believed trip 5s and even 5s-full would lose. But, backdoor low-hands and backdoor high-hands occur frequently in Stud-Eight. You need to quickly switch your thought processes when they occur and think about the new tactical possibilities that they present.
Thursday, October 14, 2010
The problem with one-way low-hands
Many players who have four low-cards after Fourth Street in Seven-Card Stud High-Low will stay until the end, and even raise before completing a low-hand. However, a hand with four low-cards and nothing else going for it has more problems than you might imagine. Consider a common scenario in which four unconnected low-cards are heads-up against a high pair. Consider this example:
You: (8, 7) 3, 2
Alice: (Q, K) K, 9
If your hand is completely live there are 16 outs to complete low-hand. That means that by the end you should complete a low-hand 73% of the time. The majority of hands, in which you split the pot, will return your money plus half the money already present from the antes and bring-ins. However, 27% of the time you lose all the money you invested on the later streets. Clearly this is a negative expectation contest because you win no additional money from your later bets the 73% of the time that you succeed, but 27% of the time you will lose all the money you invested. The money that already exists in the pot from antes and Third Street betting is rarely so large that half will offset this negative expectation.
In a three-way pot, your expectation is positive, but not as high as you might think.
You: (8, 7) 3, 2
Alice: (Q, K) K, 9
Bob: (J, J) 10, Q
If you, Alice, and Bob, each contribute $50 to see the final three cards, there will be $150 at stake. If this situation is played 100 times, you will have spent a total of $5000 to win half of $150, or $75 for the 73 low-pots that you will win on average. Your total return is $5475, which is less than a 10% return on your investment, barely enough to cover the rake. However, if you are in a hand such as this against two one-way high hands, there is the possibility of making your low-hand early and being able to freeroll on later streets.
However, a dangerous situation arises when you have a one-way low-draw against a high hand and another low-draw. In this case the probability of making a low-hand decreases because your draw is usually not completely live. The reduction in outs can be exacerbated by mucked low-cards after the deal. Consider a deal in which a 5 and 6 are mucked on Third Street and the following three-way hand develops:
You: (8, 7) 3, 2
Alice: (7, 5) 4, A
Bob: (J, J) 10, Q
This is a terrible situation to be in. Alice has three of your outs and two other outs are dead. There are only 11 cards available to complete your low-hand, which means the probability has decreased to 59%. While this is still a better than even chance it shifted your expectation to negative. If you, Alice, and Bob, each contribute $50 to see the final three cards, there will be $150 at stake. If this situation is played 100 times, you will have spent a total of $5000 to win $75 for the 59 low-pots that you will win on average. Your total return is $4425, a loss of $575 or 11.5%. That figure optimistically assumes that you win the low-pot each time that you make a qualifying low-hand. In fact, Alice is drawing to a better low-hand than yours, and a significant fraction of the time she will win the low-pot even if you qualify. That means that your expected losses will be much worse than 11.5%.
However, if your hand has scoop potential the expectation shifts to your favor. Consider having connected low-cards:
You: (3, 4) 5, 6
Alice: (Q, K) K, 9
Alice will still scoop the 27% of the time that you fail to make a low-hand. But 44% of the time you will complete a straight that most likely will scoop, and 29% of the time you will win the low pot. If you and Alice each contribute $50, there will be $100 at stake. Consider 100 trials of this scenario. At $50 for each trial it will cost you $5000 total. On average, you will win $100 the 44 times you hit the straight, and $50 the 29 times you make a low-hand only. Your total winnings over 100 trials will average $5850,which is a return of 17%
In a three-way pot against two high hands your positive expectation is even greater if both high hands stay until the end and a straight holds up for high. Consider this example:
You: (3, 4) 5, 6
Alice: (Q, K) K, 9
Bob: (J, J) 10, Q
If you, Alice and Bob each contribute $50, it will cost you $5000 to play this scenario 100 times. On average, you will win $150 the 44 times you hit the straight, and $75 the 29 times you make a low-hand only. Your total winnings over 100 trials will average $8775,which is a return of 75%. In practice this large positive expectation will be offset by the times when the high-hands improve to better than a 7-high straight which will still result in a split-pot.
