May 31, 2020

*** This hand was suggested by Ras2829
0-0*  ?
30%
27%
11%
10%
6%
5%
4%
2%
1%
1%
0%
0%
0%
Total votes: 199
thelawnet
262 votes

Joined: January 2020
 
 
 
Sunday 3:32 AM
to continue yesterday's discussion about how the bot calculates our % chance of winning, while it's not clear to me that the Hal bot does it this way, in general there are only 14,641 (121 numbers between 0 and 120 for each player) possible starting positions before each game.

This is a very small number in computational terms, and we could look at actual outcomes e.g. here

https://blog.cribbagepro.net/2012/11/cribbage-strategy-and-board-position.html

where we see that based on millions of *actual* games played, we can know that at 90-90, there is a 50/50 chance of either player winning. Indeed at any of the 14,641 possible positions there will be a fixed % chance that we win. And near the end of the game, there will be a significant difference between say 110-110 and 109-109.

For a bot, we could ignore those numbers and instead use 'best possible play', where we calculate the chance of winning based on our play. For example, at 116-116 there is a 100% chance of the game ending this deal. At 100-100 that is not so, but if we have already calculated that the chances from 116-116 are, say 70% pone, 30%, dealer, 0% new deal, the exact winning % for us at 100-100 simply becomes a matter of recursing through the possible outcomes, weighted by the chance that they occur, and then using our already calculated win % for each possible score outcome at the end of the deal.

Here we might note that if we are playing against humans, then humans make mistakes, so it might be in some ways better for the bot to use the human winning %, for matches between players who win overall (don't look at the results for poor players), in order to determine outcomes.

Then when we have a hand like today's A248910, it follows that for each of the 15 possible discards/hands, then each one will have a distribution of outcomes after the deal.

For example, if we toss 29 today, then we have at least 4 points, while pone may have any number of points. If play were to proceed 4K8A, then there is some chance the bot plays AA for 7.

Overall, there will be for any given toss therefore a table of possible outcomes, together with the likelihood they occur.

E.g.

4-12 occurs 1.6% of the time for the toss 29


If we multiply the frequency of the chance of a given post-hand score occurring by our known (calculated above) binary chances of winning or losing from any given position, it follows that we can calculate the exact chance to win or lose with any given toss, providing we peg the same way as the bot.

At 0-0, there should be a relatively straightforward correlation between the difference between our points and pone's points and our winning chance, but clearly as with yesterday as we approach critical score zones, then both players for instance adding 4 points will benefit one player more than another and indeed we might profit by scoring average 8.2 to pone's 10.2 points (-2 points) rather than say 12.2 points to 12.2 points (+0 points), in that position is important.

I do not think this is particularly important at 0-0, but some seem to think it is. I have not seen numbers supporting this though?!

Anyway, with today's hand we have trash. A4 tends to peg well, i.e. AX4. No 6 though. 489T pegs poorly.

So I'm holding A249 and will try to out-peg pone. The cut is interesting - if pone has a 4, he will certainly pair us. However 74% of the time, he was not dealt a 4, so the lead of a 4 looks like a conservative lead and we may get the X back for our 15-2. If we lead the A then pone probably won't pair, but he may spot the trap and reply with 7,8, etc.

So will lead the 4, and then we have 4-9-2, 4-X-A, etc.

