Basic Math Facts for Fifth Street in Seven-Card Stud High-Low Eight or Better

In keeping with last month's theme on math facts for Sixth Street draws, here are tables with information on Fifth Street draws. Here is an executive summary of some basic math facts to remember on Fifth Street with two cards still to come:

• There can never be more than 44 unseen cards by Sixth Street because you must have at least one opponent with 3 cards showing, and you will know 5 of your cards. That means that you will always know the value and location of at least 8 cards by Sixth Street.

• If there were 7 players dealt into the hand at the beginning and 5 folded on Third Street leaving you with heads-up play against one opponent, there are 39 unseen cards.

• The percentages below are based on 44 unseen cards with totally live draws. For each missing out subtract 2.3% (1/44) from the calculated percentage.

• For every 4 additional seen cards that do not include outs that you need, the numeric values of percentages shift upward by about 10%. For example the probability of hitting an open-ended straight draw (8 outs) by the end increases from 33.4% to 36.4% which is an increase of 3 percentage points (the number 3 is a little less than 10% of 33.4).

High Hand Draws (probability of improvement by drawing two cards)

Holding	Improve to:	Outs	Percentage
Trips	Quads	1	4.5 %
Open-ended Straight-Flush Draw	Straight-Flush	2	9.0 %
Pair	Trips	2	9.0 %
Inside straight-draw	Straight	4	17.5 %
Flush-draw with four-low	Low Flush	4	17.5 %
Two Pair	Full	4	17.5 %
Flush-draw with four-low	High Flush	5	21.7 %
Open-ended Straight-Draw	Straight	8	33.4%
Flush-draw	Flush	9	37.1 %

Low Hand Draws (probability of improvement by drawing two cards)

Holding	Improve to:	Outs	Percentage
8-high draw	8-high	16	60 %
7-high draw	8-high	4	17.5 %
7-high draw	7-high	12	47.6 %
6-high draw	8-high	4	17.5 %
6-high draw	7-high	4	17.5 %
6-high draw	6-high	8	33.4 %
5-high draw	8-high	4	17.5 %
5-high draw	7-high	4	17.5 %
5-high draw	6-high	4	17.5 %
5-high draw	5-high	4	17.5 %

Basic Math Facts for Sixth Street in Seven-Card Stud High-Low Eight or Better

I've been busy doing some math calculations for my upcoming book on Seven-Card Stud High-Low. The tables that follow contain information on Sixth Street draws. Here is an executive summary of some basic math facts to remember:

• There can never be more than 42 unseen cards by Sixth Street because you must have at least one opponent with 4 cards showing, and you will know 6 of your cards. That means that you will always know the value and location of at least 10 cards by Sixth Street.

• If there were 7 players dealt into the hand at the beginning and 5 folded on Third Street leaving you with heads-up play against one opponent, there are 37 unseen cards.

• The percentages below are based on 42 unseen cards with totally live draws. For each missing out subtract 2.4% (1/42) from the calculated percentage.

• For every 4 additional seen cards (additional opponent) that do not include outs that you need, the numeric values of percentages shift upward by about 10%. For example the probability of hitting an open-ended straight draw (8 outs) shifts from 19.0% to 21.1%, an increase of 2.1 percentage points (the number 2.1 is a little more than 10% of 19.0).

High Hand Draws

Holding	Improve to:	Outs	Percentage
Trips	Quads	1	2.4 %
Open-ended Straight-Flush Draw	Straight-Flush	2	4.8 %
Pair	Trips	2	4.8 %
Inside straight-draw	Straight	4	9.5 %
Flush-draw with four-low	Low Flush	4	9.5 %
Two Pair	Full	4	9.5 %
Flush-draw with four-low	High Flush	5	11.9 %
Open-ended Straight-Draw	Straight	8	19.0 %
Flush-draw	Flush	9	21.4 %
Trips	Full	9	21.4 %
Pair	Two Pair	12	28.6 %
6 unmatched	Pair	18	42.9 %

Low Hand Draws

Holding	Improve to:	Outs	Percentage
8-high draw	8-high	16	38.1 %
7-high draw	8-high	4	9.5 %
7-high draw	7-high	12	28.6 %
6-high draw	8-high	4	9.5 %
6-high draw	7-high	4	9.5 %
6-high draw	6-high	8	19.0 %
5-high draw	8-high	4	9.5 %
5-high draw	7-high	4	9.5 %
5-high draw	6-high	4	9.5 %
5-high draw	5-high	4	9.5 %

Stealing Antes

Of the Stud variants of poker, Seven-Card Stud High/Low is the hardest game in which to steal antes. For example in Razz, the bring-in is usually a face card, and very rarely will the bring-in defend against a complete bet from an exposed low card. In Razz you can have wired kings with an exposed 2 and steal the antes. It is more difficult to steal antes in high-only Seven-Card Stud because bring-ins can have many possible card combinations worth defending. In fact, in high-only Seven-Card Stud a bring-in with a pair or high-cards in the hole, will usually defend against someone she believes is trying to steal the antes. However, a bring-in in high-only Stud will usually not defend complete garbage.

