Using Dealt Pat Hand Distributions to Make Decisions in 2-7 No Limit Single Draw

Deuce to Seven No Limit Single draw is often described as the purest of all poker forms. Often times you are faced with huge decisions that hinge on one simple question; “Does he have it or not? But there are some decisions that are based almost purely on mathematics. Those who are interested in picking up the game either on its own or as part of mixed games must have a sound base in certain fundamentals to aid in their decision making. One of these is knowledge of the dealt pat hand distribution. A pat hand is one that is complete in that you do not intend to break in an attempt to make a better hand. The distribution suggests that mathematically you are much more likely to be dealt a weaker pat hand as opposed to a strong one. This article will detail that distribution and provide examples on how one would use this information.

Straights and flushes count against you so the best possible hand in the game is 23457 unsuited. There are only three other seven hands, 23467, 23567, and the 24567. But there are many more pat eights, nines, tens, and jacks:

 # Pat Hands
Sevens4
Eights14
Nines34
Tens69
Jacks125

Note how many more hand combinations there are towards the bottom of the table. There are more dealt pat jacks then sevens, eights, nines, and tens put together. An important fact in NL 2-7 Single Draw is that a pat jack is typically a favorite over any one card draw. Thus many players will play all of these hands strongly before the first draw. And to avoid playing with their hands face up many players will play all of their pat hands in a similar manner.

Thus if both you and your opponent are pat, it is often useful to know how your likely it is that you have the best hand against a range of pat jacks or better:

Vs Pat Jack or Better   
# Hands# Hands Beat or "Tie"% Win or "Tie"
75XXX1246100.00%
76XXX324599.60%
85XXX124298.40%
86XXX424198.00%
87XXX923796.30%
95XXX122892.70%
96XXX422792.30%
97XXX1022390.70%
98XXX1921386.60%
T5XXX119478.90%
T6XXX419378.50%
T7XXX1018976.80%
T8XXX2017972.80%
T9XXX3415964.60%
J5XXX112550.80%
J6XXX412450.40%
J7XXX1012048.80%
J8XXX2011044.70%
J9XXX359036.60%
JTXXX555522.40%
246

This chart is somewhat simplistic in that it does not account for card removal based upon your holding. For example, if you are holding 96542 it is less likely that your opponent is also holding a nine. It also lumps several hands together in the same grouping. For example this chart assumes 98432 “ties” with 98764. But it is very handy as a quick guide.

Let’s look at some examples where we will take a more precise treatment:
Hand in Action 1: Keep the Ten or Draw to the Nuts vs the All In Player?

$10/$20 $5 ante Cash Game

The action folds to you on the button with T7432. You raise to $60. The small blind folds and the big blind goes all in for $80 total. You make the trivially easy call of $20. The big blind stands pat. The big blind is on a short stack and a jack is generally favored over any one card draw. Thus it is assumed that the big blind can hold any pat seven through a jack. Do you stand pat or draw one?

You lose to all sevens (4), eights (14), and nines (34). That is 52 hands. You beat all jacks and there are 125 of them.

Only five tens beat you, the T5432, T6432, T6532, T6542, and the T6543. You tie another T7432 but beat the remaining 63 tens. But since you hold a ten, it is 25% less likely that your opponent hold one due to card removal. Thus the tens must also be discounted due to the fact you hold a ten. This is the reason for the 75% factor applied below.

Total Hands that beat you: 4 ‘sevens’ + 14 ‘eights’ +34 ‘nines’ + 5(.75) ‘better tens’ = 55.75

Total Hands that you beat: 63(.75) ‘worse tens’ + 125 ‘jacks’ = 172.25

Thus by staying pat you will win [172.25/(55.75 + 172.25)] = 75.56% of the time.

But if you drew one card you would be an underdog. Even against JT986 you would only win 45.2% of the time.

Note: You also hold a seven so technically you should also discount the sevens, 87XXX, 97XXX, and J7XXX hands. But we don’t need to get that detailed with this example. Incorporating this would only increase the win percentage here. In addition, we should also acknowledge that the big blind could also hold a queen.

This result may surprise some new players as they don’t realize how many more pat tens and jacks there are in the initial pat distribution compared with sevens, eights, and nines. They logic that they lose to all pat nines and better and some tens and see their super smooth draw and decide to take a card. That would be a major mistake.

