A key feature of lowball games such as Razz or Deuce-to-Seven Triple Draw is that all players are competing to make hands using the same set of low cards. Thus it is fairly common that some of the cards which you need to complete a hand will be held in another player’s hand. An unsophisticated estimate may overvalue your hand’s strength, even if you properly take into account the probability that you are drawing “dead” to a better made low.

Calling and Drawing One on the Last Round

In Deuce to Seven Triple Draw, you will often begin with a very strong starter, play it aggressively, and then be confronted with a bet or a raise from a player who is likely to be pat on the last draw with a hand that is a favorite to beat yours. In practice the pot has usually grown to the point where it is correct to continue. But when the pot is small or your opponent is predictable, calculating the correct odds of drawing becomes very important. It is also useful to approach a hand from the opposite perspective; given a correct understanding of 3rd-round odds, you can evaluate the effects of manipulating the pot size.

Suppose you hold 7c 4c 3d 2h Jc; this is a very strong starting hand, and you raise it from UTG at a six-handed table. It is folded around to the big blind, a fairly passive and straightforward player, who calls. There are 5 small bets in the pot. The big blind draws one and you draw one as well, getting another jack.

The big blind checks and you decide to check as well. Again you both draw one, and you get a T. The big blind bets out, offering you 3.5 to 1 pot odds. Your read is that he will only do so with an 8-low or better in this situation. (However, any 9, T, and usually a J is a favorite to win if you are still drawing) How many outs in how many unknown cards?

The naïve estimate is to say that we have somewhere between 8 and 12 outs in 45 cards. We have seen 7 cards total (the original five and two drawn cards), and any 5 or 6 will almost certainly win, while an 87432 will beat some of the rougher possible pat hands, perhaps 50% of the time. We may well decide on 10 outs in 45 unknown cards, or 3.5 to 1 odds, making it a break-even call, if we ignore the implied odds of making a 75432 (a “wheel”) or 76432 on the end.

But in this case the true odds are significantly enough higher that we should expect to lose about 12% of a bet on these calls. You know that your opponent has to have five cards between 8 and 2 in his hand, so it is likely that some of the cards you need are dead.

The method for calculating a better answer is to enumerate each of the possible holdings, assign them a probability, and calculate the number of outs based on that particular holding. Calculating the weighted average of our expectation in each case results in the final answer.

This process can really only feasibly be done on a spreadsheet (or with a computer program), because the distribution of the opponent’s hands is weighted by the cards we hold. For example, because we hold a 7, 4, 3, and 2, there are fewer available cards to make hands such as 76432. (Only 3 of each of the wheel cards, plus 4 sixes =
324 possibilities.) We have not seen an 8, 6, or 5, so our opponent is more likely to be holding hands such as 86532 (4 * 4 * 4 * 3 * 3 = 576 possibilities, nearly twice as likely.)

In this analysis I will be ignoring any additional information we might have about the strength of the opponent’s holding, so all pat 8’s or better are weighted equally according to the available cards. I will also neglect to subtract the possibilities that are flushes, which has only a small effect on the answer. Finally, I assume that we have seen three face cards (J, J, and T in the example above) rather than a low card that would influence the probabilities of various opponent holdings.

Here are the results of carrying out this exercise:

Hand	Weight (# of possibilities)	Relative Probability	Outs in 40 unknown cards	Expectation
75432	324	0.0396	1.5	0.0375
76432	324	0.0396	5.5	0.1375
76532	432	0.0529	6	0.15
76542	432	0.0529	6	0.15
85432	432	0.0529	7	0.175
86432	432	0.0529	7	0.175
86532	576	0.0705	6	0.15
86542	576	0.0705	6	0.15
86543	576	0.0705	6	0.15
87432	324	0.0396	9.5	0.2375
87532	432	0.0529	10	0.25
87542	432	0.0529	10	0.25
87543	432	0.0529	10	0.25
87632	432	0.0529	10	0.25
87642	432	0.0529	10	0.25
87643	432	0.0529	10	0.25
87652	576	0.0705	9	0.225
87653	576	0.0705	9	0.225
Total / Average	8172	1.0000	7.7	0.193

Outs for a tie are counted as half outs. “Relative probabilities” shows the expected distribution of holdings. Note that we sometimes have more outs to beat a stronger holding, and that the outs are relative to 40 unknown cards because we have specified all five of the opponent’s cards. (But not his earlier discards. We cannot tell whether they are bricks that are advantageous to us to have been discarded, or paired cards that might be useless to him but advantageous to us.)