However, in Seven-Card Stud High/Low, the bring-in will often defend against a player that she believes is stealing the antes. If the bring-in has an exposed low-card and picks up a second exposed low-card, she can become very aggressive. An opponent trying to steal antes with garbage might find himself in an uncomfortable situation, because the bring-in’s low-cards could allow her to freeroll on later streets. In Seven-Card Stud High/Low, low-cards because of the their dual roles in both high and low hands, have greater value than mid-range cards, which can only contribute to high-hands. A bring-in with an exposed 2 and 6, 7, in the hole has a better starting hand than an opponent with an exposed K and Q, 8, in the hole, even though the bring-in has no over cards. Attempting to steal antes with uncoordinated high-cards will not work in Stud-High/Low as often as it will in Stud-high. A significant fraction of the time, the bring-in will have a hand worth defending, or even raising with in response to a completed bet.

Calling versus Raising on Seventh Street

If you are debating whether to call or raise, it often means that you have a chance to scoop or it is a multi-way pot for which you expect to win half and therefore seek to maximize the pot size. If you have a lock on both halves of the pot, raising is automatic and no decision needs to be made. But, when the outcome is not clear, you need to consider which action is most likely to maximize your earnings. Consider these situations:

Nut low-hand versus a better high-hand in a heads-up pot: This is a common occurrence and often the low-hand cannot beat the board, which means that you know that a split-pot will result at showdown. Many players in this situation simply call. However, this is a mistake. If you have the nut-low you should raise for two reasons. First it often forces your opponent to make a difficult decision because his high holding might not be that strong. For example, a low pair showing in his hand that you cannot beat might be all he has for high. His initial bet might be a bluff and a raise could induce a fold. Second you do not want to develop a pattern of raising in situations in which your low-hand scoops and calling when it does not. Obviously, when your low-hand scoops you want your raise called. Only raising when you want to be called will soon result in no callers in those circumstances. That might be useful for inducing a fold later on when you need an opponent to fold. But, in general it is better to keep up the pressure on your opponents. It is better over the long run to have a table image in which you are feared because opponents will check to you more often and allow you to drive the action.


Nut low-hand versus a better high-hand in a multi-way pot: Raising might not be the way to extract maximum value in a multi-way pot if it drives out paying customers. If an opponent acts after you, and has to cold-call two large bets, she is more likely to fold than if only one bet is required. Consider a three-way pot in which you act second after an initial bet from Bob but before Alice. Consider these possible outcomes:

1. You call, Alice folds (no additional gain).
2. You call, Alice calls (you gain half a large bet).
3. You raise, Alice folds (no additional gain).
4. You raise, Alice calls, and Bob calls (you gain one large bet).
5. You raise, Alice calls, Bob re-raises, you cap the action, and Alice and Bob both call (you gain two large bets).

Obviously, since you have the nut low-hand, scenario 5 is the one you desire. But, that requires a raise on your part, and with it a risk of scenario 3 in which Alice folds leaving you with no additional gain. On the other hand, if you call Alice might still fold (scenario 1) leaving you with no gain. That means that if you decide to call you are hoping for scenario 2 as opposed to scenario 1.

You need to judge how much Alice likes her hand. She will need a lot of confidence for scenarios 4 or 5 to play out. But, if Alice doesn't like her hand it will not matter if you call or raise. Therefore, the only reason to call is if you think Alice is unsure of her hand but willing to make a crying call at the end if the price stays low.

The situation is of course different if you act after Alice, and she already called Bob's initial bet. In that case you have nothing to lose by raising and everything to gain.

Non-nut low-hand versus a potential better low-hand: As a general rule you should call with any qualified low-hand at the end. Pots are usually large by the end and it is never a certainty that anyone else has a qualified low-hand. A single incorrect fold can be very costly in relation to the cost of calling one last bet.

However, cold-calling raises from players with potentially better low-hands is problematic. These can be some of the toughest decisions that you will make. Suppose you have a weak low-hand such as an 8-6 high against an opponent showing a 6, 4, J, 9 on the board who suddenly raises on the end. In this circumstance the chances are high that your low-hand is beat. If the player who initiated betting appears confident in a high-hand you have to weigh the loss of four large bets against the reduced chance of winning back anything. Folding in these situations will most likely save money over the long run.

It is to avoid these expensive traps that you should not routinely play weak low-hands such as 8-7 or 8-6 high. When you play 8-high low cards you will face these kind of tough decisions often. There will of course be times that you back into an 8-high low with legitimate starting cards such as 3, 4, 5, 6, but those will be less frequent occurrences.

