PLO8 Probabilities
My home game has gradually drifted away from hold-em over the years and started playing more mixed games. Recently, we have played several sessions of exclusively PLO8 which is definitely a fun game.
Most in the group are good with math and understand poker probabilities, but nobody (including me) has a firm grasp on what the "normal" distribution of pot winnings are in PLO8. So I volunteered to ask this august forum and undertake a few simple simulations.
Let's start slow and easy. Consider heads-up PLO8 and suppose every deal goes to showdown. What are the expected frequencies of all the possible ways the pot can be split in this case? One player scoops both high and low pots, two players chop the pot (one wins high and the other wins low), one player scoops the entire pot when neither player makes a low (this case is probably not called a scoop), etc.
Since I am fairly new to this game, my intuition is virtually non-existent. I have ideas on how often one or both players make a low, but that is about the extent of my "knowledge".
I guess there are conflicting forces working for and against scoops (supposing the board makes a low available). On one hand aces work both ways so a player with an ace may have a slight edge in scooping. On the other hand, winning the high typically uses high cards so may work against that player winning the low since low cards are needed to win the low pot.
I tried to write a simple HU O8 simulator. I will present the preliminary results below. Since this is a brand new program, there is a chance that I made a mistake in programming somewhere along the line, and since I am such a new O8 player I cannot tell if these results are sensible or not.
Here are the results of a simulation of 100,000 HU O8 deals (both players go to showdown on every deal) broken down by which player won the high pot (including the possibility of a tie) and which player won the low pot (including the possibilities or a tie or neither player having a low).
Hopefully, everyone can read the above table. Scoops of both high and low pots would be cells [1,1] and [2,2]. Chopped pots where one player wins high and the other player wins low would be cells [1,2] and [2,1]. Scoops (so-called) where neither player has a low would be cells [1,4] and [2,4].
From the table we see that:
(1) In roughly 50% of the pots neither player has a low. Is this believable?
(2) In the other 50% of the pots, roughly 21% are scooped and roughly 26% are chopped (roughly 3% of the pots have either the high or low pot chopped). Again, is this believable?
Any thoughts would be appreciated.
I will kick off a simulation of more deals later tonight, but I wanted to post preliminary results in case they are way off.
Most in the group are good with math and understand poker probabilities, but nobody (including me) has a firm grasp on what the "normal" distribution of pot winnings are in PLO8. So I volunteered to ask this august forum and undertake a few simple simulations.
Let's start slow and easy. Consider heads-up PLO8 and suppose every deal goes to showdown. What are the expected frequencies of all the possible ways the pot can be split in this case? One player scoops both high and low pots, two players chop the pot (one wins high and the other wins low), one player scoops the entire pot when neither player makes a low (this case is probably not called a scoop), etc.
Since I am fairly new to this game, my intuition is virtually non-existent. I have ideas on how often one or both players make a low, but that is about the extent of my "knowledge".
I guess there are conflicting forces working for and against scoops (supposing the board makes a low available). On one hand aces work both ways so a player with an ace may have a slight edge in scooping. On the other hand, winning the high typically uses high cards so may work against that player winning the low since low cards are needed to win the low pot.
I tried to write a simple HU O8 simulator. I will present the preliminary results below. Since this is a brand new program, there is a chance that I made a mistake in programming somewhere along the line, and since I am such a new O8 player I cannot tell if these results are sensible or not.
Here are the results of a simulation of 100,000 HU O8 deals (both players go to showdown on every deal) broken down by which player won the high pot (including the possibility of a tie) and which player won the low pot (including the possibilities or a tie or neither player having a low).
HIGH WINNER | P1 Wins Low | P2 Wins Low | Tie for Low | Neither has a Low |
---|---|---|---|---|
P1 Wins High | 10,531 | 13,176 | 774 | 24,620 |
P2 Wins High | 13,122 | 10,696 | 744 | 24,885 |
Tie for High | 245 | 254 | 183 | 770 |
Hopefully, everyone can read the above table. Scoops of both high and low pots would be cells [1,1] and [2,2]. Chopped pots where one player wins high and the other player wins low would be cells [1,2] and [2,1]. Scoops (so-called) where neither player has a low would be cells [1,4] and [2,4].
From the table we see that:
(1) In roughly 50% of the pots neither player has a low. Is this believable?
(2) In the other 50% of the pots, roughly 21% are scooped and roughly 26% are chopped (roughly 3% of the pots have either the high or low pot chopped). Again, is this believable?
Any thoughts would be appreciated.
I will kick off a simulation of more deals later tonight, but I wanted to post preliminary results in case they are way off.
Here are the results of a simulation of 1,000,000 deals of HU O8.
Edited to add the following:
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = (248,301+248,416+7,307) / 1,000,000 = 50.4024%
(2) Pct of deals for which one player scoops both high and low pots = (105,542+105,493) / 1,000,000 = 21.1035%
(3) Pct of deals for which two different players win high and low pots =(131,826+131,534) / 1,000,000 = 26.3360%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = (2,496+2,423+7,487+7,361+1,814) / 1,000,000 = 2.1581%
HIGH WINNER | P1 Wins Low | P2 Wins Low | Tie for Low | Neither has a Low |
---|---|---|---|---|
P1 Wins High | 105,542 | 131,534 | 7,487 | 248,301 |
P2 Wins High | 131,826 | 105,493 | 7,361 | 248,416 |
Tie for High | 2,496 | 2,423 | 1,814 | 7,307 |
Edited to add the following:
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = (248,301+248,416+7,307) / 1,000,000 = 50.4024%
(2) Pct of deals for which one player scoops both high and low pots = (105,542+105,493) / 1,000,000 = 21.1035%
(3) Pct of deals for which two different players win high and low pots =(131,826+131,534) / 1,000,000 = 26.3360%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = (2,496+2,423+7,487+7,361+1,814) / 1,000,000 = 2.1581%
I will attempt to run similar simulations with different numbers of players but the runs take forever on my crappy PC.
Here are the results of 1,000,000 deals for 3-player O8. Note to keep things relatively simple, for each half of the pot I have lumped all ties into one bucket, no matter how many or which players tie.
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = (143,382+142,729+143,336+11,341) / 1,000,000 = 44.0788%
(2) Pct of deals for which one player scoops both high and low pots = (51,373+51,677+51,616) / 1,000,000 = 15.4666%
(3) Pct of deals for which two different players win high and low pots =(59,222+59,742+59,609+59,789+59,158+60,055) / 1,000,000 = 35.7575%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = (3,607+3,695+3,615+10,576+10,909+10,687+3,882) / 1,000,000 = 4.6971%
I suppose I can keep track of these high-level summary figures as we vary the number of players at the table.
Here are the results of 1,000,000 deals for 3-player O8. Note to keep things relatively simple, for each half of the pot I have lumped all ties into one bucket, no matter how many or which players tie.
HIGH WINNER | P1 Wins Low | P2 Wins Low | P3 Wins Low | Any Tie for Low | Nobody has a Low |
---|---|---|---|---|---|
P1 Wins High | 51,373 | 59,222 | 59,742 | 10,576 | 143,382 |
P2 Wins High | 59,609 | 51,677 | 59,789 | 10,909 | 142,729 |
P3 Wins High | 59,158 | 60,055 | 51,616 | 10,687 | 143,336 |
Any Tie for High | 3,607 | 3,695 | 3,615 | 3,882 | 11,341 |
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = (143,382+142,729+143,336+11,341) / 1,000,000 = 44.0788%
(2) Pct of deals for which one player scoops both high and low pots = (51,373+51,677+51,616) / 1,000,000 = 15.4666%
(3) Pct of deals for which two different players win high and low pots =(59,222+59,742+59,609+59,789+59,158+60,055) / 1,000,000 = 35.7575%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = (3,607+3,695+3,615+10,576+10,909+10,687+3,882) / 1,000,000 = 4.6971%
I suppose I can keep track of these high-level summary figures as we vary the number of players at the table.
I probably should have started even before Heads-Up.
Here are two fairly recent threads on how often a Low is Available in O8 (three or more distinct ranks 8 or lower on the board where, of course, Ace is considered low for this).
https://forumserver.twoplustwo.com/2...-plo8-1630125/
https://forumserver.twoplustwo.com/2...ow-o8-1690200/
We found that the probability of a Low being Available in O8 is 60.0905%. Obviously, this is independent of the number of players at the table (considering all deals as we are in this thread). And, furthermore, this means that the probability of a Low being Made in O8 is capped at 60.0905%.
Our intuition and common sense suggest that the probability of a Low being Made will increase (with a cap at 60.0905%) as the number of players at the table increases.
__________
Okay, with those preliminaries taken care of, how about we look at what happens if there is only one player at the table. How often is a Low Made then, again, as above, considering all possible deals (all deals go to showdown)?
I took three different approaches to answer this simple question. (1) Simulation; (2) Combinatorics; (3) Brute-force computer program.
Simulation
Simulating a one-player O8 table is very straightforward when all deals go to showdown.
In a simulation of 1,000,000 deals I found that in 347,589 deals did Hero make a Low. This, of course, is 34.7589%.
The standard formulas for standard errors of a binomial probability show that the true percentage should be around this value +/- 0.095% (as per usual, I report twice the "standard error" to reflect two "standard deviations" in both directions).
Combinatorics
Since this one-player question seems so easy, I thought I could crank out the answer in short order using combinatorics.