However, due to card removal you should break a J7432. Referring to the chart above we see that without card removal the win/”tie” percentage is 48.8%. Card removal would remove 75% of the jacks you beat and thus make it clearly correct to draw.

Hand in Action 2: Check or Bet the Drawn 98432

$5/$10 Cash Game with $10 button ante; effective stacks of $1000

The action folds to you in the small blind with K8432 and you raise to $30. The big blind re-raises to $110. Re-raising is an option but you decide to call and draw one. Villain stays pat. You make a 98432.

What is your plan post-draw?

We first need to make some assumptions with regards to the big blind’s holding. He has the advantage of acting last and his range is thus uncapped. That means he can either have a monster (seven or eight) or a marginal hand (tens and jacks) that he would be happy to just showdown with no more betting after the draw. He could also be on a pure bluff.

Assuming he would re-raise all Pat Jacks or better (along with some bluffs) you should immediately recognize that you are a huge favorite to hold the best hand.

What if we tighten up his re-raising range to all tens or better?

You beat all of the tens. And with a 98432 you also beat 18 of the 34 different nines which are 98532 through 98764.

Due to card removal we should apply the 75% factor to all pat eights and nines. In addition, let’s also apply the 75% factor the T8XXX and T9XXX hands of which there are 54.

Total Hands that beat you: 4 ‘sevens’ + 14(.75) ‘eights’ + 15(.75) ‘better nines’ = 25.75

Total Hands that you beat: 18(.75) ‘worse nines’ + 15 ‘T6XXX and better tens’ + 54(.75) ‘T8XXX and T9XXX tens’ = 69

Thus if even an opponent re-raises with a pat ten or better you are approximately an 73.6% favorite [69/( 24.75 + 69)] to have the best hand.

However most players would re-raise with any pat Jack or better so all things considered you are usually over an 80% favorite to have the best holding in this situation. If your opponent just calls and stands pat than their hand is basically face up as a very marginal weak holding. That is why we assume that most opponents re-raise with a lot of hands weaker than a ten or better.

So after the draw do we check/call, check/fold, bet out or go for a check/raise?

If you check, your opponent will likely check back the vast majority of hands that you beat. With a jack and most tens your opponent will very often be happy to just rap the table and see if they collect the pot. Most of them would only value bet hands that beat you and thus they would have to have a large percentage of “snows” in their range to make this call profitable. (A “snow” is a bluff when an opponent stays pat with a hand that has no chance to win in a showdown.) So a check/call does not seem ideal. Check/folding is risky if your opponent may possibly bet worse pat hands for value or if he is tricky and “snows” a lot. Bottom line is that if you are checking you are often guessing what to do when faced with a big bet after the draw. And when you are guessing in poker you are not usually winning.

Since you beat the vast majority of your opponent’s hands, you should disappoint them by betting out. This bet needs to target your opponents tens and jacks and put them to a tough decision. The bet can’t be too big that it will scare off your opponent nor should it be so small that it makes it easier for your opponent to raise you. This would be either for either thin value or as a bluff and would put you to a very tough decision.

Thus it is usually right to bet around 2/3 pot in this situation. A 2/3 pot bet gives your opponent 2.5 to 1 odds on his call. In order to balance this bet there should also be some bluffs in your range when you miss. Bluffing with the 6 cards that pair your two highest cards (in this case the eights or fours) is a reasonable bluffing strategy for this situation. Of course if you feel your opponent can be exploited one way or another you would adjust this strategy.

A check/raise could be creative and bold but it would seem to be a very unprofitable line. You would collect bets from his “snows” and it is within the realm of possibility that he would bet/fold hands slightly better hands than you hold such as 97652. You hold a nine so it is less likely that he has a pat nine. It is way more likely that he holds pat tens and jacks anyway and as we are already discussed he will often check back these hands and you lose potential value.

The vast majority of players tend to check/call in this situation. But knowledge of the dealt pat hand distribution combined with a solid background in poker theory will lead to more profitable lines in this hand and many other situations.

Summary

Some of your playing decisions come down to simple mathematical exercises and this low hanging fruit should be grasped. However, No Limit Single Draw can often turn into a battle of psychological warfare. So you must enter these conflicts heavily armed with a working knowledge of what range of hands your opponent is likely to hold. This is vital in this game as it is in any form of poker.