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.

Probabilities for Simultaneous High and Low-Draws: Weighted Outs

The big-money hands in Seven-Card Stud High-Low are the low straights and low flushes. These hands often scoop and better yet, usually get paid. Any player with at least two-pair will pay if the pot is large, because it is not possible to know for certain that the hand qualifies for low, let alone know that it is a straight or flush.

However, if you start with connected or suited cards, and then pick up a helpful card on Fourth Street, you still do not have a winning hand. Here is a summary of the outs available to complete the low-hand, high-hand, or both for some common Fifth Street-draws. For these tabulations I will introduce the concept of "weighted outs". In many cases the total number of outs to make a high-hand, low-hand, or both, are the same. But some circumstances are more favorable than others, because the numbers of outs that can make both a high-hand and a low-hand are greater than the number that make high-hand only or low-hand only.

To reflect this discrepancy, I define a "weighted out" as equal to 1 if the out completes a high-hand and a low-hand, and one-half (0.5) if the out completes a high-hand or a low-hand. Totaling "weighted outs" is a better measure of your actual equity in the pot. For example, if you have seen 16 cards after Fourth Street, there are 36 unseen cards. If your total weighted outs are 12, you have 33% equity in the pot. Some of the outs might result in winning only half the pot, and some the entire pot, but on average you can expect to pick up 33% of the money on the table if the scenario is repeated many times.

Here are the tabulations:

Open-ended straight-flush with low-draw

Example: 3, 4, 5, 6 all in Spades

Improve to:	Outs	Weighted Outs
Straight-Flush	2	2
Straight	6	6
Low-flush	2	2
High-flush	5	2.5
Low-hand	6	3
Totals	21	15.5

Four-flush with inside straight and low-draw

Example: A, 3, 4, 5 all in Hearts

Improve to:	Outs	Weighted Outs
Straight-Flush	1	1
Straight	3	3
Low-flush	3	3
High-flush	5	2.5
Low-hand	9	4.5
Totals	21	14

Four-flush with low-draw

Example: A, 3, 5, 6 in Hearts

Improve to:	Outs	Weighted Outs
Low-flush	4	4
High-flush	5	2.5
Low-hand	12	6
Totals	21	12.5

Open-ended straight with low-draw

Example: 3, 4, 5, 6 rainbow

Improve to:	Outs	Weighted Outs
Straight	8	8
Low-hand	8	4
Totals	16	12

Inside straight with low-draw

Example: 3, 4, 6, 7 rainbow

Improve to:	Outs	Weighted Outs
Straight	4	4
Low-hand	12	6
Totals	16	10

All of these tabulations count the maximum number of outs possible, which only occurs when the draw is completely live. It is rare that a Fifth Street draw, in which 10 to 15 cards are already exposed, would be completely live. That is why counting weighted outs on the later streets is so important. Consider the most favorable draw-an open-ended straight-flush with low-draw. If the 6 outs to a low-hand are missing, the weighted outs drop by 3 to a total of 12.5. But, if the 6 outs for the straight are missing, the total weighted outs drops by 6 to 9.5.

The tabulations above are still valid on later streets. In all likelihood there would be fewer unseen cards on later streets, but more missing outs. As is usually the case in Seven-Card Stud High-Low, pot equity can change drastically from street-to-street as cards are exposed.