The answer this process gives us a pot equity of about 19.3%. This corresponds to odds of 4.2:1, or about 8 and a half outs in 45 cards, almost as if we had entirely discounted our “8” outs. But it is not that the 8’s have little value. Nearly half of the opponent’s holdings in this situation are 87432 or worse, so our estimate that an 8 would be good half the time was pretty much correct. It is that the simple estimate does not properly take into account the dead cards held in the opponent’s hand. Notice that all but one of the opponent’s hands must contain at least one five or six! We were about one and a half outs too high; in the next sections I will show that about a half-out per rank is an
appropriate rule of thumb to use (as long as the possibility that a made hand still loses is taken into account.)

Evaluating Outs Before the First Draw

A similar procedure can be used to enumerate possible starting hands and calculate the average number of dead outs in an opponent’s hand. This process is somewhat more complex because a three-card starting hand (such as 752xx) may be “stronger” than average by containing low paired cards (like 75522) instead of “bricks”
(like 752KQ.)

Again, for simplicity the small number of flushes is ignored. In this section I also ignore any specific cards that we might be holding. The starting range I will examine here is drawn from the recommendations by Daniel Negreanu in Super System II:

• All pat 7s and 8s
• Any one-card draw to a 7 that includes a 2 (7xx2)
• Any one-card draw to an 8 that includes a 2 or 3
• The one-card draws without a 7 or an 8 (2345, 2346, 2356, 2456)
• Two-card draws that have a deuce but not an 8 (234, 235, 236, 237, …,
  267.)
  *The strong two-card 8 draws 238, 248, and 258.

A pat hand like 76432 has 1024 possibilities (4 suits for each of 5 cards) and “kills” one each of the five cards it holds.

A one-card draw like 7632x is more complicated. “x” cannot be 8, 4, or 5, as those would provide a different, pat, starting hand. There are 24 possible “bricks” (each of 9TJQKA). This gives 6144 possibilities for the starting hand 7632x where “x” is a brick (9 or higher), because 6144 = 4 * 4 * 4 * 4 * 24. But there are also 1536 additional
possibilities where ‘x’ pairs one of the other cards. For example, there are 384 = 4 * 4 * 4 * (4C2) ways of selecting 76632. These additional possibilities “kill” more than one out.

A two-card draw requires another level of consideration. 432xx where xx are both bricks provides 17,664 = 4 * 4 * 4 * (24C2) possibilities. 4322x, 4332x, and 4432x provide an addition 2304 = 4 *4 * (4C2) * 24 possibilities each. And 43222, 43322, 44322, 43332, 44332, and 44432 each provide an additional 624 possibilities. The total count is 546,736 starting hands in 2,598,960 5-card hands (about 21%.) Of these, about 3.4% are pat hands, 36.7% are one-card draws, and the remaining 59.9% are two-card draws. (In practice, depending on position and the pre-draw action it is of course appropriate to play more or fewer hands than this, including some three-card
draws.)

The measure of the most interest is the weighted sum of dead outs in each hand. Across this range, the expected number of dead cards is as follows:

Rank	Expected number of cards
2	0.99
3	0.52
4	0.47
5	0.47
6	0.42
7	0.42
8	0.42
Total	3.70

That is, if your opponent has a quality starting hand, the expected number of deuces he holds is about one (i.e., he almost always has one) while there is about 40-50% expectation of him holding any other low card. Thus, in the absence of other information, a reasonable rule of thumb may be to assume that the true number of outs is lower by about half an out per rank. For example, a 5432 does not have 8 outs three times to make a strong pat hand, but closer to 7 outs. (But remember, since we have put the opponent on a hand range we must also decrease the number of “unknown” cards appropriately.)