These examples show how important the possibility of a scoop is to determining expectation. The challenge when you play the high side of these scenarios is to judge if your opponent has scoop potential so that you can avoid playing a hand in which you have a negative expectation. In the examples discussed, I specified the hole cards so that I could present a precise calculation of expectation. In practice you don't see your opponent's hole cards and must infer the values. In you fold a high pair any time that your opponent has two exposed low-cards, you are giving up in a situation in which you have positive expectation. However, anytime you are playing into a sequence of four connected low-cards, or four suited low-cards, you have a negative expectation.
Here are some guidelines for making that judgment.
You: (8, 7) 3, 2
Alice: (Q, K) K, 9
If your hand is completely live there are 16 outs to complete low-hand. That means that by the end you should complete a low-hand 73% of the time. The majority of hands, in which you split the pot, will return your money plus half the money already present from the antes and bring-ins. However, 27% of the time you lose all the money you invested on the later streets. Clearly this is a negative expectation contest because you win no additional money from your later bets the 73% of the time that you succeed, but 27% of the time you will lose all the money you invested. The money that already exists in the pot from antes and Third Street betting is rarely so large that half will offset this negative expectation.
In a three-way pot, your expectation is positive, but not as high as you might think.
You: (8, 7) 3, 2
Alice: (Q, K) K, 9
Bob: (J, J) 10, Q
If you, Alice, and Bob, each contribute $50 to see the final three cards, there will be $150 at stake. If this situation is played 100 times, you will have spent a total of $5000 to win half of $150, or $75 for the 73 low-pots that you will win on average. Your total return is $5475, which is less than a 10% return on your investment, barely enough to cover the rake. However, if you are in a hand such as this against two one-way high hands, there is the possibility of making your low-hand early and being able to freeroll on later streets.
However, a dangerous situation arises when you have a one-way low-draw against a high hand and another low-draw. In this case the probability of making a low-hand decreases because your draw is usually not completely live. The reduction in outs can be exacerbated by mucked low-cards after the deal. Consider a deal in which a 5 and 6 are mucked on Third Street and the following three-way hand develops:
You: (8, 7) 3, 2
Alice: (7, 5) 4, A
Bob: (J, J) 10, Q
This is a terrible situation to be in. Alice has three of your outs and two other outs are dead. There are only 11 cards available to complete your low-hand, which means the probability has decreased to 59%. While this is still a better than even chance it shifted your expectation to negative. If you, Alice, and Bob, each contribute $50 to see the final three cards, there will be $150 at stake. If this situation is played 100 times, you will have spent a total of $5000 to win $75 for the 59 low-pots that you will win on average. Your total return is $4425, a loss of $575 or 11.5%. That figure optimistically assumes that you win the low-pot each time that you make a qualifying low-hand. In fact, Alice is drawing to a better low-hand than yours, and a significant fraction of the time she will win the low-pot even if you qualify. That means that your expected losses will be much worse than 11.5%.
However, if your hand has scoop potential the expectation shifts to your favor. Consider having connected low-cards:
You: (3, 4) 5, 6
Alice: (Q, K) K, 9
Alice will still scoop the 27% of the time that you fail to make a low-hand. But 44% of the time you will complete a straight that most likely will scoop, and 29% of the time you will win the low pot. If you and Alice each contribute $50, there will be $100 at stake. Consider 100 trials of this scenario. At $50 for each trial it will cost you $5000 total. On average, you will win $100 the 44 times you hit the straight, and $50 the 29 times you make a low-hand only. Your total winnings over 100 trials will average $5850,which is a return of 17%
In a three-way pot against two high hands your positive expectation is even greater if both high hands stay until the end and a straight holds up for high. Consider this example:
You: (3, 4) 5, 6
Alice: (Q, K) K, 9
Bob: (J, J) 10, Q
If you, Alice and Bob each contribute $50, it will cost you $5000 to play this scenario 100 times. On average, you will win $150 the 44 times you hit the straight, and $75 the 29 times you make a low-hand only. Your total winnings over 100 trials will average $8775,which is a return of 75%. In practice this large positive expectation will be offset by the times when the high-hands improve to better than a 7-high straight which will still result in a split-pot.