From the links posted above, we know that there are 7 different cases as to how a Low can be Available in O8. Let's try to "solve" the first case via combinatorics.
Case 1 of Low Available: Board has 3 Distinct Low Ranks with counts [1,1,1] and 2 High cards
How can a player make a low in this case?
Case 1A: Player has 2 low cards and 2 high cards (low cards are distinct and non-overlapping with low cards on board)
= C(5,2)*C(4,1)*C(4,1)*C(18,2)
= 24,480
Case 1B: Player has 3 low cards with a pair and 1 high card (two distinct low card ranks are non-overlapping with low cards on board)
= C(5,2)*C(2,1)*C(4,2)*C(4,1)*C(18,1)
= 8,640
Case 1C: Player has 3 low cards without a pair and 1 high card (three distinct low card ranks are non-overlapping with low cards on board)
= C(5,3)*C(4,1)*C(4,1)*C(4,1)*C(18,1)
= 11,520
Case 1D: Player has 3 low cards without a pair and 1 high card (one overlapping rank with low cards on board)
= C(5,2)*C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(18,1)
= 25,920
Case 1E: Player has 4 low cards with trips (two distinct low card ranks non-overlapping with low cards on board)
= C(5,2)*C(2,1)*C(4,3)*C(4,1)
= 320
Case 1F: Player has 4 low cards with two pairs (two distinct low card ranks non-overlapping with low cards on board)
= C(5,2)*C(4,2)*C(4,2)
= 360
Case 1G: Player has 4 low cards with 1 pair of overlapping rank and 2 singles non-overlapping ranks
= C(3,1)*C(3,2)*C(5,2)*C(4,1)*C(4,1)
= 1,440
Case 1H: Player has 4 low cards with 1 pair of non-overlapping rank, 1 single overlapping rank, and 1 single non-overlapping rank
= C(3,1)*C(3,1)*C(5,2)*C(2,1)*C(4,2)*C(4,1)
= 4,320
Case 1I: Player has 4 low cards with 1 pair of non-overlapping rank, 2 singles non-overlapping ranks
= C(5,3)*C(3,1)*C(4,2)*C(4,1)*C(4,1)
= 2,880
Case 1J: Player has 4 low cards without a pair, 2 overlapping ranks and 2 non-overlapping ranks
= C(3,2)*C(3,1)*C(3,1)*C(5,2)*C(4,1)*C(4,1)
= 4,320
Case 1K: Player has 4 low cards without a pair, 1 overlapping rank and 3 non-overlapping ranks
= C(3,1)*C(3,1)*C(5,3)*C(4,1)*C(4,1)*C(4,1)
= 5,760
Case 1L: Player has 4 low cards without a pair, 4 non-overlapping ranks
= C(5,4)*C(4,1)*C(4,1)*C(4,1)*C(4,1)
= 1,280
Subtotal = 91,240
As you might imagine, slogging through the combinatorics of these subcases loses its appeal after awhile so, given that this was just the first of 7 cases, I looked around and hoped to find an easier way to do this. Which brings us to:
Computer Brute-Force
Brute-forcing every possible deal of even one-person O8 is pushing the limits of my crappy PC, so I didn't like that idea. However, since we know the 7 cases (and their frequencies) in which a board makes a Low Available, brute-forcing those 7 cases one at a time is straightforward and eminently doable.
The following table summarizes these results.
A few comments on the table before I present the key result.
(1) The pcts in the far right column are brand new to me since I am a new O8 player. I was surprised by many of them but don't have anything else to say at this point.
(2) It makes sense to this novice O8 player that the number of hands that make a low would be identical (91,240) in the first four cases above. I think the reason is that in Omaha, of course, you must use exactly three cards from the board and two cards from your hand. So in each of the first four cases above, you must use the same three ranks from the board to make a low. In these cases, then, a "duplicate" of one of those ranks is equivalent to a high card -- it cannot be used to make a low anyway.
(3) On first sight, I was puzzled why the number of hands that make a low in case 5 (126,984) is not identical to the number of hands that make a low in case 6 (123,776). My initial reasoning was the same as above. A "duplicate" of a low card on board is tantamount to a high card. But this reasoning must be wrong since these two cases give different results.
If anyone wants to chime in here and help out a novice O8 player better understand this result, I would appreciate it.
(4) The number of hands that make a low in the first case (91,240) is the value we attained via our tedious combinatoric approach, so I guess I am glad about that. The two approaches reinforce/confirm each other's results.
Okay, now to the key result. Using the information contained in the above table, it is straightforward to obtain the following result:
Pct of One Player making a Low in O8:
= 161,374,983,168 / 463,563,500,400
= 34.8118%
Remember that a Low is Available 60.0905% so, on average, one-player makes a low when a low is available approximately 58% of the time. I suppose experienced O8 players know stuff like this off the top of their head.
Note that the result from the modified brute-force approach that 34.8118% of all deals of one-player O8 have a low is consistent with the result from our simulation of 1,000,000 deals (34.7589%).
To match the way the results of the multiple-player tables have been presented in the thread, this means that there is No Low Made in 65.1882% of deals in one-player O8.
As I stated in an earlier post in this thread, since I am an O8 novice, I could easily present results in this thread that are completely wrong, but I would not even know they are wrong. So feel free to confirm/refute any results presented herein.
Here are two fairly recent threads on how often a Low is Available in O8 (three or more distinct ranks 8 or lower on the board where, of course, Ace is considered low for this).
https://forumserver.twoplustwo.com/2...-plo8-1630125/
https://forumserver.twoplustwo.com/2...ow-o8-1690200/
We found that the probability of a Low being Available in O8 is 60.0905%. Obviously, this is independent of the number of players at the table (considering all deals as we are in this thread). And, furthermore, this means that the probability of a Low being Made in O8 is capped at 60.0905%.
Our intuition and common sense suggest that the probability of a Low being Made will increase (with a cap at 60.0905%) as the number of players at the table increases.
__________
Okay, with those preliminaries taken care of, how about we look at what happens if there is only one player at the table. How often is a Low Made then, again, as above, considering all possible deals (all deals go to showdown)?
I took three different approaches to answer this simple question. (1) Simulation; (2) Combinatorics; (3) Brute-force computer program.
Simulation
Simulating a one-player O8 table is very straightforward when all deals go to showdown.
In a simulation of 1,000,000 deals I found that in 347,589 deals did Hero make a Low. This, of course, is 34.7589%.
The standard formulas for standard errors of a binomial probability show that the true percentage should be around this value +/- 0.095% (as per usual, I report twice the "standard error" to reflect two "standard deviations" in both directions).
Combinatorics
Since this one-player question seems so easy, I thought I could crank out the answer in short order using combinatorics.
From the links posted above, we know that there are 7 different cases as to how a Low can be Available in O8. Let's try to "solve" the first case via combinatorics.
Case 1 of Low Available: Board has 3 Distinct Low Ranks with counts [1,1,1] and 2 High cards
How can a player make a low in this case?
Case 1A: Player has 2 low cards and 2 high cards (low cards are distinct and non-overlapping with low cards on board)
= C(5,2)*C(4,1)*C(4,1)*C(18,2)
= 24,480
Case 1B: Player has 3 low cards with a pair and 1 high card (two distinct low card ranks are non-overlapping with low cards on board)
= C(5,2)*C(2,1)*C(4,2)*C(4,1)*C(18,1)
= 8,640
Case 1C: Player has 3 low cards without a pair and 1 high card (three distinct low card ranks are non-overlapping with low cards on board)
= C(5,3)*C(4,1)*C(4,1)*C(4,1)*C(18,1)
= 11,520
Case 1D: Player has 3 low cards without a pair and 1 high card (one overlapping rank with low cards on board)
= C(5,2)*C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(18,1)
= 25,920
Case 1E: Player has 4 low cards with trips (two distinct low card ranks non-overlapping with low cards on board)
= C(5,2)*C(2,1)*C(4,3)*C(4,1)
= 320
Case 1F: Player has 4 low cards with two pairs (two distinct low card ranks non-overlapping with low cards on board)
= C(5,2)*C(4,2)*C(4,2)
= 360
Case 1G: Player has 4 low cards with 1 pair of overlapping rank and 2 singles non-overlapping ranks
= C(3,1)*C(3,2)*C(5,2)*C(4,1)*C(4,1)
= 1,440
Case 1H: Player has 4 low cards with 1 pair of non-overlapping rank, 1 single overlapping rank, and 1 single non-overlapping rank
= C(3,1)*C(3,1)*C(5,2)*C(2,1)*C(4,2)*C(4,1)
= 4,320
Case 1I: Player has 4 low cards with 1 pair of non-overlapping rank, 2 singles non-overlapping ranks
= C(5,3)*C(3,1)*C(4,2)*C(4,1)*C(4,1)
= 2,880
Case 1J: Player has 4 low cards without a pair, 2 overlapping ranks and 2 non-overlapping ranks
= C(3,2)*C(3,1)*C(3,1)*C(5,2)*C(4,1)*C(4,1)
= 4,320
Case 1K: Player has 4 low cards without a pair, 1 overlapping rank and 3 non-overlapping ranks
= C(3,1)*C(3,1)*C(5,3)*C(4,1)*C(4,1)*C(4,1)
= 5,760
Case 1L: Player has 4 low cards without a pair, 4 non-overlapping ranks
= C(5,4)*C(4,1)*C(4,1)*C(4,1)*C(4,1)
= 1,280
Subtotal = 91,240
As you might imagine, slogging through the combinatorics of these subcases loses its appeal after awhile so, given that this was just the first of 7 cases, I looked around and hoped to find an easier way to do this. Which brings us to:
Computer Brute-Force
Brute-forcing every possible deal of even one-person O8 is pushing the limits of my crappy PC, so I didn't like that idea. However, since we know the 7 cases (and their frequencies) in which a board makes a Low Available, brute-forcing those 7 cases one at a time is straightforward and eminently doable.
