Counting the low and high cards in the deck

To calculate the overall probability on the flop, turn, or river, we need to know how many total board combinations exist at each of these stages, and the types of boards possible.  To know the board combinations we need to know how many high cards and low cards exist in the deck.  There are 52 cards in the deck, 48 of which you haven't seen.  You're holding one high cards, an Ace a Deuce, and a three (or any three low card combinations).  That means there are 19 high cards left in the deck (four-9's, 10's, J's, Q's, K's minus the one high card already in your hand).  There are also three remaining aces, deuces, and threes (the actual rank of these cards doesn't matter -- whether they're aces, deuces, threes, or aces, fives, and eights -- the odds below will be the same).  We'll group these low cards into a set called "Group-1."  There is also a second set of low cards:  four each remaining low cards of the other ranks (four 4's, 5's, 6's, 7's, and 8's).  Well group these into a set called "Group-2." 

Grouping these two sets of low cards is necessary because unlike the A2HH case where drawing any one of our hole cards will counterfeit us, A2LH holdings may be drawn upon on the flop, turn, and river without nullifying the possibilities of making our low hand.  Therefore, we need to group these cards separately for the purposes of generating separate sets of odds for each.

As a reminder, the flop can come in one of four combinations:  HHH, HHL, HLL, LLL.  The HHH flop kills our hand, unless we flop trips, quads, a straight, etc.  With 52 cards in the deck, of which 48 you haven't seen, the total number of board combinations are at the flop are:

Flopping 1-Low Card (Board=HHL):

Unlike holding a bare A2HH in your hand, calculating the odds when you hold multiple low cards isn't nearly as straight forward -- not even when the board is HHL.  This happens because there are two sets of low cards to choose from (Group-1 and Group-2), and we must calculate the odds for each of these separately, then combine them at the end to obtain the total odds.

Group-1 Odds:

In Group-1, there are three low cards to choose from, and three ways to choose each of these low cards.  Therefore the total number of combinations of flopping HHL from Group-1 are:

Odds of flopping one low card from Group-1:

Group-2 Odds:

In Group-2, there are five low cards to choose from, and four ways to choose each of these low cards.  Therefore the total number of combinations of flopping HHL from Group-2 are:

Odds of flopping one low card from Group-2:

Aggregate Odds:

Flopping 2-Low Card (Board=HLL):

When we start selecting two low cards, this is where the math and odds seem to spiral out of control.  But if we keep everything straight in our heads, it shouldn't be too bad.  With a board like HLL, there are two low cards to choose.  These low cards may come both from Group-1, both from Group-2, or one each from Group-1 and Group-2.  As we did above, we must calculate the odds separately for each, and then add them up at the end to obtain the total odds for making this board combination.

Group-1 + Group-1 Odds:

There are only three ranks of cards in Group-1, and three each of these cards.  We must select two of them, and one high card.  This means there are 19 high cards to select one, three low cards to select two, and of these three low cards there are 3² ways to choose them.  The odds are therefore:

Odds of flopping two low cards, each from Group-1:

Group-1 + Group-2 Odds:

There are only three ranks of cards in Group-1, and three ways to choose one of these cards; we must select one of these.  There are five ranks of cards in Group-2, and four ways to choose one of those; and we must select one of those also.  We also must select one of the 19 high cards in the deck.  The odds are therefore:

Odds of flopping one low card from Group-1 and one low card from Group-2:

Group-2 + Group-2 Odds:

The math behind flopping two cards from Group-2 will look almost identical to the math from flopping two cards from Group-1 (above).  There are five ranks in Group-2 and four ways to select each of these cards; and we must also select one of the 19 high cards left in the deck.  This means we must select one of 19 high cards, and two of five Group-2 cards with 4² ways to select those two cards.  The odds are therefore:

Odds of flopping two low cards, each from Group-1:

Aggregate Odds:

Flopping the nut low (Board=LLL):

Since I already built the foundation and established the complexity of combining the difference groups of cards together to calculate the odds, I won't bother to do it again.  Instead, I'm going to simply assume familiarity with the math by now, how many ranks are selected, and how many ways those selections can be made.  Therefore I will enumerate the different types of three-low-card combinations that may exist for each, and present the math and odds behind it without elaborate explanations.

Group-1 + Group-1 + Group-1:

This group doesn't exist because there are only three ranks of cards in group one.  Therefore we cannot select one of each of them without counterfeiting the cards in our hand and still make a nut low.