These examples show how important the possibility of a scoop is to determining expectation. The challenge when you play the high side of these scenarios is to judge if your opponent has scoop potential so that you can avoid playing a hand in which you have a negative expectation. In the examples discussed, I specified the hole cards so that I could present a precise calculation of expectation. In practice you don't see your opponent's hole cards and must infer the values. In you fold a high pair any time that your opponent has two exposed low-cards, you are giving up in a situation in which you have positive expectation. However, anytime you are playing into a sequence of four connected low-cards, or four suited low-cards, you have a negative expectation.
Here are some guidelines for making that judgment.
- Count your opponent's outs for a low-hand. If many of the low-cards needed are dead the probability that your opponent will qualify for the low-pot by the end drops considerably.
- Note possible implied outs. A hand with a low door-card that limped in on Third Street and mucked after catching a high card on Fourth Street, probably removed two additional low-cards from play, not just the one exposed.
- Pay attention to the blockers and take special note of the 4s and 5s. As explained in the previous section if the either rank-4s or 5s-are dead, low straights cannot occur.
- Note gaps in exposed low-cards. An 8, 2 showing is much less of a threat than an exposed 3, 2.
- Most importantly, track your opponent's tendencies. A tough, aggressive opponent who always plays to scoop is much more likely to have connected low-cards than an opponent who consistently limps in with any random set of low-cards.
Wednesday, September 15, 2010
Counting Outs in Seven-Card Stud High-Low
Having identified the different kinds of drawing hands last month, we can now count the outs available in order to determine likelihood of improvement. However, Stud games, in contrast to flop games, usually have many more exposed cards to account for when adding up outs. As a result, counting outs is much more situational in Seven-Card Stud High-Low than in Hold'em.
Consider the following extreme example. For the coordinated high example of 2-clubs, 3- clubs, 4-clubs, 5- clubs, 5-diamonds, 5-spades there are a total of 31 cards that will improve its high ranking, qualify the hand for the low-pot, or do both. But the 31 total is an upper limit on the number of outs available to improve the hand. Suppose the 9- clubs, 7- clubs, 4-diamonds, and 6-spades are exposed other hands. All these cards are no longer available as outs, so the count needs to be reduced by four to a total of 27
Let us also suppose that another player has all four Aces exposed in her hand. Now not only must the Aces be removed from the total outs, but most of the high draws for this hand are now dead. There is only one out available for this hand to take the high-pot-the 6-clubs-and for the low pot all that remain are the two other 6s, three other 7s, and the four 8s. Instead of 31 outs available, there are actually only 10 cards that matter, because if someone already has quad Aces hitting quad 5s or any of the full houses, or high flushes, will not win anything.
The consideration of this extremely unlikely scenario is useful because it illustrates four important points about counting outs. You need to subtract from the total for the following reasons.
Dead cards: You must subtract from the total available outs all the helpful cards that you see in other hands because they are no longer available. This is different than in Hold'em, in which every card that you see is potentially playable.
Dead draws: You must subtract from the total available outs all the cards that you see that improve your hand but do not win. Again in contrast to Hold'em, you can often see in a Stud game when you are drawing dead, because certain improvements will not beat your opponent's exposed cards ("beat the board").
Weighted outs: Not all outs are of equal value. In high-low games some outs win the entire pot while other outs win just half. In the example of the straight flush draw versus quads, the 6-clubs wins it all while the nine available low cards win just half the pot. To account for this difference, in this book I introduce the concept of counting "weighted outs." To count weighted outs, add 1 for each out that scoops, and 1/2 for each out that only wins half the pot. For this example there is one out that scoops and nine outs that take the low-pot, so the total number of weighted outs is 1 + 9 x (1/2), or 5.5. I will show in a future post how a count of weighted outs can be used to determine pot equity, which is a measure of how much you should invest when betting on a hand.
