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.