I have seen players automatically raise on the end with any qualified low-hand, no matter how weak. This is often a costly mistake. Weak low-hands lose frequently in Stud-Eight so you should not get too confident with an 8-high or even 7-high low-hand unless no other low-hands are possible. Remember that it is possible for any player with two wheel cards on the board to have a wheel. Also remember that if you are showing cards such as 8, 5, J, K, a player with a 7-high low or better can raise back because he will know that his low-hand is better.

Calling versus Folding on Seventh Street

If you are deciding between calling and folding on Seventh Street, it means that you do not like your holdings. One of the toughest decisions in Stud-Eight is whether to call on the end with a hand that finished “one-way,” meaning that the hand can only take half the pot. By Seventh Street, the pot can be very large, so losing even half the pot to an incorrect fold can be costly. In many cases, weak one-way hands can hold up for half the pot, if the action came from players vying for the other half. Here are some of the more uncomfortable circumstances to be in at the end.

A weak low versus a potential low: This situation arises with hands such as (5, 6) 7, A, K, 9 (8). By picking up an 8 on the end you are saddled with one of the worse possible low-hands and a terrible high-hand. Some one showing (X, X) 7, 4, 2, A (X) can easily have you beat. However, as a general rule, if you have a qualifying low-hand, and there is a large pot at the end, you should call because it never certain that four scary looking low-cards in an opponent’s hand are part of a qualified low-hand. An exception can be made, if it is a multi-way pot with a lot of raising, in which case it is not a bad play to fold. Good hand reading ability is vital in making the decision because there are times when the player with the low cards has a wired high-pair and little chance completing a low-hand. In heads-up play don’t fold if your opponent appears to be playing for low and you can beat the board for high. In the example above, the A, K might hold up for the high-pot if the other hand hasn’t paired. If the other hand has paired, it will be less likely to qualify for the low-pot.

Two pair versus a potential straight: If the straight cards are high and the player holding them bets or raises into a threatening board, you can confidently muck the two pair. Some one checking and calling with (X, X) J, 10, 9, K, (X) who suddenly bets on the end into a possible low-hand, made the straight. A more difficult situation is when the straight cards are low. For example, you have (A, 2) A, 3, 6, 2 (J) against a player with (X, X) 4, 5, 6, 2 (X), who bets out on the end without having made any prior raises. You lost your chance for a low-hand when the Jack hit, and there is a good chance your Aces-up are no longer good for high. Generally you should call here, unless it is a multi-way pot with a lot of jamming on the end, in which case your hand is clearly no good for high. Again, in making this decision, watch for the 4s and 5s and the connectedness of the cards. For example, your chances are much better for taking the high-pot if you have (A, 4) A, 4, 5, 6 (J) and your opponent has (X, X) 2, 3, 7, 9 (X), than in first example in which all your opponent needs is a hidden 3 to scoop.


One small pair versus a potential low: This is often an apparently hopeless situation, but surprisingly it might not be and you can give up a lot of equity by routinely folding. Suppose you have (3, 4) 5, 6, J, 9, (5) versus (X, X) 7, 3, 2, K (X) and your opponent bets. You missed both the straight draw and the low-hand draw and have nothing but a pair of 5s for a high-hand. However, if your opponent played only for low he might not have any paired cards, which means that your 5s are good for high. In fact, I’ve been in hands in which calling with the 5s scooped for me because the bet was a semi-bluff from an opponent who also missed the low-hand and had an even lower pair.

Trips or less versus another hand betting into a low-hand: A frustrating occurrence is to start with rolled trips or two small pair that do not fill up or make a low-hand. Finishing with (3, 3) 3, 4, 5, J (K) or (A, A) 2, 2, 5, 6, (Q) are frequent outcomes from promising starting cards. These kinds of hands are still strong, but qualify only for the high-pot which limits your implied pot odds on the end. In a heads-up pot, you would simply call. The difficult decision arises in multi-way pots in which a player bets into a potential low-hand, who either raises or might raise when her turn comes. What to do depends on your judgment of whether the player making the bet is representing a high-hand or a low-hand.

If the bettor is representing a high-hand, she has you beat because she wouldn’t bet unless she has better than a low straight. There are exceptions, such as players who bet two pair aggressively in this situation anticipating that for this board you and the other player each have low-hands. However, if you played your early cards aggressively, which you should have, that usually dissuades most players from betting into you on the end unless they hit a monster hand. As painful as it might be, it is better to fold in this situation rather than get jammed by the low-hand.

An extremely uncomfortable situation arises if the bettor is representing a better low-hand than the other player’s low-hand. If each opponent thinks that his or her low-hand is the best, each will want to cap the raising. In that case you will have to go along and hope that neither holds a low straight. Again your aggressive play early in the hand because of the strength of your starting cards means that the pot prior to Seventh Street betting is already large.

A Clever Bluff

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.

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.

• 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.