One of the smartest Seventh Street plays I’ve witnessed occurred in a three-way hand in which I had a lock on the low-pot, against two players vying for the high-pot. On Fifth and Sixth Streets the action began with Player 1, showing a high-hand on the board, who checked to Player 2, who checked to me. I bet and both players called. But on Seventh Street, Player 1 checked, and Player 2, who appeared to be on some sort of draw, bet. I raised because I thought Player 1, who clearly had the best hand all along, would call. But, Player 1, who did not want to get jammed by Player 2 and me, folded. I split the pot with Player 2 who had a high hand that could not have beat the high-hand Player 1 showed on the board. Player 2 said to me: “I was hoping that you would raise.”
Of course, I would not have raised had I known Player 1’s response would be to fold. Who won the high pot made no difference to me, I just wanted the pot to be as big as possible and driving Player 1 out reduced my winnings. Had Player 1 called the raise it would have increased my winnings. It is always difficult predicting your opponent’s response.
However, Player 2’s bluff was a smart play. In limit poker it is difficult to pull off successful bluffs late in the hand because the pot is usually large in relation to the cost of the single bet needed to call. Had Player 1 and 2 been heads-up, or playing high-only Seven-Card Stud, the bluff would not have worked. But, with me in between them holding the power to cap raises with no risk to myself, Player 1 had to consider the possibility of calling four large bets before showdown, which is a significant cost. By manipulating my action in the way that Player 2 did, he essentially used one bet to make a four-bet bluff. That is leverage not available in no-limit games in which a bluff with say an amount equal to half the pot, requires a bet equal to half the pot. Player 2 made a clever play that must have high expected value.
This events of this hand also illustrate the dangers inherent in playing a one-way high-hand in a situation in which everyone knows that it is a one-way high-hand. Player 1 did not bet, or check-raise my bets for two reasons. First he needed Player 2 to stay in order to make any money at all. Second, I would certainly re-raise, and Player 1 did not want to be heads-up against me for a huge pot, because I might be freerolling. The net result was that Player 1 was trapped into tentative play. He wanted Player 2 in the hand, but he did not want to be outdrawn for the high-pot. He wanted a large pot because he can only win half, but not too large because he might get scooped. He repeatedly checked and called, hoping for the best, but in the end was run off the pot by the more aggressive players.
If Player 1 had had a one-way high-hand with appearances of possibly qualifying for the low-pot, it would have discouraged Player 2 from trying his bluff, and caused me to think more carefully about raising given that I might not have a lock on the low-pot.
Tuesday, April 26, 2011
Thursday, March 31, 2011
Fifth Street Decisions: Betting into Three Exposed Low-Cards
The balance of power can shift dramatically on Fifth Street because it is now possible for a player to have a qualifying low-hand and potentially a lock on half the pot if no one else can make a better low. Prior to Fifth Street it is usually the high hands that drive the action. A player holding the best high hand will not want to give free draws to the low hands prior to Fifth Street. But, if at Fifth Street a player has accumulated three low-cards the high hand needs to be careful. The player with the best high-hand has a conundrum with two mutually exclusive goals for continuing.
Many players lean toward the cautious side and automatically check to a hand with three exposed low cards. However, that kind of routine play becomes predictable and cedes too much power to opponents with hands that could be mediocre at best. The highest hand scoops by default if no low-hand qualifies, and not every hand with three exposed low-cards on Fifth Street is a qualified low.
Consider a hand with four unpaired low-cards after Fourth Street such as (2, 3) 5, 7. If the low cards are live, 22 remain that are not 5s or 7s. Of these 22 cards, six will be 2s and 3s and will pair the hole cards. That means 6 out of 22 times, or 27% of the time, a low-card falling on Fifth Street will pair a hole card. If you are looking at an opponent's hand that consists of (X, X) 5, 7, 2 and believe that the hole cards are low and unpaired, the 2 represents a completed low-hand at most 73% of the time. Depending on the cards already played and your opponent's prior hand history, that percentage might be less.
For example, consider the hand (2, 3) 5, 7 with a board showing that two of the 6s, two of the 4s, and two of the 8s, dead. That leaves 16 remaining low-cards, not ranked 5 or 7, of which 6, or 37.5% will pair the 2 or the 3. That means in many hands in which significant numbers of low-cards are dead, the odds could tilt more towards the possibility that (X, X) 5, 7, 2 is a small pair rather than a qualified low-hand. To know how much the odds tilt requires some knowledge of your opponent because she might be more likely to draw with a four-low if many of the dead cards don't match her hole cards and reduce the possibility of pairing.