The following table summarizes these results.
Low Available Case | Description | Number of Boards | Hands One Player Makes Low | Pct of Possible One-Player Hands |
---|---|---|---|---|
1 | 3 Distinct Low Ranks [1,1,1] | 680,960 | 91,240 | 51.15% |
2 | 3 Distinct Low Ranks [2,1,1] | 322,560 | 91,240 | 51.15% |
3 | 3 Distinct Low Ranks [2,2,1] | 24,192 | 91,240 | 51.15% |
4 | 3 Distinct Low Ranks [3,1,1] | 10,752 | 91,240 | 51.15% |
5 | 4 Distinct Low Ranks [1,1,1,1] | 358,400 | 126,984 | 71.19% |
6 | 4 Distinct Low Ranks [2,1,1,1] | 107,520 | 123,776 | 69.39% |
7 | 5 Distinct Low Ranks [1,1,1,1,1] | 57,344 | 136,127 | 76.32% |
A few comments on the table before I present the key result.
(1) The pcts in the far right column are brand new to me since I am a new O8 player. I was surprised by many of them but don't have anything else to say at this point.
(2) It makes sense to this novice O8 player that the number of hands that make a low would be identical (91,240) in the first four cases above. I think the reason is that in Omaha, of course, you must use exactly three cards from the board and two cards from your hand. So in each of the first four cases above, you must use the same three ranks from the board to make a low. In these cases, then, a "duplicate" of one of those ranks is equivalent to a high card -- it cannot be used to make a low anyway.
(3) On first sight, I was puzzled why the number of hands that make a low in case 5 (126,984) is not identical to the number of hands that make a low in case 6 (123,776). My initial reasoning was the same as above. A "duplicate" of a low card on board is tantamount to a high card. But this reasoning must be wrong since these two cases give different results.
If anyone wants to chime in here and help out a novice O8 player better understand this result, I would appreciate it.
(4) The number of hands that make a low in the first case (91,240) is the value we attained via our tedious combinatoric approach, so I guess I am glad about that. The two approaches reinforce/confirm each other's results.
Okay, now to the key result. Using the information contained in the above table, it is straightforward to obtain the following result:
Pct of One Player making a Low in O8:
= 161,374,983,168 / 463,563,500,400
= 34.8118%
Remember that a Low is Available 60.0905% so, on average, one-player makes a low when a low is available approximately 58% of the time. I suppose experienced O8 players know stuff like this off the top of their head.
Note that the result from the modified brute-force approach that 34.8118% of all deals of one-player O8 have a low is consistent with the result from our simulation of 1,000,000 deals (34.7589%).
To match the way the results of the multiple-player tables have been presented in the thread, this means that there is No Low Made in 65.1882% of deals in one-player O8.
As I stated in an earlier post in this thread, since I am an O8 novice, I could easily present results in this thread that are completely wrong, but I would not even know they are wrong. So feel free to confirm/refute any results presented herein.
I am not interested in adapting my hold'em evaluator for this right now, but I can help with one thing. I am probably not much less of a novice O8 player than you, but I believe I can explain observation (3). Case 7 has the same number of 2-low combinations as case 6. However, case 7 has more unpaired 2-low combinations than Case 6. Consider the trivial case of a four card deck, {1, 1, 2, 2}. If the board is {2, 2} we can only have {1, 1} in our hand, and a pair can't qualify for low. If the board is {1, 2} then we have {1, 2} in our hand, which is a two card low.
I am not interested in adapting my hold'em evaluator for this right now, but I can help with one thing. I am probably not much less of a novice O8 player than you, but I believe I can explain observation (3). Case 7 has the same number of 2-low combinations as case 6. However, case 7 has more unpaired 2-low combinations than Case 6. Consider the trivial case of a four card deck, {1, 1, 2, 2}. If the board is {2, 2} we can only have {1, 1} in our hand, and a pair can't qualify for low. If the board is {1, 2} then we have {1, 2} in our hand, which is a two card low.
In the four cases where there are exactly three distinct low cards on board (Cases 1-4), here was my thinking.
Suppose you have a list of the 91,240 hands that make a low in Case 1 [1,1,1], three low singletons on board. Then consider Case 2 [2,1,1] where one of the three low ranks on board is paired.
For concreteness, suppose Board 1 is 8s 6s 4s Kh Qh and Board 2 is 8s 8h 6s 4s Kh (swapping the Qh for the 8h). Then if you simply replace the 8h with the Qh wherever it appears in the list of 91,240 hands which make a low on Board 1, you will have automatically have a list of 91,240 hands which make a low on Board 2. There is a one-to-one mapping between the two lists, so, of course, the total number in the two lists is identical.
As browni3141 suggests, the same "mapping" idea does not work when there are more than 3 low ranks on board as is the case for Cases 5 and 6.
Again, for concreteness, suppose Board 5 is 8s 6s 4s 2s Kh and Board 6 is 8s 8h 6s 4s 2s (swapping the Kh for the 8h). Then simply replacing the 8h with the Kh does not automatically give you a hand that makes a low on Board 6.
I suppose this is obvious. Consider the hand Qd Jc 8h 3c in the list of hands that make a low on Board 5. If you replace the 8h with the Kh you now have the hand Qd Jc Kh 3c which clearly does NOT make a low on Board 6.
I have attempted to construct other arguments related to this question but they all encounter this same difficulty. An additional duplicated low card on board surely reduces the number of hands that make a low whenever there is "slack" in the system (i.e., 4 low ranks on board) since there are hands that rely upon that rank to make the low. So removing one of that rank reduces the number of possible ways that could happen in a version of card removal.
This is not the situation of Case 1 vs. Case 2 since the 8h can never be critical to making a low in either case. If the 8h appears in a hand which makes a low on Board 1, then it must be true that there are at least 2 other low ranks (other than 8) appearing in the hand since there are exactly 3 low ranks on board (one of them being an 8). So removing the 8h from that hand (via the card removal/swapping mapping) can never turn a hand from making a low into one that no longer makes a low.
Despite my reluctance, I tried to derive the other case tallies via combinatorics. Recall that the question is: given a specific board type, how many 4-card Omaha hands make a Low with that Board (using the normal Omaha rules where you must use exactly 3 cards from the board and exactly two cards from your hand).
The following is extremely tedious and may very well not be the most efficient way to proceed. Also, another possible approach in this type of question is to apply the Principle of Inclusion-Exclusion (PIE) of which I am merely a dabbler.