Group-1 + Group-1 + Group-2:

Odds of flopping Group-1 + Group-1 + Group-2:

Group-1 + Group-2 + Group-2:

Odds of flopping Group-1 + Group-2 + Group-2:

Group-2 + Group-2 + Group-2:

Odds of flopping Group-2 + Group-2 + Group-2:

Aggregate Odds:

In case you didn't notice: there's only a 15.15% chance of flopping the nut low when you're holding A2LH type of hand.  That's the bad news; the good news is that these odds are nearly double those of flopping the nut low with an A2HH hand (7.40%).

Chasing the nut low:

In the previous article (Raising with A2HH), we found that chasing the nut low depends on how much money there is in the pot and the type of flop (HHL or HLL).  When there's only a 6.19% chance of getting a runner-runner nut low, there's virtually no situation that justifies the chase.  So it will be interesting to see how these numbers compare against our A2LH hand.

Recall from above how the different groups of low cards and the combinations of those cards composing the flop generate different sets of odds, and those odds must be combined to obtain the total set of odds.  The same holds true for the turn and river.  Since the low cards can be generated from either Group-1 or Group-2, we must calculate different sets of odds for each, and add them all up to determine the total odds of obtaining our low.

As we accumulate these odds, it will also be useful to establish some type of notation to keep track of the board combinations in conjunction with the groups of cards from which they were derived.  Therefore, the following notation will be used:

Low(A.B.C) -- where A, B, and C are numbers representing which low card groups are used to generate Low-Card #1, #2, and #3.  For example, Low(1.1.2) would mean 2-cards from Group-1 and 1-card from Group-2; Low(2.2.2) would mean all three cards are derived from Group-2; etc.

Chasing on a 2-low flop (Board=HLL):

From above, we saw three sets of odds for the low card combinations:  Group-1+Group-1, Group-1+Group-2, and Group-2+Group-2.  We must now generate separate sets of Group-1 and Group-2 odds for each of these combinations.

Group-1+Group-1 (Low 1.1.X):

Since there are only three ranks of cards in Group-1, we cannot use any more Group-1 cards for the turn or river because doing so would counterfeit the cards in our hand.  Therefore, the only turn and river combinations must come from Group-2.

There are five ranks of cards in Group-2, and four combinations of ways to chose one card from each rank.  Therefore, the odds are calculated as follows:

Since there are 45 unseen cards at the turn, and 44 unseen cards at the river, the odds of hitting one of these are:

(20/45) = 44.44% +
(20/44) = 45.45%
=============  89.89% chance of making the low after the flop.

Therefore the overall odds of hitting a low with card combinations Low(1.1.2) are roughly the following:

Group-1+Group-2 (Low1.2.X):

Unlike Low(1.1.X) above where we couldn't use any cards from Group-1, Low(1.2.X) can select either from Group-1 or Group-2.  At the point of the turn and river, there are two ranks of cards available from Group-1 and 4 ranks of cards available from Group-2.

For turn and river cards from Group-1

(6/45) = 13.33% +
(6/44) = 13.64%
=============  26.97% chance of making the low after the flop.

For turn and river cards from Group-2

(16/45) = 35.56% +
(16/44) = 36.36%
=============  71.92% of making the low after the flop.

Therefore the overall odds of hitting a low with card combinations Low(1.2.X) are roughly the following:

Group-2+Group-2 (Low2.2.X):

Just like Low(1.2.X) above we can select turn and river cards either from Group-1 or Group-2.  At the point of the turn and river, there are three ranks of cards available from Group-1 and 3 ranks of cards available from Group-2.

For turn and river cards from Group-1

(9/45) = 20.00% +
(9/44) = 20.45%
=============  40.45% chance of making the low after the flop.

For turn and river cards from Group-2

(12/45) = 26.67% +
(12/44) = 27.27%
=============  53.94% of making the low after the flop.

