Wednesday, September 15, 2010

Counting Outs in Seven-Card Stud High-Low

Having identified the different kinds of drawing hands last month, we can now count the outs available in order to determine likelihood of improvement. However, Stud games, in contrast to flop games, usually have many more exposed cards to account for when adding up outs. As a result, counting outs is much more situational in Seven-Card Stud High-Low than in Hold'em.

Consider the following extreme example. For the coordinated high example of 2-clubs, 3- clubs, 4-clubs, 5- clubs, 5-diamonds, 5-spades there are a total of 31 cards that will improve its high ranking, qualify the hand for the low-pot, or do both. But the 31 total is an upper limit on the number of outs available to improve the hand. Suppose the 9- clubs, 7- clubs, 4-diamonds, and 6-spades are exposed other hands. All these cards are no longer available as outs, so the count needs to be reduced by four to a total of 27

Let us also suppose that another player has all four Aces exposed in her hand. Now not only must the Aces be removed from the total outs, but most of the high draws for this hand are now dead. There is only one out available for this hand to take the high-pot-the 6-clubs-and for the low pot all that remain are the two other 6s, three other 7s, and the four 8s. Instead of 31 outs available, there are actually only 10 cards that matter, because if someone already has quad Aces hitting quad 5s or any of the full houses, or high flushes, will not win anything.

The consideration of this extremely unlikely scenario is useful because it illustrates four important points about counting outs. You need to subtract from the total for the following reasons.

Dead cards: You must subtract from the total available outs all the helpful cards that you see in other hands because they are no longer available. This is different than in Hold'em, in which every card that you see is potentially playable.

Dead draws: You must subtract from the total available outs all the cards that you see that improve your hand but do not win. Again in contrast to Hold'em, you can often see in a Stud game when you are drawing dead, because certain improvements will not beat your opponent's exposed cards ("beat the board").

Weighted outs: Not all outs are of equal value. In high-low games some outs win the entire pot while other outs win just half. In the example of the straight flush draw versus quads, the 6-clubs wins it all while the nine available low cards win just half the pot. To account for this difference, in this book I introduce the concept of counting "weighted outs." To count weighted outs, add 1 for each out that scoops, and 1/2 for each out that only wins half the pot. For this example there is one out that scoops and nine outs that take the low-pot, so the total number of weighted outs is 1 + 9 x (1/2), or 5.5. I will show in a future post how a count of weighted outs can be used to determine pot equity, which is a measure of how much you should invest when betting on a hand.

Implied outs: You can often subtract from the total number of outs, cards that you cannot see because your opponent's actions imply the contents of their hole cards. Suppose three players in the hand after Fourth Street show 2-3, 5-7, and A-2, and all act as if they are on draws to low-hands. If you are looking for a low-card, it is clear that six of them are dead, but in this situation you can imply that 12 are dead, because the six unseen hole cards are most likely low.

Once an accurate count of outs is determined, the probability for improvement can be found by dividing the total outs by the number of unseen cards.