Case 5: Board = 4 Distinct Low Cards [1,1,1,1] plus 1 High Card
Case 5A. 2 low cards and 2 high cards (2 non-overlapping ranks from low ranks on board)
= C(4,2)*C(4,1)*C(4,1)*C(19,2)
= 16,416
Case 5B. 2 low cards and 2 high cards (1 non-overlapping, 1 overlapping)
= C(4,1)*C(4,1)*C(4,1)*C(3,1)*C(19,2)
= 32,832
Case 5C. 3 low cards without a pair and 1 high card (3 non-overlapping)
= C(4,3)*C(4,1)*C(4,1)*C(4,1)*C(19,1)
= 4,864
Case 5D. 3 low cards without a pair and 1 high card (2 non-overlapping, 1 overlapping)
= C(4,2)*C(4,1)*C(4,1)*C(4,1)*C(3,1)*C(19,1)
= 21,888
Case 5E. 3 low cards without a pair and 1 high card (1 non-overlapping, 2 overlapping)
= C(4,1)*C(4,1)*C(4,2)*C(3,1)*C(3,1)*C(19,1)
= 16,416
Case 5F. 3 low cards with a pair and 1 high card (2 non-overlapping)
= C(4,2)*C(2,1)*C(4,2)*C(4,1)*C(19,1)
= 5,472
Case 5G. 3 low cards with a pair and 1 high card (1 non-overlapping with pair, 1 single overlapping)
= C(4,1)*C(4,2)*C(4,1)*C(3,1)*C(19,1)
= 5,472
Case 5H. 3 low cards with a pair and 1 high card (1 single non-overlapping, 1 overlapping pair)
= C(4,1)*C(4,1)*C(4,1)*C(3,2)*C(19,1)
= 3,648
Case 5I. 4 low cards without a pair (4 non-overlapping)
= C(4,4)*C(4,1)*C(4,1)*C(4,1)*C(4,1)
= 256
Case 5J. 4 low cards without a pair (3 non-overlapping, 1 overlapping)
= C(4,3)*C(4,1)*C(4,1)*C(4,1)*C(4,1)*C(3,1)
= 3,072
Case 5K. 4 low cards without a pair (2 non-overlapping, 2 overlapping)
= C(4,2)*C(4,1)*C(4,1)*C(4,2)*C(3,1)*C(3,1)
= 5,184
Case 5L. 4 low cards without a pair (1 non-overlapping, 3 overlapping)
= C(4,1)*C(4,1)*C(4,3)*C(3,1)*C(3,1)*C(3,1)
= 1,728
Case 5M. 4 low cards with a pair (3 non-overlapping)
= C(4,3)*C(3,1)*C(4,2)*C(4,1)*C(4,1)
= 1,152
Case 5N. 4 low cards with a pair (2 non-overlapping w/pair, 1 single overlapping)
= C(4,2)*C(2,1)*C(4,2)*C(4,1)*C(4,1)*C(3,1)
= 3,456
Case 5O. 4 low cards with a pair (2 non-overlapping, 1 overlapping pair)
= C(4,2)*C(4,1)*C(4,1)*C(4,1)*C(3,2)
= 1,152
Case 5P. 4 low cards with a pair (1 non-overlapping pair, 2 single overlapping)
= C(4,1)*C(4,2)*C(4,2)*C(3,1)*C(3,1)
= 1,296
Case 5Q. 4 low cards with a pair (1 single non-overlapping, 2 overlapping w/pair)
= C(4,1)*C(4,1)*C(4,2)*C(2,1)*C(3,2)*C(3,1)
= 1,728
Case 5R. 4 low cards with 2 pairs (2 non-overlapping)
= C(4,2)*C(4,2)*C(4,2)
= 216
Case 5S. 4 low cards with 2 pairs (1 non-overlapping, 1 overlapping)
= C(4,1)*C(4,2)*C(4,1)*C(3,2)
= 288
Case 5T. 4 low cards with trips (2 non-overlapping)
= C(4,2)*C(2,1)*C(4,3)*C(4,1)
= 192
Case 5U. 4 low cards with trips (1 non-overlapping trips, 1 single overlapping)
= C(4,1)*C(4,3)*C(4,1)*C(3,1)
= 192
Case 5V. 4 low cards with trips (1 non-overlapping single, 1 overlapping trips)
= C(4,1)*C(4,1)*C(4,1)*C(3,3)
= 64
Total for Case 5 = 126,984 [confirmed by computer brute-force]
Case 6: Board = 4 Distinct Low Cards [2,1,1,1]
Case 6A. 2 low cards and 2 high cards (2 non-overlapping ranks from low ranks on board)
= C(4,2)*C(4,1)*C(4,1)*C(20,2)
= 18,240
Case 6B. 2 low cards and 2 high cards (1 non-overlapping, 1 overlapping not board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(20,2)
= 27,360
Case 6C. 2 low cards and 2 high cards (1 non-overlapping, 1 overlapping board pair)
= C(4,1)*C(4,1)*C(1,1)*C(2,1)*C(20,2)
= 6,080
Case 6D. 3 low cards without a pair and 1 high card (3 non-overlapping)
= C(4,3)*C(4,1)*C(4,1)*C(4,1)*C(20,1)
= 5,120
Case 6E. 3 low cards without a pair and 1 high card (2 non-overlapping, 1 overlapping not board pair)
= C(4,2)*C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(20,1)
= 17,280
Case 6F. 3 low cards without a pair and 1 high card (2 non-overlapping, 1 overlapping board pair)
= C(4,2)*C(4,1)*C(4,1)*C(1,1)*C(2,1)*C(20,1)
= 3,840
Case 6G. 3 low cards without a pair and 1 high card (1 non-overlapping, 2 overlapping not board pair)
= C(4,1)*C(4,1)*C(3,2)*C(3,1)*C(3,1)*C(20,1)
= 8,640
Case 6H. 3 low cards without a pair and 1 high card (1 non-overlapping, 2 overlapping with board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(1,1)*C(2,1)*C(20,1)
= 5,760
Case 6I. 3 low cards with a pair and 1 high card (2 non-overlapping)
= C(4,2)*C(2,1)*C(4,2)*C(4,1)*C(20,1)
= 5,760
Case 6J. 3 low cards with a pair and 1 high card (1 non-overlapping w/pair, 1 single overlapping not board pair)
= C(4,1)*C(4,2)*C(3,1)*C(3,1)*C(20,1)
= 4,320
Case 6K. 3 low cards with a pair and 1 high card (1 non-overlapping w/pair, 1 single overlapping board pair)
= C(4,1)*C(4,2)*C(1,1)*C(2,1)*C(20,1)
= 960
Case 6L. 3 low cards with a pair and 1 high card (1 single non-overlapping, 1 paired overlapping not board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,2)*C(20,1)
= 2,880
Case 6M. 3 low cards with a pair and 1 high card (1 single non-overlapping, 1 paired overlapping board pair)
= C(4,1)*C(4,1)*C(1,1)*C(2,2)*C(20,1)
= 320
Case 6N. 4 low cards without a pair (4 non-overlapping)
= C(4,4)*C(4,1)*C(4,1)*C(4,1)*C(4,1)
= 256
Case 6O. 4 low cards without a pair (3 non-overlapping, 1 overlapping not board pair)
= C(4,3)*C(4,1)*C(4,1)*C(4,1)*C(3,1)*C(3,1)
= 2,304
Case 6P. 4 low cards without a pair (3 non-overlapping, 1 overlapping board pair)
= C(4,3)*C(4,1)*C(4,1)*C(4,1)*C(1,1)*C(2,1)
= 512
Case 6Q. 4 low cards without a pair (2 non-overlapping, 2 overlapping not board pair)
= C(4,2)*C(4,1)*C(4,1)*C(3,2)*C(3,1)*C(3,1)
= 2,592
Case 6R. 4 low cards without a pair (2 non-overlapping, 2 overlapping w/board pair)
= C(4,2)*C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(1,1)*C(2,1)
= 1,728
Case 6S. 4 low cards without a pair (1 non-overlapping, 3 overlapping no board pair)
= C(4,1)*C(4,1)*C(3,3)*C(3,1)*C(3,1)*C(3,1)
= 432
Case 6T. 4 low cards without a pair (1 non-overlapping, 3 overlapping w/board pair)
= C(4,1)*C(4,1)*C(3,2)*C(3,1)*C(3,1)*C(1,1)*C(2,1)
= 864
Case 6U. 4 low cards with a pair (3 non-overlapping)
= C(4,3)*C(3,1)*C(4,2)*C(4,1)*C(4,1)
= 1,152
Case 6V. 4 low cards with a pair (2 non-overlapping w/pair, 1 single overlapping no board pair)
= C(4,2)*C(2,1)*C(4,2)*C(4,1)*C(3,1)*C(3,1)
= 2,592
Case 6W. 4 low cards with a pair (2 non-overlapping w/pair, 1 single overlapping board pair)
= C(4,2)*C(2,1)*C(4,2)*C(4,1)*C(1,1)*C(2,1)
= 576
Case 6X. 4 low cards with a pair (2 non-overlapping, 1 pair overlapping no board pair)
= C(4,2)*C(4,1)*C(4,1)*C(3,1)*C(3,2)
= 864
Case 6Y. 4 low cards with a pair (2 non-overlapping, 1 pair overlapping board pair)
= C(4,2)*C(4,1)*C(4,1)*C(1,1)*C(2,2)
= 96
Case 6Z. 4 low cards with a pair (1 non-overlapping w/pair, 2 singles overlapping no board pair)
= C(4,1)*C(4,2)*C(3,2)*C(3,1)*C(3,1)
= 648
Case 6AA. 4 low cards with a pair (1 non-overlapping w/pair, 2 singles overlapping w/board pair)
= C(4,1)*C(4,2)*C(3,1)*C(3,1)*C(1,1)*C(2,1)
= 432
Case 6AB. 4 low cards with a pair (1 non-overlapping single, 2 overlapping w/pair and no board pair)
= C(4,1)*C(4,1)*C(3,2)*C(2,1)*C(3,2)*C(3,1)
= 864
Case 6AC. 4 low cards with a pair (1 non-overlapping single, 2 overlapping w/pair, single=board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,2)*C(1,1)*C(2,1)
= 288
Case 6AD. 4 low cards with a pair (1 non-overlapping single, 2 overlapping w/pair, pair=board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(1,1)*C(2,2)