Implied outs: You can often subtract from the total number of outs, cards that you cannot see because your opponent's actions imply the contents of their hole cards. Suppose three players in the hand after Fourth Street show 2-3, 5-7, and A-2, and all act as if they are on draws to low-hands. If you are looking for a low-card, it is clear that six of them are dead, but in this situation you can imply that 12 are dead, because the six unseen hole cards are most likely low.
Once an accurate count of outs is determined, the probability for improvement can be found by dividing the total outs by the number of unseen cards.
Consider the following extreme example. For the coordinated high example of 2-clubs, 3- clubs, 4-clubs, 5- clubs, 5-diamonds, 5-spades there are a total of 31 cards that will improve its high ranking, qualify the hand for the low-pot, or do both. But the 31 total is an upper limit on the number of outs available to improve the hand. Suppose the 9- clubs, 7- clubs, 4-diamonds, and 6-spades are exposed other hands. All these cards are no longer available as outs, so the count needs to be reduced by four to a total of 27
Let us also suppose that another player has all four Aces exposed in her hand. Now not only must the Aces be removed from the total outs, but most of the high draws for this hand are now dead. There is only one out available for this hand to take the high-pot-the 6-clubs-and for the low pot all that remain are the two other 6s, three other 7s, and the four 8s. Instead of 31 outs available, there are actually only 10 cards that matter, because if someone already has quad Aces hitting quad 5s or any of the full houses, or high flushes, will not win anything.
The consideration of this extremely unlikely scenario is useful because it illustrates four important points about counting outs. You need to subtract from the total for the following reasons.
Dead cards: You must subtract from the total available outs all the helpful cards that you see in other hands because they are no longer available. This is different than in Hold'em, in which every card that you see is potentially playable.
Dead draws: You must subtract from the total available outs all the cards that you see that improve your hand but do not win. Again in contrast to Hold'em, you can often see in a Stud game when you are drawing dead, because certain improvements will not beat your opponent's exposed cards ("beat the board").
Weighted outs: Not all outs are of equal value. In high-low games some outs win the entire pot while other outs win just half. In the example of the straight flush draw versus quads, the 6-clubs wins it all while the nine available low cards win just half the pot. To account for this difference, in this book I introduce the concept of counting "weighted outs." To count weighted outs, add 1 for each out that scoops, and 1/2 for each out that only wins half the pot. For this example there is one out that scoops and nine outs that take the low-pot, so the total number of weighted outs is 1 + 9 x (1/2), or 5.5. I will show in a future post how a count of weighted outs can be used to determine pot equity, which is a measure of how much you should invest when betting on a hand.
Implied outs: You can often subtract from the total number of outs, cards that you cannot see because your opponent's actions imply the contents of their hole cards. Suppose three players in the hand after Fourth Street show 2-3, 5-7, and A-2, and all act as if they are on draws to low-hands. If you are looking for a low-card, it is clear that six of them are dead, but in this situation you can imply that 12 are dead, because the six unseen hole cards are most likely low.
Once an accurate count of outs is determined, the probability for improvement can be found by dividing the total outs by the number of unseen cards.
Sunday, August 15, 2010
Kinds of Drawing Hands in Seven-Card Stud High-Low (Stud-Eight)
In high only poker the concept of drawing to improve the rank of your hand is simple--the hand can only increase in rank. Therefore, cards drawn either raise the rank of your hand or they do not. But, in high-low forms of poker, defining improvement is more complicated because there are three kinds of improvement--increasing the rank of your high-hand, qualifying for a low-hand, and improving your low-hand. All hands, no matter how low, have the potential to win the high-pot, but not all hands qualify for the low-pot. To complicate matters further, some drawn cards can simultaneously improve your high and low hands, while some drawn cards can improve your high hand at the expense of disqualifying you from holding a low-hand. To account for these complexities we will define two categories of hands--half-made hands and drawing hands, and within each of these categories define three kinds of hands. That makes a total of six different kinds of drawing hands.
Half-made hands are holdings that can win half the pot, but need improvement to win the other half. The three kinds of half-made hands in order of desirability are:
Coordinated highs are hands that still need to qualify for the low-pot, but have outs for a low-hand that will improve the rank of the high hand. For example consider the best possible coordinated high that you could hold on Sixth Street--2-clubs, 3-clubs, 4-clubs, 5-clubs, 5-diamonds, 5-spades. This hand is already trip 5s, but it has four sequential suited low-cards. It can improve its high ranking to quads, a full house, or a 9-high flush or better without qualifying for the low-pot. But, it can also improve it high ranking to a 5-high or 6-high straight flush, 7-high or 8-high flush, 5-high or 6-high straight and at the same time qualify for a low-hand. It can also qualify for low without improving its high ranking. For example it can become a 7-high or 8-high low-hand, and remain trip 5s.
Uncoordinated highs still need to qualify for the low-pot, but in doing so cannot improve their high ranking. While coordinated highs might have some outs to qualify for the low-pot that do not improve the high ranking, for uncoordinated highs none of the outs for a low-hand improve the high hand. Consider the best uncoordinated high-hand that you can hold--A-spades, 2-spades, 3-spades, 5-diamonds, Q-spades, K-spades. This hand is an A-K-Q-high flush with four low-cards. There are 16 outs to qualify this hand for the low-pot--any of the 4s, 6s, 7s, or 8s--but none of these outs will change the fact that the high hand is a flush.
Lows are hands that qualify for the low-pot, but rank poorly as high hand. For these hands you are drawing for the high half of the pot. For example the holding--2-hearts, 3-hearts, 4-hearts, 5-hearts, 7-diamonds, 8-spades is a qualified 7-high low-hand, but only an 8-high high hand. However, it has many outs available to improve its high ranking. An A-hears or 6-hearts would result in a straight flush, a K-hearts, Q-hearts, J-hearts, 10-hearts, 9-hearts, 8-hearts, or 7-hearts, would make it a flush, while an A-clubs, A-diamonds, A-spades, 6-clubs, 6-diamonds, or 6-spades, would make it a straight. Any of the remaining 2s, 3s, 4s, 5s, or unsuited 8s or 7s would make it one pair.
Drawing hands are holdings that need improvement to win any part of the pot. In order of desirability the kinds or drawing hands are:
Scoop draws have outs available that can simultaneously win the high and low pots. Consider a Sixth Street holding of 2-clubs, 3-clubs, 4-clubs, 5-clubs, 9-hearts, 10-spades. This open-ended straight flush-draw has outs that will improve to a high-only flush (K-clubs, Q-clubs, J-clubs, 9-clubs, 10-clubs), outs that will qualify for the low-pot (7-hearts, 7-spades, 7-diamonds, 8-hearts, 8-spades, 8-diamonds) along with outs that qualify for the low-pot and improve the high hand. The 7-clubs or 8-clubs completes a flush and qualifies for low; the A-clubs or 6-clubs completes a straight flush and qualifies for low; the A-diamonds, A-hearts, A-spades, 6-diamonds, 6-hearts, 6-spades, would all complete a straight and qualify for low.
One-way high-draws can never qualify for the low-pot. The only outs available improve the high-ranking. Consider K-spades, K-diamonds, K-hearts, Q-hearts, J-hearts, 10-hearts, played against a 3-hearts, 4-diamonds, 5-spades, 6-clubs, 7-diamonds, 8-hearts, 8-spades. The trips Kings will lose to the 8-high straight, but the hand has many outs for a better high hand. It can improve to a straight flush, quads, full house, flush, or straight, and all of these hands would beat a low straight for the high-pot. However, the 8-high straight has a lock on the low-pot because a hand with trip Kings can never qualify.
One-way low-draws are drawing dead for the high-pot, but can still qualify for the low-pot. For example the hand A-clubs, 2-spades, 4-diamonds, 7-hearts, J-clubs, K-spades can never make a high hand better than one pair. If it plays against an opponent showing two pair--10s and 9s--on the board the hand is drawing dead for the high-pot, but drawing any of the remaining 3s, 5s, 6s, or 8s will win the low-pot.