Hope to get a better hand next time.
thelawnet says: to continue on the bot theme, we should note that a good bot must defeat all humans. When we say 'beat', we must consider the sample size. For example, a human winning 520 out of 1000 games would not constitute proof that the human is better. Generally the task of proving superiority is harder (requires more time) the more closely matched the players are. As I understand it, halscrib does not beat all players over such a horizon. So we should take care. I am not very good at cribbage, but still win a lot. This is because in my opinion humans make (even more than I) egregious errors. A bot cannot make such errors, because it is infallible, and in more popular games, such as chess or Go, computers crush humans. So I'm slightly suspicious as to the bot's methodology and numbers although it's likely the best we have - as it is my assessment that with such a mathematical game as cribbage, a sufficiently smart computer program should grind humans into meat paste as each number is calculated to infinitesimal precision.
thelawnet says: Further, to look at the bot's numbers for example yesterday it quotes W4%. To me this doesn't exactly make sense in that, as noted, our chance of winning at any given score is a precise and exact number, and every time we play at 100-100, the chance of winning will be the exact same %. So I don't quite follow the logic of 'W3' and 'W4' in that this suggests some sort of unnecessary simulation process (may be wrong). Unnecessary because we simply need to know the distribution of possible outcomes, the % chance of each occurring, and then the binary W/L % for each hand, which again, is EXACTLY THE SAME for any given score pair.
thelawnet says: should say ALWAYS THE SAME. Continuing, in general we can influence proceedings in various ways: 1. as non-dealer we can choose to maximise our own hand, maximise our hand minus dealer's crib, or some combination in between. 2. as either player we can firstly hold cards that will tend to peg more points for both players, or cards that will tend to peg more points for us. For example, 5555 as non-dealer will peg almost nothing, while 2233 will peg many. 3. As either player we can attempt to play our cards DIFFERENTLY. For example, if pone leads a 4, at 0*-0, we might pair it, but at 110-112* probably not (depending on the rest of our hand)
thelawnet says: If we look at the 28th May, the score was 30-40*. If we refer to the chart in the blog entry above, it seems that at scores such as this, it makes a very small difference whether the hand ends up at, say, 45-50, or 47-52. Clearly it makes SOME difference, but it should be extremely obvious that the further from 121 we are the SIZE of the importance of the difference must be less important. Here we see the 5567 hand scored 12.22 points, while the 88 toss averaged 5.58. This brings the scores, obviously, to 42.22 - 45.58. However we can see here an obvious problem with these numbers. Pone has a hand as well, and while the average dealer hand averages, say 7 points, it's necessarily the case that that will be true here, as we are holding 2 5s, for example. In addition, it's not really clear to me whether the value of dealer's crib in isolation is that meaningful. We know that dealer tosses certain cards MORE, and other cards LESS. This is because he has the task of maximising the total points from two hands, while non-dealer wants to maximise one hand, and minimise another. So the information that a certain toss gives pone a higher crib might be slightly misleading if that higher crib average results from dealer downsizing his own hand. It may be that this particular hand/crib jostling is not that big of an effect, but at least we should want to know the average size of pone's hand (which is likely helped from our perspective by our holding the 5s)
thelawnet says: Continuing, we can see that the 5567 is worth net (our hand minus their crib) 6.64 points while the 6788 is worth net of only 4.98 points. This is a big difference. But the bot has decided that we should play a defense strategy for pegging. This means that we should not trade pegs with pone, because we are far behind, and makes perfect sense. What doesn't make sense is how the numbers are quoted, and nor indeed are they comparable. Because it's 'defense', the peg scores quoted as 'pone's pegs'. Not pone - our pegs. Thus for example, the non-dealer 5567 has human data for thousands of hands showing average 4.41 pegs for pone, and 2.17 for us. This is a score of -2.24, obviously. The bot quotes a 'defense' score of -3.89 points. That is to say that by pegging defensively it thinks it can incur only 3.89 pegs. Which is better than the human average. Meanwhile it quotes 6788 as -2.61 pegs. Here human data suggests -3.47 pegs, but we expect the bot to be