It's also possible that your opponent already has a pair-split or wired-going into Fifth Street and cannot complete a low-hand on that street. Sometimes the pair consists of wired high cards, in which case completing a low-hand at all is unlikely. If you have a better high hand you definitely want to bet. That means knowing your opponent's range of playable hands is necessary.
Given these considerations, here are some general guidelines for betting into a hand with three exposed low-cards on Fifth Street.
Situations to check:
* Your opponent has a very tight hand range that does not include small pairs and uncoordinated low-cards.
* The exposed cards include a 4 and a 5. Because these cards are necessary to complete low straights, hands with 4s and 5s can be dangerous holdings to bet into.
* The exposed cards are sequential.
* The exposed cards are suited
* It is a multi-way pot and your high holding is vulnerable in relation to the other hands in play.
Situations to bet:
* Your opponent has a wide hand range that includes any pair and any four low-cards.
* The 4s and 5s are dead or nearly so, in which case a split-pot is more likely if you have a solid high hand and your opponent does complete a qualified low-hand.
* You are playing heads-up, which reduces that chance of being out-drawn for the high-pot by the other players.
* You still have a chance to win the low-pot yourself. If you have Aces-up containing a low-pair and in addition a low kicker, or you have trips low-cards with two additional low-cards, you could still scoop against a qualified low-hand.
In summary, the decision to check or bet should be carefully considered. Do not get into the predictable habit of automatically checking to any hand with three exposed low-cards.
- Betting to protect the high hand and building a pot.
- Checking to reduce risk if the low-hand is freerolling for a better high-hand in order to scoop.
Many players lean toward the cautious side and automatically check to a hand with three exposed low cards. However, that kind of routine play becomes predictable and cedes too much power to opponents with hands that could be mediocre at best. The highest hand scoops by default if no low-hand qualifies, and not every hand with three exposed low-cards on Fifth Street is a qualified low.
Consider a hand with four unpaired low-cards after Fourth Street such as (2, 3) 5, 7. If the low cards are live, 22 remain that are not 5s or 7s. Of these 22 cards, six will be 2s and 3s and will pair the hole cards. That means 6 out of 22 times, or 27% of the time, a low-card falling on Fifth Street will pair a hole card. If you are looking at an opponent's hand that consists of (X, X) 5, 7, 2 and believe that the hole cards are low and unpaired, the 2 represents a completed low-hand at most 73% of the time. Depending on the cards already played and your opponent's prior hand history, that percentage might be less.
For example, consider the hand (2, 3) 5, 7 with a board showing that two of the 6s, two of the 4s, and two of the 8s, dead. That leaves 16 remaining low-cards, not ranked 5 or 7, of which 6, or 37.5% will pair the 2 or the 3. That means in many hands in which significant numbers of low-cards are dead, the odds could tilt more towards the possibility that (X, X) 5, 7, 2 is a small pair rather than a qualified low-hand. To know how much the odds tilt requires some knowledge of your opponent because she might be more likely to draw with a four-low if many of the dead cards don't match her hole cards and reduce the possibility of pairing.
It's also possible that your opponent already has a pair-split or wired-going into Fifth Street and cannot complete a low-hand on that street. Sometimes the pair consists of wired high cards, in which case completing a low-hand at all is unlikely. If you have a better high hand you definitely want to bet. That means knowing your opponent's range of playable hands is necessary.
Given these considerations, here are some general guidelines for betting into a hand with three exposed low-cards on Fifth Street.
Situations to check:
* Your opponent has a very tight hand range that does not include small pairs and uncoordinated low-cards.
* The exposed cards include a 4 and a 5. Because these cards are necessary to complete low straights, hands with 4s and 5s can be dangerous holdings to bet into.
* The exposed cards are sequential.
* The exposed cards are suited
* It is a multi-way pot and your high holding is vulnerable in relation to the other hands in play.
Situations to bet:
* Your opponent has a wide hand range that includes any pair and any four low-cards.
* The 4s and 5s are dead or nearly so, in which case a split-pot is more likely if you have a solid high hand and your opponent does complete a qualified low-hand.
* You are playing heads-up, which reduces that chance of being out-drawn for the high-pot by the other players.
* You still have a chance to win the low-pot yourself. If you have Aces-up containing a low-pair and in addition a low kicker, or you have trips low-cards with two additional low-cards, you could still scoop against a qualified low-hand.
In summary, the decision to check or bet should be carefully considered. Do not get into the predictable habit of automatically checking to any hand with three exposed low-cards.
Monday, February 28, 2011
Betting versus Checking on Seventh Street
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.
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.
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.
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