One point illustrated here, which all solid Deuce-to-Seven players understand, is the importance of starting with a deuce. Not only does this prevent you from making a straight, except for draws like 2345, but you are much less likely to draw a deuce (against solid opponents) than any other card. A hand like 3457x thus labors under a double disadvantage of having more dead outs to a wheel (since the other players will likely have deuces) and of lacking the “6” outs since they produce a very weak holding, a straight.

A player trying to calculate outs will already know the contents of his own hand, which changes the probabilities for the opponents’ holdings, as demonstrated in the previous section. It is also unknown at this point how well this rule of thumb extends to multiple players.

Dead Outs Versus a Possibly Rough Pat Player

Here is a third example of dead outs, calculated for the case where we hold 2348 and have again seen three high cards, versus a player who is pat with any 9 or better (including ones derived from poor starting hands):

Rank	Expected number of cards
2	0.64
3	0.62
4	0.60
5	0.66
6	0.66
7	0.48
8	0.62
9	0.72
Total	5.00

Notice that the expected number of dead cards now totals to 5, since our opponent no longer has a possibility of holding any face cards or pairs.

The half a dead out per card estimate is a reasonable first approximation here, as the average is 9/5 = 0.55 dead outs per card. But, we need to discount our outs yet further to handle the cases where we may be drawing dead, or our highest rank out is no good. A uniform distribution of pat hands such as the one I assume here is heavily weighted
toward 9’s (72% of the possible holdings), but even so the possibility of drawing a nine and winning contributes only fractionally more than 1 out. A five, on the other hand, is usually good and is worth a bit more than 3 outs. A detailed calculation shows that we have 10 outs (in 40 cards) in this situation, or about 3 to 1 odds.

Can we estimate this knowing only the general shape of the pat hand distribution?

• 98xxx’s account for about 40% of the pat hands, but all these hands have one 9 already dead. Add this to our estimate of 10.5 outs for 5’s, 6’s, and 7’s (using the half-out rule of thumb) and we get 13.5 outs for this range.
• Next, better pat nines account for another 30%, and 87’s are a further 10%; in these cases any 5, 6, or 7 is good, for an estimated 10.5 live outs.
• 86s and 85’s constitute another 10% of hands but only a 5 or 6 is good and possibly gets half, so we can have no better than 6 outs.
• The remaining 10% of pat hands are sevens, to which we are drawing dead.

Putting these out counts together gives us 0.40 * 13.5 + 0.40 * 10.5 + 0.10 * 6 = 10.2 effective outs, pretty close to the value arrived at via a more complicated analysis. This hand distribution is probably a reasonable estimate to use whenever an opponent may have an arbitrary 9 or better. (However, an opponent who started with a decent hand
will not make extremely rough 9s such as 98654, nor hands such as 96543, so the assumption of uniform distribution does not strictly hold--- which explains why deuces do not appear more dead than other cards in this distribution.)

Conclusion

In lowball games, some of your outs are nearly always dead since all players are competing for the same set of cards. Simple methods of out-counting overestimate your draw’s strength, even if you correctly estimate the chance that your made hand will be good. Although it’s generally not possible to evaluate, in real-time, what your true odds of winning a hand are, it is possible to develop rules of thumb to adjust for the cards held by other players and be aware of their impact on the odds.

The primary rule of thumb introduced here is to discount each of your possible ranks by a half an out against one opponent; this seems like a successful way to estimate how many of your outs are in his hand.

A second rule of thumb is the incredible importance of holding a deuce in your starting hand. Deuces are twice as likely to be dead as any other card, against standard opponents, so even a playable starter such as 8643 is weaker than a simple out count would suggest.

Finally, a good triple draw player must have a ready estimate for how likely his higher-rank outs are to be good, because they may hit yet lose the hand. In the examples given here, the highest out was good no more than 50% of the time, and I suggest this as a third rule of thumb.

(This article originally appeared in the October 2006 issue of Two Plus Two Magazine.)