lower as it's playing defensively. However, neither the -3.89 numbers nor the -2.61 pegs are entirely meaningful on their own. In particular, at 0-0, it clearly would NOT make sense to say that if hand A incurs mean 4 pegs, and hand B incurs mean 2 pegs, while hand A SCORES mean 5 pegs, and hand B mean 0.5 pegs, that hand A is better. This cannot possibly be correct and it is completely certain that the sums given by the bot are misleading at best, because in general when we are sufficiently far from home, incurring 4 points and scoring 6 points is FAR better than incurring 2 and scoring 0. There is no way that this is NOT correct. So I am not sure if the sum comes directly from the bot or are entered by the human user, but it's not exactly meaningful. Clearly the choice between 'defense' and 'offense' is a valid idea in general, just not the way the numbers are being presented, and they cannot possibly be absolutes. In other words it cannot be correct to absolutely play 'pegging defense' at 30-40*. A more nuanced position (possibly taken by the bot), is that given the choice between 2 pegs us, 3 pegs pone, and 5 pegs us 6 pegs pone, then we should prefer the former. That is very obvious. But that will NEVER be the case. What we will in fact have on a mathematical level is something like a choice between 2-3.1 and say 3-4. And the REAL picture is in fact a distribution of pegging outcomes each with a % chance of happening. E.g. 10% to score 6 and concede 9, 10% to score 2 and concede 2, etc. In general, conceding pegs is NOT bad, in early game, providing we peg relatively MORE (or less than average as pone....)
thelawnet says: So the bot tends to give misleading numbers in that it makes NO SENSE to deduct pone's pegs without adding our own, when are FAR from 121, and what we need is MORE POINTS. So the 'expected averages' aren't really that.... We have OUR expected average, which is Our hand + our pegs + our crib And then pone's, which is the opposite. But if we only consider HALF the pegging, we will give numbers that are nonsensical....
thelawnet says: After the 3 cut on 28th May, we see the bot goes to optimal as presumably it has decided it's not too far behind now. Then it quotes 'net pegging points'. Here, the 5 is the 'cheapest' lead, though all leads are likely to result in pone out-pegging us (which we expect any way). However it may be that following 5-X 15-2 for pone, the play goes 7-go-6 for 4 and then X-5 15-2 for 2. Of course that won't necessarily happen, but the bot has determined that 5 has the lowest mean points cost. However 7 may incur FEWER points, but there's nothing to say that this makes us more likely to win! In general, with the 5 coming out net cost 0.52 points cheaper it seems unlikely that even if the 7 incurs fewer points, that it gives us a greater winning % from this distance out.
thelawnet says: if we look at yesterday's puzzle, and IGNORE pegs, then we see in fact: A278 - 15.13 points hand + crib W 37.2% (NOTE: 11.6% not specified!) A678 - 14.90 points hand + crib W 38.4% 2678 - 14.88 points hand + crib W 38.5% 5678 - 13.74 points hand + crib W 35.0% So the bad hold of 5678 does poorly as we expect, and A678 and 2678 are identical for all reasonable purposes, which is not surprising given how similar they are. So WHY is the flush poor here? It's actually quite simple - although the bot HIDES the information from us. A678 and 2678 peg almost exactly the same, but A278 simply pegs FEWER points. And that it does should be obvious by looking at it. So even though the bot says 'defense', and only shows us the pegs AGAINST us, it's the pegs we are NOT scoring with this defensive flush that are holding us back. At least that's what I BELIEVE, from the fact that larger databases show that A278, 2678 and A678 all INCUR about the same number of pegs, but A278 SCORES significantly fewer. So much for defensive! If we were quoted the true 'expected average', of ALL OUR POINTS SCORED minus ALL PONE'S POINTS SCORED, then MOST of the time the highest number is going to give us the biggest W%. Unfortunately the bot numbers seem almost OBFUSCATED, even if the W% itself is accurate!
Gougie00 says: Reduce the caffeine - Greg
james500 says: Hi Lawnet, a great deal of food for thought in your post today. You've added a very analytical voice to the site.