= 144
Case 6AE. 4 low cards with 2 pairs (2 non-overlapping)
= C(4,2)*C(4,2)*C(4,2)
= 216
Case 6AF. 4 low cards with 2 pairs (1 non-overlapping, 1 overlapping not board pair)
= C(4,1)*C(4,2)*C(3,1)*C(3,2)
= 216
Case 6AG. 4 low cards with 2 pairs (1 non-overlapping, 1 overlapping board pair)
= C(4,1)*C(4,2)*C(1,1)*C(2,2)
= 24
Case 6AH. 4 low cards with trips (2 non-overlapping)
= C(4,2)*C(2,1)*C(4,3)*C(4,1)
= 192
Case 6AI. 4 low cards with trips (1 non-overlapping trips, 1 overlapping not board pair)
= C(4,1)*C(4,3)*C(3,1)*C(3,1)
= 144
Case 6AJ. 4 low cards with trips (1 non-overlapping trips, 1 overlapping board pair)
= C(4,1)*C(4,3)*C(1,1)*C(2,1)
= 32
Case 6AK. 4 low cards with trips (1 non-overlapping single, 1 trips overlapping not board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,3)
= 48
TOTAL for Case 6 = 123,776 [confirmed by computer brute-forced]
Case 7: Board = 5 Distinct Low Cards [1,1,1,1,1]
Case 7A. 2 low cards and 2 high cards (2 non-overlapping ranks from board)
= C(3,2)*C(4,1)*C(4,1)*C(20,2)
= 9,120
Case 7B. 2 low cards and 2 high cards (1 non-overlapping, 1 overlapping)
= C(3,1)*C(4,1)*C(5,1)*C(3,1)*C(20,2)
= 34,200
Case 7C. 2 low cards and 2 high cards (2 overlapping)
= C(5,2)*C(3,1)*C(3,1)*C(20,2)
= 17,100
Case 7D. 3 low cards without a pair and 1 high card (3 non-overlapping)
= C(3,3)*C(4,1)*C(4,1)*C(4,1)*C(20,1)
= 1,280
Case 7E. 3 low cards without a pair and 1 high card (2 non-overlapping, 1 overlapping)
= C(3,2)*C(4,1)*C(4,1)*C(5,1)*C(3,1)*C(20,1)
= 14,400
Case 7F. 3 low cards without a pair and 1 high card (1 non-overlapping, 2 overlapping)
= C(3,1)*C(4,1)*C(5,2)*C(3,1)*C(3,1)*C(20,1)
= 21,600
Case 7G. 3 low cards without a pair and 1 high card (3 overlapping)
= C(5,3)*C(3,1)*C(3,1)*C(3,1)*C(20,1)
= 5,400
Case 7H. 3 low cards with a pair and 1 high card (2 non-overlapping)
= C(3,2)*C(2,1)*C(4,2)*C(4,1)*C(20,1)
= 2,880
Case 7I. 3 low cards with a pair and 1 high card (1 non-overlapping pair, 1 overlapping single)
= C(3,1)*C(4,2)*C(5,1)*C(3,1)*C(20,1)
= 5,400
Case 7J. 3 low cards with a pair and 1 high card (1 non-overlapping single, 1 overlapping pair)
= C(3,1)*C(4,1)*C(5,1)*C(3,2)*C(20,1)
= 3,600
Case 7K. 3 low cards with a pair and 1 high card (2 overlapping)
= C(5,2)*C(2,1)*C(3,2)*C(3,1)*C(20,1)
= 3,600
Case 7L. 4 low cards without a pair (3 non-overlapping 1 overlapping)
= C(3,3)*C(4,1)*C(4,1)*C(4,1)*C(5,1)*C(3,1)
= 960
Case 7M. 4 low cards without a pair (2 non-overlapping, 2 overlapping)
= C(3,2)*C(4,1)*C(4,1)*C(5,2)*C(3,1)*C(3,1)
= 4,320
Case 7N. 4 low cards without a pair (1 non-overlapping, 3 overlapping)
= C(3,1)*C(4,1)*C(5,3)*C(3,1)*C(3,1)*C(3,1)
= 3,240
Case 7O. 4 low cards without a pair (4 overlapping)
= C(5,4)*C(3,1)*C(3,1)*C(3,1)*C(3,1)
= 405
Case 7P. 4 low cards with a pair (3 non-overlapping)
= C(3,3)*C(3,1)*C(4,2)*C(4,1)*C(4,1)
= 288
Case 7Q. 4 low cards with a pair (2 non-overlapping w/pair, 1 single overlapping)
= C(3,2)*C(2,1)*C(4,2)*C(4,1)*C(5,1)*C(3,1)
= 2,160
Case 7R. 4 low cards with a pair (2 non-overlapping, 1 overlapping pair)
= C(3,2)*C(4,1)*C(4,1)*C(5,1)*C(3,2)
= 720
Case 7S. 4 low cards with a pair (1 non-overlapping w/pair, 2 overlapping)
= C(3,1)*C(4,2)*C(5,2)*C(3,1)*C(3,1)
= 1,620
Case 7T. 4 low cards with a pair (1 non-overlapping, 2 overlapping w/pair)
= C(3,1)*C(4,1)*C(5,2)*C(2,1)*C(3,2)*C(3,1)
= 2,160
Case 7U. 4 low cards with a pair (3 overlapping)
= C(5,3)*C(3,1)*C(3,2)*C(3,1)*C(3,1)
= 810
Case 7V. 4 low cards with 2 pairs (2 non-overlapping)
= C(3,2)*C(4,2)*C(4,2)
= 108
Case 7W. 4 low cards with 2 pairs (1 non-overlapping, 1 overlapping)
= C(3,1)*C(4,2)*C(5,1)*C(3,2)
= 270
Case 7X. 4 low cards with 2 pairs (2 overlapping)
= C(5,2)*C(3,2)*C(3,2)
= 90
Case 7Y. 4 low cards with trips (2 non-overlapping)
= C(3,2)*C(2,1)*C(4,3)*C(4,1)
= 96
Case 7Z. 4 low cards with trips (1 non-overlapping trips, 1 single overlapping)
= C(3,1)*C(4,3)*C(5,1)*C(3,1)
= 180
Case 7AA. 4 low cards with trips (1 single non-overlapping, 1 overlapping trips)
= C(3,1)*C(4,1)*C(5,1)*C(3,3)
= 60
Case 7AB. 4 low cards with trips (2 overlapping)
= C(5,2)*C(2,1)*C(3,3)*C(3,1)
= 60
TOTAL for Case 7 = 136,127 [confirmed by computer brute-force]
The following is extremely tedious and may very well not be the most efficient way to proceed. Also, another possible approach in this type of question is to apply the Principle of Inclusion-Exclusion (PIE) of which I am merely a dabbler.
Case 5: Board = 4 Distinct Low Cards [1,1,1,1] plus 1 High Card
Case 5A. 2 low cards and 2 high cards (2 non-overlapping ranks from low ranks on board)
= C(4,2)*C(4,1)*C(4,1)*C(19,2)
= 16,416
Case 5B. 2 low cards and 2 high cards (1 non-overlapping, 1 overlapping)
= C(4,1)*C(4,1)*C(4,1)*C(3,1)*C(19,2)
= 32,832
Case 5C. 3 low cards without a pair and 1 high card (3 non-overlapping)
= C(4,3)*C(4,1)*C(4,1)*C(4,1)*C(19,1)
= 4,864
Case 5D. 3 low cards without a pair and 1 high card (2 non-overlapping, 1 overlapping)
= C(4,2)*C(4,1)*C(4,1)*C(4,1)*C(3,1)*C(19,1)
= 21,888
Case 5E. 3 low cards without a pair and 1 high card (1 non-overlapping, 2 overlapping)
= C(4,1)*C(4,1)*C(4,2)*C(3,1)*C(3,1)*C(19,1)
= 16,416
Case 5F. 3 low cards with a pair and 1 high card (2 non-overlapping)
= C(4,2)*C(2,1)*C(4,2)*C(4,1)*C(19,1)
= 5,472
Case 5G. 3 low cards with a pair and 1 high card (1 non-overlapping with pair, 1 single overlapping)
= C(4,1)*C(4,2)*C(4,1)*C(3,1)*C(19,1)
= 5,472
Case 5H. 3 low cards with a pair and 1 high card (1 single non-overlapping, 1 overlapping pair)
= C(4,1)*C(4,1)*C(4,1)*C(3,2)*C(19,1)
= 3,648
Case 5I. 4 low cards without a pair (4 non-overlapping)
= C(4,4)*C(4,1)*C(4,1)*C(4,1)*C(4,1)
= 256
Case 5J. 4 low cards without a pair (3 non-overlapping, 1 overlapping)
= C(4,3)*C(4,1)*C(4,1)*C(4,1)*C(4,1)*C(3,1)
= 3,072
Case 5K. 4 low cards without a pair (2 non-overlapping, 2 overlapping)
= C(4,2)*C(4,1)*C(4,1)*C(4,2)*C(3,1)*C(3,1)
= 5,184
Case 5L. 4 low cards without a pair (1 non-overlapping, 3 overlapping)
= C(4,1)*C(4,1)*C(4,3)*C(3,1)*C(3,1)*C(3,1)
= 1,728
Case 5M. 4 low cards with a pair (3 non-overlapping)
= C(4,3)*C(3,1)*C(4,2)*C(4,1)*C(4,1)
= 1,152
Case 5N. 4 low cards with a pair (2 non-overlapping w/pair, 1 single overlapping)
= C(4,2)*C(2,1)*C(4,2)*C(4,1)*C(4,1)*C(3,1)
= 3,456
Case 5O. 4 low cards with a pair (2 non-overlapping, 1 overlapping pair)
= C(4,2)*C(4,1)*C(4,1)*C(4,1)*C(3,2)
= 1,152
Case 5P. 4 low cards with a pair (1 non-overlapping pair, 2 single overlapping)
= C(4,1)*C(4,2)*C(4,2)*C(3,1)*C(3,1)
= 1,296
Case 5Q. 4 low cards with a pair (1 single non-overlapping, 2 overlapping w/pair)
= C(4,1)*C(4,1)*C(4,2)*C(2,1)*C(3,2)*C(3,1)
= 1,728
Case 5R. 4 low cards with 2 pairs (2 non-overlapping)
= C(4,2)*C(4,2)*C(4,2)
= 216
Case 5S. 4 low cards with 2 pairs (1 non-overlapping, 1 overlapping)
= C(4,1)*C(4,2)*C(4,1)*C(3,2)
= 288
Case 5T. 4 low cards with trips (2 non-overlapping)
= C(4,2)*C(2,1)*C(4,3)*C(4,1)
= 192
Case 5U. 4 low cards with trips (1 non-overlapping trips, 1 single overlapping)
= C(4,1)*C(4,3)*C(4,1)*C(3,1)
= 192
Case 5V. 4 low cards with trips (1 non-overlapping single, 1 overlapping trips)
= C(4,1)*C(4,1)*C(4,1)*C(3,3)
= 64