Half-made hands are holdings that can win half the pot, but need improvement to win the other half. The three kinds of half-made hands in order of desirability are:
Coordinated highs are hands that still need to qualify for the low-pot, but have outs for a low-hand that will improve the rank of the high hand. For example consider the best possible coordinated high that you could hold on Sixth Street--2-clubs, 3-clubs, 4-clubs, 5-clubs, 5-diamonds, 5-spades. This hand is already trip 5s, but it has four sequential suited low-cards. It can improve its high ranking to quads, a full house, or a 9-high flush or better without qualifying for the low-pot. But, it can also improve it high ranking to a 5-high or 6-high straight flush, 7-high or 8-high flush, 5-high or 6-high straight and at the same time qualify for a low-hand. It can also qualify for low without improving its high ranking. For example it can become a 7-high or 8-high low-hand, and remain trip 5s.
Uncoordinated highs still need to qualify for the low-pot, but in doing so cannot improve their high ranking. While coordinated highs might have some outs to qualify for the low-pot that do not improve the high ranking, for uncoordinated highs none of the outs for a low-hand improve the high hand. Consider the best uncoordinated high-hand that you can hold--A-spades, 2-spades, 3-spades, 5-diamonds, Q-spades, K-spades. This hand is an A-K-Q-high flush with four low-cards. There are 16 outs to qualify this hand for the low-pot--any of the 4s, 6s, 7s, or 8s--but none of these outs will change the fact that the high hand is a flush.
Lows are hands that qualify for the low-pot, but rank poorly as high hand. For these hands you are drawing for the high half of the pot. For example the holding--2-hearts, 3-hearts, 4-hearts, 5-hearts, 7-diamonds, 8-spades is a qualified 7-high low-hand, but only an 8-high high hand. However, it has many outs available to improve its high ranking. An A-hears or 6-hearts would result in a straight flush, a K-hearts, Q-hearts, J-hearts, 10-hearts, 9-hearts, 8-hearts, or 7-hearts, would make it a flush, while an A-clubs, A-diamonds, A-spades, 6-clubs, 6-diamonds, or 6-spades, would make it a straight. Any of the remaining 2s, 3s, 4s, 5s, or unsuited 8s or 7s would make it one pair.
Drawing hands are holdings that need improvement to win any part of the pot. In order of desirability the kinds or drawing hands are:
Scoop draws have outs available that can simultaneously win the high and low pots. Consider a Sixth Street holding of 2-clubs, 3-clubs, 4-clubs, 5-clubs, 9-hearts, 10-spades. This open-ended straight flush-draw has outs that will improve to a high-only flush (K-clubs, Q-clubs, J-clubs, 9-clubs, 10-clubs), outs that will qualify for the low-pot (7-hearts, 7-spades, 7-diamonds, 8-hearts, 8-spades, 8-diamonds) along with outs that qualify for the low-pot and improve the high hand. The 7-clubs or 8-clubs completes a flush and qualifies for low; the A-clubs or 6-clubs completes a straight flush and qualifies for low; the A-diamonds, A-hearts, A-spades, 6-diamonds, 6-hearts, 6-spades, would all complete a straight and qualify for low.
One-way high-draws can never qualify for the low-pot. The only outs available improve the high-ranking. Consider K-spades, K-diamonds, K-hearts, Q-hearts, J-hearts, 10-hearts, played against a 3-hearts, 4-diamonds, 5-spades, 6-clubs, 7-diamonds, 8-hearts, 8-spades. The trips Kings will lose to the 8-high straight, but the hand has many outs for a better high hand. It can improve to a straight flush, quads, full house, flush, or straight, and all of these hands would beat a low straight for the high-pot. However, the 8-high straight has a lock on the low-pot because a hand with trip Kings can never qualify.
One-way low-draws are drawing dead for the high-pot, but can still qualify for the low-pot. For example the hand A-clubs, 2-spades, 4-diamonds, 7-hearts, J-clubs, K-spades can never make a high hand better than one pair. If it plays against an opponent showing two pair--10s and 9s--on the board the hand is drawing dead for the high-pot, but drawing any of the remaining 3s, 5s, 6s, or 8s will win the low-pot.
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