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.