Total for Case 5 = 126,984 [confirmed by computer brute-force]
Case 6: Board = 4 Distinct Low Cards [2,1,1,1]
Case 6A. 2 low cards and 2 high cards (2 non-overlapping ranks from low ranks on board)
= C(4,2)*C(4,1)*C(4,1)*C(20,2)
= 18,240
Case 6B. 2 low cards and 2 high cards (1 non-overlapping, 1 overlapping not board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(20,2)
= 27,360
Case 6C. 2 low cards and 2 high cards (1 non-overlapping, 1 overlapping board pair)
= C(4,1)*C(4,1)*C(1,1)*C(2,1)*C(20,2)
= 6,080
Case 6D. 3 low cards without a pair and 1 high card (3 non-overlapping)
= C(4,3)*C(4,1)*C(4,1)*C(4,1)*C(20,1)
= 5,120
Case 6E. 3 low cards without a pair and 1 high card (2 non-overlapping, 1 overlapping not board pair)
= C(4,2)*C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(20,1)
= 17,280
Case 6F. 3 low cards without a pair and 1 high card (2 non-overlapping, 1 overlapping board pair)
= C(4,2)*C(4,1)*C(4,1)*C(1,1)*C(2,1)*C(20,1)
= 3,840
Case 6G. 3 low cards without a pair and 1 high card (1 non-overlapping, 2 overlapping not board pair)
= C(4,1)*C(4,1)*C(3,2)*C(3,1)*C(3,1)*C(20,1)
= 8,640
Case 6H. 3 low cards without a pair and 1 high card (1 non-overlapping, 2 overlapping with board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(1,1)*C(2,1)*C(20,1)
= 5,760
Case 6I. 3 low cards with a pair and 1 high card (2 non-overlapping)
= C(4,2)*C(2,1)*C(4,2)*C(4,1)*C(20,1)
= 5,760
Case 6J. 3 low cards with a pair and 1 high card (1 non-overlapping w/pair, 1 single overlapping not board pair)
= C(4,1)*C(4,2)*C(3,1)*C(3,1)*C(20,1)
= 4,320
Case 6K. 3 low cards with a pair and 1 high card (1 non-overlapping w/pair, 1 single overlapping board pair)
= C(4,1)*C(4,2)*C(1,1)*C(2,1)*C(20,1)
= 960
Case 6L. 3 low cards with a pair and 1 high card (1 single non-overlapping, 1 paired overlapping not board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,2)*C(20,1)
= 2,880
Case 6M. 3 low cards with a pair and 1 high card (1 single non-overlapping, 1 paired overlapping board pair)
= C(4,1)*C(4,1)*C(1,1)*C(2,2)*C(20,1)
= 320
Case 6N. 4 low cards without a pair (4 non-overlapping)
= C(4,4)*C(4,1)*C(4,1)*C(4,1)*C(4,1)
= 256
Case 6O. 4 low cards without a pair (3 non-overlapping, 1 overlapping not board pair)
= C(4,3)*C(4,1)*C(4,1)*C(4,1)*C(3,1)*C(3,1)
= 2,304
Case 6P. 4 low cards without a pair (3 non-overlapping, 1 overlapping board pair)
= C(4,3)*C(4,1)*C(4,1)*C(4,1)*C(1,1)*C(2,1)
= 512
Case 6Q. 4 low cards without a pair (2 non-overlapping, 2 overlapping not board pair)
= C(4,2)*C(4,1)*C(4,1)*C(3,2)*C(3,1)*C(3,1)
= 2,592
Case 6R. 4 low cards without a pair (2 non-overlapping, 2 overlapping w/board pair)
= C(4,2)*C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(1,1)*C(2,1)
= 1,728
Case 6S. 4 low cards without a pair (1 non-overlapping, 3 overlapping no board pair)
= C(4,1)*C(4,1)*C(3,3)*C(3,1)*C(3,1)*C(3,1)
= 432
Case 6T. 4 low cards without a pair (1 non-overlapping, 3 overlapping w/board pair)
= C(4,1)*C(4,1)*C(3,2)*C(3,1)*C(3,1)*C(1,1)*C(2,1)
= 864
Case 6U. 4 low cards with a pair (3 non-overlapping)
= C(4,3)*C(3,1)*C(4,2)*C(4,1)*C(4,1)
= 1,152
Case 6V. 4 low cards with a pair (2 non-overlapping w/pair, 1 single overlapping no board pair)
= C(4,2)*C(2,1)*C(4,2)*C(4,1)*C(3,1)*C(3,1)
= 2,592
Case 6W. 4 low cards with a pair (2 non-overlapping w/pair, 1 single overlapping board pair)
= C(4,2)*C(2,1)*C(4,2)*C(4,1)*C(1,1)*C(2,1)
= 576
Case 6X. 4 low cards with a pair (2 non-overlapping, 1 pair overlapping no board pair)
= C(4,2)*C(4,1)*C(4,1)*C(3,1)*C(3,2)
= 864
Case 6Y. 4 low cards with a pair (2 non-overlapping, 1 pair overlapping board pair)
= C(4,2)*C(4,1)*C(4,1)*C(1,1)*C(2,2)
= 96
Case 6Z. 4 low cards with a pair (1 non-overlapping w/pair, 2 singles overlapping no board pair)
= C(4,1)*C(4,2)*C(3,2)*C(3,1)*C(3,1)
= 648
Case 6AA. 4 low cards with a pair (1 non-overlapping w/pair, 2 singles overlapping w/board pair)
= C(4,1)*C(4,2)*C(3,1)*C(3,1)*C(1,1)*C(2,1)
= 432
Case 6AB. 4 low cards with a pair (1 non-overlapping single, 2 overlapping w/pair and no board pair)
= C(4,1)*C(4,1)*C(3,2)*C(2,1)*C(3,2)*C(3,1)
= 864
Case 6AC. 4 low cards with a pair (1 non-overlapping single, 2 overlapping w/pair, single=board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,2)*C(1,1)*C(2,1)
= 288
Case 6AD. 4 low cards with a pair (1 non-overlapping single, 2 overlapping w/pair, pair=board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,1)*C(1,1)*C(2,2)
= 144
Case 6AE. 4 low cards with 2 pairs (2 non-overlapping)
= C(4,2)*C(4,2)*C(4,2)
= 216
Case 6AF. 4 low cards with 2 pairs (1 non-overlapping, 1 overlapping not board pair)
= C(4,1)*C(4,2)*C(3,1)*C(3,2)
= 216
Case 6AG. 4 low cards with 2 pairs (1 non-overlapping, 1 overlapping board pair)
= C(4,1)*C(4,2)*C(1,1)*C(2,2)
= 24
Case 6AH. 4 low cards with trips (2 non-overlapping)
= C(4,2)*C(2,1)*C(4,3)*C(4,1)
= 192
Case 6AI. 4 low cards with trips (1 non-overlapping trips, 1 overlapping not board pair)
= C(4,1)*C(4,3)*C(3,1)*C(3,1)
= 144
Case 6AJ. 4 low cards with trips (1 non-overlapping trips, 1 overlapping board pair)
= C(4,1)*C(4,3)*C(1,1)*C(2,1)
= 32
Case 6AK. 4 low cards with trips (1 non-overlapping single, 1 trips overlapping not board pair)
= C(4,1)*C(4,1)*C(3,1)*C(3,3)
= 48
TOTAL for Case 6 = 123,776 [confirmed by computer brute-forced]
Case 7: Board = 5 Distinct Low Cards [1,1,1,1,1]
Case 7A. 2 low cards and 2 high cards (2 non-overlapping ranks from board)
= C(3,2)*C(4,1)*C(4,1)*C(20,2)
= 9,120
Case 7B. 2 low cards and 2 high cards (1 non-overlapping, 1 overlapping)
= C(3,1)*C(4,1)*C(5,1)*C(3,1)*C(20,2)
= 34,200
Case 7C. 2 low cards and 2 high cards (2 overlapping)
= C(5,2)*C(3,1)*C(3,1)*C(20,2)
= 17,100
Case 7D. 3 low cards without a pair and 1 high card (3 non-overlapping)
= C(3,3)*C(4,1)*C(4,1)*C(4,1)*C(20,1)
= 1,280
Case 7E. 3 low cards without a pair and 1 high card (2 non-overlapping, 1 overlapping)
= C(3,2)*C(4,1)*C(4,1)*C(5,1)*C(3,1)*C(20,1)
= 14,400
Case 7F. 3 low cards without a pair and 1 high card (1 non-overlapping, 2 overlapping)
= C(3,1)*C(4,1)*C(5,2)*C(3,1)*C(3,1)*C(20,1)
= 21,600
Case 7G. 3 low cards without a pair and 1 high card (3 overlapping)
= C(5,3)*C(3,1)*C(3,1)*C(3,1)*C(20,1)
= 5,400
Case 7H. 3 low cards with a pair and 1 high card (2 non-overlapping)
= C(3,2)*C(2,1)*C(4,2)*C(4,1)*C(20,1)
= 2,880
Case 7I. 3 low cards with a pair and 1 high card (1 non-overlapping pair, 1 overlapping single)
= C(3,1)*C(4,2)*C(5,1)*C(3,1)*C(20,1)
= 5,400
Case 7J. 3 low cards with a pair and 1 high card (1 non-overlapping single, 1 overlapping pair)
= C(3,1)*C(4,1)*C(5,1)*C(3,2)*C(20,1)
= 3,600
Case 7K. 3 low cards with a pair and 1 high card (2 overlapping)
= C(5,2)*C(2,1)*C(3,2)*C(3,1)*C(20,1)
= 3,600
Case 7L. 4 low cards without a pair (3 non-overlapping 1 overlapping)
= C(3,3)*C(4,1)*C(4,1)*C(4,1)*C(5,1)*C(3,1)
= 960
Case 7M. 4 low cards without a pair (2 non-overlapping, 2 overlapping)
= C(3,2)*C(4,1)*C(4,1)*C(5,2)*C(3,1)*C(3,1)
= 4,320
Case 7N. 4 low cards without a pair (1 non-overlapping, 3 overlapping)
= C(3,1)*C(4,1)*C(5,3)*C(3,1)*C(3,1)*C(3,1)
= 3,240
Case 7O. 4 low cards without a pair (4 overlapping)
= C(5,4)*C(3,1)*C(3,1)*C(3,1)*C(3,1)
= 405
Case 7P. 4 low cards with a pair (3 non-overlapping)
= C(3,3)*C(3,1)*C(4,2)*C(4,1)*C(4,1)
= 288
Case 7Q. 4 low cards with a pair (2 non-overlapping w/pair, 1 single overlapping)
= C(3,2)*C(2,1)*C(4,2)*C(4,1)*C(5,1)*C(3,1)
= 2,160
Case 7R. 4 low cards with a pair (2 non-overlapping, 1 overlapping pair)
= C(3,2)*C(4,1)*C(4,1)*C(5,1)*C(3,2)
= 720
Case 7S. 4 low cards with a pair (1 non-overlapping w/pair, 2 overlapping)
= C(3,1)*C(4,2)*C(5,2)*C(3,1)*C(3,1)
= 1,620
Case 7T. 4 low cards with a pair (1 non-overlapping, 2 overlapping w/pair)
= C(3,1)*C(4,1)*C(5,2)*C(2,1)*C(3,2)*C(3,1)
= 2,160
Case 7U. 4 low cards with a pair (3 overlapping)
= C(5,3)*C(3,1)*C(3,2)*C(3,1)*C(3,1)
= 810
Case 7V. 4 low cards with 2 pairs (2 non-overlapping)
= C(3,2)*C(4,2)*C(4,2)
= 108
Case 7W. 4 low cards with 2 pairs (1 non-overlapping, 1 overlapping)
= C(3,1)*C(4,2)*C(5,1)*C(3,2)
= 270
Case 7X. 4 low cards with 2 pairs (2 overlapping)
= C(5,2)*C(3,2)*C(3,2)
= 90
Case 7Y. 4 low cards with trips (2 non-overlapping)
= C(3,2)*C(2,1)*C(4,3)*C(4,1)
= 96
Case 7Z. 4 low cards with trips (1 non-overlapping trips, 1 single overlapping)
= C(3,1)*C(4,3)*C(5,1)*C(3,1)
= 180
Case 7AA. 4 low cards with trips (1 single non-overlapping, 1 overlapping trips)
= C(3,1)*C(4,1)*C(5,1)*C(3,3)
= 60
Case 7AB. 4 low cards with trips (2 overlapping)
= C(5,2)*C(2,1)*C(3,3)*C(3,1)
= 60
TOTAL for Case 7 = 136,127 [confirmed by computer brute-force]
I'm way too lazy to do all of that by hand. Respect.
I will present the results of the simulations which have now completed in the next few posts.
Here are the results of 1,000,000 deals for 4-player O8. Note to keep things relatively simple, for each half of the pot I have lumped all ties into one bucket, no matter how many or which players tie.
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 41.5255%
(2) Pct of deals for which one player scoops both high and low pots = 12.0654%
(3) Pct of deals for which two different players win high and low pots = 39.1825%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 7.2266%
Here are the results of 1,000,000 deals for 4-player O8. Note to keep things relatively simple, for each half of the pot I have lumped all ties into one bucket, no matter how many or which players tie.
HIGH WINNER | P1 Wins Low | P2 Wins Low | P3 Wins Low | P4 Wins Low | Any Tie for Low | Nobody has a Low |
---|---|---|---|---|---|---|
P1 Wins High | 30,248 | 32,534 | 32,953 | 32,793 | 12,279 | 99,957 |
P2 Wins High | 32,471 | 30,280 | 32,450 | 32,710 | 12,405 | 100,726 |
P3 Wins High | 32,238 | 32,806 | 30,171 | 32,287 | 12,411 | 99,691 |
P4 Wins High | 32,909 | 32,978 | 32,696 | 29,955 | 12,329 | 99,613 |
Any Tie for High | 4,051 | 4,167 | 4,238 | 4,288 | 6,098 | 15,268 |
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 41.5255%
(2) Pct of deals for which one player scoops both high and low pots = 12.0654%
(3) Pct of deals for which two different players win high and low pots = 39.1825%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 7.2266%
Here are the results of 1,000,000 deals for 5-player O8. Note to keep things relatively simple, for each half of the pot I have lumped all ties into one bucket, no matter how many or which players tie.
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 40.4838%
(2) Pct of deals for which one player scoops both high and low pots = 9.8581%
(3) Pct of deals for which two different players win high and low pots = 40.0020%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 9.6561%
HIGH WINNER | P1 Wins Low | P2 Wins Low | P3 Wins Low | P4 Wins Low | P5 Wins Low | Any Tie for Low | Nobody has a Low |
---|---|---|---|---|---|---|---|
P1 Wins High | 19,586 | 19,802 | 19,815 | 20,129 | 20,107 | 13,149 | 76,955 |
P2 Wins High | 19,950 | 19,584 | 19,980 | 19,861 | 20,286 | 13,085 | 77,176 |
P3 Wins High | 20,041 | 19,912 | 19,781 | 20,034 | 19,949 | 13,249 | 77,058 |
P4 Wins High | 20,011 | 20,006 | 19,845 | 19,556 | 20,128 | 13,077 | 77,646 |
P5 Wins High | 19,971 | 19,997 | 20,166 | 20,030 | 20,074 | 13,120 | 76,904 |
Any Tie for High | 4,395 | 4,460 | 4,540 | 4,436 | 4,400 | 8,650 | 19,099 |
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 40.4838%
(2) Pct of deals for which one player scoops both high and low pots = 9.8581%
(3) Pct of deals for which two different players win high and low pots = 40.0020%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 9.6561%
Here are the results of 1,000,000 deals for 6-player O8. Note to keep things relatively simple, for each half of the pot I have lumped all ties into one bucket, no matter how many or which players tie.
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 40.0866%
(2) Pct of deals for which one player scoops both high and low pots = 8.1116%
(3) Pct of deals for which two different players win high and low pots = 39.7663%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 12.0355%
Note here at N=6 players it becomes more likely, when a low is made, for a person to scoop the pot compared to that same person splitting the pot with any other specific opponent. That is, for the first time the diagonal elements of the above matrix are larger than the off-diagonal elements.
I am guessing this is related to the added value that Aces provide in O8 as the number of players at the table increases the quality of the winning low hand increases.
HIGH WINNER | P1 Wins Low | P2 Wins Low | P3 Wins Low | P4 Wins Low | P5 Wins Low | P6 Wins Low | Any Tie for Low | Nobody has a Low |
---|---|---|---|---|---|---|---|---|
P1 Wins High | 13,664 | 13,071 | 13,377 | 13,185 | 13,155 | 13,275 | 13,771 | 63,033 |
P2 Wins High | 13,111 | 13,373 | 13,178 | 13,221 | 13,419 | 13,201 | 13,718 | 63,020 |
P3 Wins High | 13,329 | 13,465 | 13,553 | 13,218 | 13,205 | 13,298 | 13,809 | 63,412 |
P4 Wins High | 13,425 | 13,279 | 13,211 | 13,490 | 13,214 | 13,204 | 13,707 | 63,041 |
P5 Wins High | 13,167 | 13,356 | 13,305 | 13,182 | 13,594 | 13,305 | 13,619 | 62,662 |
P6 Wins High | 13,353 | 13,114 | 13,186 | 13,258 | 13,396 | 13,442 | 13,548 | 62,967 |
Any Tie for High | 4,397 | 4,476 | 4,539 | 4,532 | 4,524 | 4,605 | 11,110 | 22,731 |
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 40.0866%
(2) Pct of deals for which one player scoops both high and low pots = 8.1116%
(3) Pct of deals for which two different players win high and low pots = 39.7663%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 12.0355%
Note here at N=6 players it becomes more likely, when a low is made, for a person to scoop the pot compared to that same person splitting the pot with any other specific opponent. That is, for the first time the diagonal elements of the above matrix are larger than the off-diagonal elements.
I am guessing this is related to the added value that Aces provide in O8 as the number of players at the table increases the quality of the winning low hand increases.
Here are the results of 1,000,000 deals for 7-player O8. Note to keep things relatively simple, for each half of the pot I have lumped all ties into one bucket, no matter how many or which players tie.
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 39.9029%
(2) Pct of deals for which one player scoops both high and low pots = 6.8723%
(3) Pct of deals for which two different players win high and low pots = 38.8484%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 14.3764%
HIGH WINNER | P1 Wins Low | P2 Wins Low | P3 Wins Low | P4 Wins Low | P5 Wins Low | P6 Wins Low | P7 Wins Low | Any Tie for Low | Nobody has a Low |
---|---|---|---|---|---|---|---|---|---|
P1 Wins High | 9,801 | 9,218 | 9,293 | 9,369 | 9,162 | 9,269 | 9,198 | 14,252 | 53,335 |
P2 Wins High | 9,276 | 9,795 | 9,188 | 9,334 | 9,319 | 9,290 | 9,241 | 13,999 | 53,400 |
P3 Wins High | 9,175 | 9,218 | 9,936 | 9,414 | 9,298 | 9,301 | 9,235 | 14,179 | 53,240 |
P4 Wins High | 9,145 | 9,298 | 9,476 | 9,654 | 9,071 | 9,091 | 9,194 | 14,319 | 53,048 |
P5 Wins High | 9,225 | 9,245 | 9,284 | 9,163 | 10,010 | 9,259 | 9,240 | 14,082 | 52,865 |
P6 Wins High | 9,440 | 9,252 | 9,037 | 9,373 | 9,356 | 9,619 | 9,218 | 14,094 | 53,136 |
P7 Wins High | 9,104 | 9,235 | 9,253 | 9,211 | 9,139 | 9,377 | 9,908 | 13,983 | 53,426 |
Any Tie for High | 4,297 | 4,450 | 4,389 | 4,348 | 4,461 | 4,461 | 4,362 | 14,088 | 26,579 |
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 39.9029%
(2) Pct of deals for which one player scoops both high and low pots = 6.8723%
(3) Pct of deals for which two different players win high and low pots = 38.8484%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 14.3764%
Here are the results of 1,000,000 deals for 8-player O8. Note to keep things relatively simple, for each half of the pot I have lumped all ties into one bucket, no matter how many or which players tie.
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 39.9700%
(2) Pct of deals for which one player scoops both high and low pots = 5.9023%
(3) Pct of deals for which two different players win high and low pots = 37.4815%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 16.6462%
HIGH WINNER | P1 Wins Low | P2 Wins Low | P3 Wins Low | P4 Wins Low | P5 Wins Low | P6 Wins Low | P7 Wins Low | P8 Wins Low | Any Tie for Low | Nobody has a Low |
---|---|---|---|---|---|---|---|---|---|---|
P1 Wins High | 7,449 | 6,599 | 6,645 | 6,790 | 6,608 | 6,749 | 6,714 | 6,739 | 14,342 | 46,044 |
P2 Wins High | 6,658 | 7,294 | 6,855 | 6,590 | 6,770 | 6,857 | 6,631 | 6,699 | 14,298 | 46,068 |
P3 Wins High | 6,519 | 6,781 | 7,375 | 6,705 | 6,622 | 6,834 | 6,685 | 6,749 | 14,473 | 46,266 |
P4 Wins High | 6,825 | 6,785 | 6,670 | 7,277 | 6,633 | 6,758 | 6,916 | 6,623 | 14,410 | 46,129 |
P5 Wins High | 6,721 | 6,651 | 6,724 | 6,742 | 7,441 | 6,546 | 6,675 | 6,700 | 14,176 | 46,229 |
P6 Wins High | 6,665 | 6,855 | 6,568 | 6,666 | 6,725 | 7,350 | 6,743 | 6,634 | 14,061 | 46,387 |
P7 Wins High | 6,527 | 6,749 | 6,626 | 6,595 | 6,707 | 6,681 | 7,514 | 6,774 | 14,396 | 46,181 |
P8 Wins High | 6,658 | 6,745 | 6,495 | 6,692 | 6,745 | 6,532 | 6,665 | 7,323 | 14,451 | 46,057 |
Any Tie for High | 4,392 | 4,308 | 4,407 | 4,265 | 4,294 | 4,216 | 4,255 | 4,339 | 17,379 | 30,339 |
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 39.9700%
(2) Pct of deals for which one player scoops both high and low pots = 5.9023%
(3) Pct of deals for which two different players win high and low pots = 37.4815%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 16.6462%
Here are the results of 1,000,000 deals for 9-player O8. Note to keep things relatively simple, for each half of the pot I have lumped all ties into one bucket, no matter how many or which players tie.
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 39.9028%
(2) Pct of deals for which one player scoops both high and low pots = 5.0732%
(3) Pct of deals for which two different players win high and low pots = 36.0300%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 18.9940%
HIGH WINNER | P1 Wins Low | P2 Wins Low | P3 Wins Low | P4 Wins Low | P5 Wins Low | P6 Wins Low | P7 Wins Low | P8 Wins Low | P9 Wins Low | Any Tie for Low | Nobody has a Low |
---|---|---|---|---|---|---|---|---|---|---|---|
P1 Wins High | 5,697 | 4,916 | 4,925 | 5,099 | 5,016 | 4,871 | 4,990 | 4,963 | 4,994 | 14,782 | 40,076 |
P2 Wins High | 5,096 | 5,627 | 4,973 | 4,997 | 5,042 | 5,009 | 5,031 | 5,089 | 4,891 | 14,518 | 40,936 |
P3 Wins High | 5,050 | 5,032 | 5,705 | 4,987 | 4,925 | 5,080 | 4,920 | 4,979 | 4,939 | 14,459 | 40,508 |
P4 Wins High | 5,045 | 5,071 | 5,071 | 5,679 | 5,072 | 5,006 | 5,099 | 5,055 | 4,933 | 14,566 | 40,364 |
P5 Wins High | 4,964 | 4,922 | 5,096 | 5,060 | 5,560 | 5,011 | 4,906 | 4,865 | 5,110 | 14,587 | 40,262 |
P6 Wins High | 4,964 | 4,983 | 4,978 | 5,152 | 4,985 | 5,610 | 4,974 | 5,033 | 5,075 | 14,708 | 40,542 |
P7 Wins High | 5,036 | 5,002 | 4,961 | 4,927 | 5,017 | 5,030 | 5,677 | 4,910 | 4,916 | 14,698 | 40,727 |
P8 Wins High | 5,020 | 5,085 | 5,031 | 5,118 | 5,098 | 4,946 | 4,899 | 5,672 | 5,011 | 14,530 | 40,804 |
P9 Wins High | 5,013 | 5,136 | 5,011 | 4,928 | 5,069 | 4,902 | 5,003 | 4,987 | 5,505 | 14,589 | 40,784 |
Any Tie for High | 4,205 | 4,159 | 4,186 | 4,152 | 4,235 | 4,250 | 4,169 | 4,127 | 4,125 | 20,895 | 34,025 |
Let's try to summarize a few high-level results:
(1) Pct of deals for which there is No Low = 39.9028%
(2) Pct of deals for which one player scoops both high and low pots = 5.0732%
(3) Pct of deals for which two different players win high and low pots = 36.0300%
(4) Pct of deals for which both high and low pots exist and there is a tie for one or more halves = 18.9940%
Okay, here is a summary table of the results posted thus far. The first row results are the exact probabilities while the other rows in the table are the results from simulations of 1,000,000 deals with that many players.
Miscellaneous comments on these results:
1. All of these results are based upon every player going to showdown on every deal. Of course, that is an extreme assumption and real-world O8 results are likely to be different from the above percentages.
2. We know that a Low is Available (based purely upon a random Board) in 60.0905% of O8 deals. The table suggests that this cap is approached in most full-ring O8 games.
3. Standard sampling theory suggests that the percentages in the table (except the first row) are within 0.1% of the true values.
4. Deals that involve ties (more than one player sharing one or both halves of the pot) could well be considered split pots and lumped in with the "Split Pot" (one player winning the high pot and another player winning the low pot) percentages above.
5. From the previously reported results, the low half of the pot is much more likely to be tied than the high half of the pot.
Players at Table | No Low Made | Low Made | Scoop Pot | Split Pot | Tied Pot |
---|---|---|---|---|---|
1 | 65.1882% | 34.8118% | na | na | na |
2 | 50.4024% | 49.5976% | 21.1035% | 26.3360% | 2.1581% |
3 | 44.0788% | 55.9212% | 15.4666% | 35.7575% | 4.6971% |
4 | 41.5255% | 58.4745% | 12.0654% | 39.1825% | 7.2266% |
5 | 40.4838% | 59.5162% | 9.8581% | 40.0020% | 9.6561% |
6 | 40.0866% | 59.9134% | 8.1116% | 39.7663% | 12.0355% |
7 | 39.9029% | 60.0971% | 6.8723% | 38.8484% | 14.3764% |
8 | 39.9700% | 60.0300% | 5.9023% | 37.4815% | 16.6462% |
9 | 39.9028% | 60.0972% | 5.0732% | 36.0300% | 18.9940% |
Miscellaneous comments on these results:
1. All of these results are based upon every player going to showdown on every deal. Of course, that is an extreme assumption and real-world O8 results are likely to be different from the above percentages.
2. We know that a Low is Available (based purely upon a random Board) in 60.0905% of O8 deals. The table suggests that this cap is approached in most full-ring O8 games.
3. Standard sampling theory suggests that the percentages in the table (except the first row) are within 0.1% of the true values.
4. Deals that involve ties (more than one player sharing one or both halves of the pot) could well be considered split pots and lumped in with the "Split Pot" (one player winning the high pot and another player winning the low pot) percentages above.
5. From the previously reported results, the low half of the pot is much more likely to be tied than the high half of the pot.
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