Therefore the overall odds of hitting a low with card combinations Low(2.2.X) are roughly the following:

Summary:  Complete odds for a Low with Board=HLL

Total Odds:  38.81%

Notice the different odds for making a low when the board=HLL.  When the two low cards were flopped from Group-1, you have nearly a 90% chance of hitting your low from any of the 20 cards in Group-2.  However, when both low cards were flopped from Group-2, you have between a 40% - 54% chance of hitting your low from any of the cards left in Group-1 or Group-2.  It would be tempting to add these two sets of odds together to mean a 94% chance of hitting the low -- but this simply doesn't make any sense.  Since the 54% is the best case scenario for hitting a low, adding the 40% case to these odds doesn't make any sense because you can't have both -- you'll either get one card from the 40% case, or one card from the 54% case -- but not both.  Therefore, the odds should never be combined or added to each other.

Chasing on a 1-low flop (Board=HHL):

Now that you've see what's involved in calculating the odds for Board=HLL, we must go through the same exact exercise for Board=HHL.  Whether we selected one card from Group-1 or Group-2, we still must select two remaining cards to make our low.

Group-1 Low(1.1.2)

Since we have already selected one low card from Group-1 and zero cards from Group-2, have two remaining ranks from Group-1 and five remaining ranks from Group-2.

Therefore the odds for Low(1.1.2) are as follows:

Group-1 Low(1.2.1)

Since we have already selected one low card from Group-1 and zero cards from Group-2, have two remaining ranks from Group-1 and five remaining ranks from Group-2.

Therefore the odds for Low(1.2.1) are as follows:

Group-1 Low(1.2.2)

In this case, we select both turn and river cards from Group-2.  There will be five rank cards to select on the turn, and 4 rank cards to select on the river.

Therefore the odds for Low(1.2.2) are as follows:

Group-2 Low(2.1.1)

Since we have already selected one low card from Group-2 and zero cards from Group-1, have three remaining ranks from Group-1 and four remaining ranks from Group-2.  In this case, we're going to select both cards from Group-1.  This means we have three ranks to select on the turn and two ranks to select on the river.

Therefore the odds for Low(2.1.1) are as follows:

Group-2 Low(2.1.2)

In this case, we're going to select one rank from Group-1 and one rank from Group-2.

Therefore the odds for Low(2.1.2) are as follows:

Group-2 Low(2.2.1)

In this case, we're going to select one rank from Group-1 and one rank from Group-2.  However in this case, the order of the selection matters because there are 45 unseen cards on the turn and 44 unseen cards on the river.

Therefore the odds for Low(2.2.1) are as follows:

Group-2 Low(2.2.2)

In this case, we're going to both ranks from Group-2.

Therefore the odds for Low(2.2.2) are as follows:

Summary:  Complete odds for a Low with Board=HHL

Odds of catching Runner-Runner low cards:

These four combinations above represent the total odds for catching a runner-runner low.  All together, they equal 31.01%.  But remember, as much as we might be tempted to add these odds together, we cannot do so because best case you have a 16.16% chance of catching a runner-runner low; adding up all of the cases together will not increase these odds. 

In order to justify a runner-runner low draw chase on a Board=HHL,  one would need to be getting in excess of 12:1 on their money.  You'll notice these are the same exact odds as the A2HH case -- and should come as no surprise because the best case scenario still only has five ranks of cards to choose from.  Without greater than 12:1 on our money, there really isn't any situation where there are sufficient odds (greater than 12:1) that would justify chasing a a runner-runner low.

Regardless, the combined odds of catching a runner-runner low are actually increased over the A2HH case (6.19%).  Even though the best case is still a 6:1 underdog, the overall odds have increased because there are many more ways to generate a runner-runner low.

Total Odds:  7.8494%

Summary:

Even though the odds are considerably more in your favor when you hold A2LH vs. A2HH, the chances are against flopping your nut low (almost 6.60:1 against).  But how far to chase the nut low still should depend on how many low cards flop, and how much money is in the pot, and how much money will be put on the pot on the flop and turn as well.  The odds of flopping a nut low are only 15.15%.  However, the odds of making a low once you've flopped two cards are as high as 90%.  Just as we found with the A2HH case, we need to get greater than 12:1 on our money to justify chasing the runner-runner low.  There are few, if any situations that would generate these kinds of return on investment, therefore chasing the runner-runner low is still just as bad an idea as it was with A2HH.

Even though the combined odds of getting a low are 61.81%, for all intents and purposes, we need to exclude the 7.85% odds on chasing the runner-runner low (because only a true idiot would chase it).  After we subtract those odds, we're still nearly a 54% favorite to hit our low draw -- thus making it very profitable for us to play hand containing A23H, and other forms of A2LH.

References:

Low Board Probability by Brian Alspach: