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.

• 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.

The common Fourth Street confrontation between a high-hand and a draw to a low-hand has a precarious balance. The player with the low-hand has more information because it is difficult to hide a high-hand, while the player with the high hand must guess at the quality of the low-hand. However, no qualified low-hand can exist on Fourth Street which means that high hand is poised to scoop unless drawn out against.

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.