JQT says: Yes Sir, Mr. 'thelawnet' I envy your energy: if you scroll back, say a decade or so, I once often posted such lengthy analyses on this site, to the consternation of perhaps some, but also to the delight of at least a few. But however, as my life's odometer has since long ago 'rolled' from past age fifty and now has reached beyond age sixty, I seem to lack that same amount of energy that I once had, which makes me wonder: What decade (if you wish to answer) were you born? And per you name, are you into "law" in some fashion, or maybe it's "lawn" care LOL (I somehow suspect the former and not the latter!) Where do you 'hail from,' and when did you begin playing 'This Game Of Ours' called Cribbage? For me, I suppose it was 1964 or thereabouts, when I was about six years of age, that I began my love for the game. I am a writer by avocation, and I was an Electrical Engineer by vocation, formerly working for over two decades at Hewlett-Packard Company as an Information Engineer, although I have been retired/disabled since age forty-two starting in the year 2000. Anyway, as james500 says, it's very nice to have someone with your expertise and acumen aboard this site and posting actively, even though we may sometimes disagree on some choices: it's all what makes Cribbage such a wonderful game! - j q t -
Ras2829 says: Hi thelawnet: I would seriously advise that challenging the genius of Hal Mueller and his tutorials is not a productive thing to do. Hal Mueller challenged the thinking of many in Cribbage World with revolutionary ideas when introduced and was a major reason that cribbage has become a lifetime pursuit in learning. Prior to that many folks had played this game a lifetime and saw nothing more to it than one experiences daubing a bingo card. RAS plays an average of 100 games of crib weekly, live and electronically. HalscribCLX, REX, and Cribbage Prof, all Mueller's creations, are the toughest three players I face. With live players I win 58-59% of games played. With the crib bots I win 51%. Those are huge differences. I sometimes question the ranking or choice made by HalscribCLX in some mild form. Most often I rationalize that I am still learning this game and there is something to learn from differing choices or views. Often advise folks that I have not played long enough to learn this game. It takes more than a lifetime and most folks don't get 300 years.
dgergens says: Thanks for answering the question! Now I have to go study it. I'll understand it eventually, but have to walk myself through it. Cheers.
thelawnet says: Hi, Ras, perhaps I was not clear enough. As you say, it takes more than a lifetime to learn this game. Nothing can be more certain than the fact that a sufficiently advanced computer must absolutely beat you and everyone else alive or dead, or at the very worst case come out 50/50. That is true of Magnus Carlsen vs any number of chess computers, Lee Sedol vs a Go computer, and it must be true of cribbage. However such has not always been true. 1980s chess computer would lose to the best human. As I understand it Halscrib is somewhat dated software? It's perfectly possible for it to be genius yet for it to be less than the ultimate player. Indeed so long as humans can beat it, we can absolutely certain that it's NOT the ultimate player, because the FIRST task is for the bot to beat humans, and then we have 'robot supremacy' and they are left to battle each other, as Bot A takes on Bot B for 1 million games or until such number as we can be certain that one bot is better than the other. I don't have a copy of the software, so beyond the logical necessity that it is not unimpeachable inasmuch that it does not beat everyone, which it must, I don't know what output it produces and to what extent the output we see here is 'raw' and to what extent there is human involvement. So I can only comment on what I see here. And in respect of THAT output I can only observe, that if for example we quote at 0-0* numbers in the format 'my score' minus 'pone's crib' minus 'pone's pegging' then THAT output is DEFINITELY misleading in that it excludes OUR pegging, which in one case might be VERY GOOD, and in another case mediocre. And yet, the bot produces W% numbers that seem to reflect OUR pegging (as of course it must!) So we must understand firstly what the data ARE and then what the data MEAN. If we don't know what we are looking at, then the numbers will just be NOISE on a screen. But by understanding the significance of what is NOT there, we can understand our tool much better. Anyway, it is clear to me that the 'Halscrib' approach of 'aggressive' vs 'optimal' vs 'defensive' is entirely logical, and represents a big advance on a fixed strategy, BUT when we are considering a computer that has NO NEED AT ALL to follow anything so simple (in the context of computers capable of BILLIONS of computations per second) as a ternary logic of defensive/optimal/aggressive. The best strategy cannot possibly be as simple in that, particularly at a position like 0-0. At 0-0, a pegging run where we score 11 points is glorious! And so what if pone scores 5 points! The 11 points just absolutely is better. So if it's true that the bot is following a rigid ternary strategy, we can see that it must be possible for the bot to employ BETTER strategy, in that while we might reject, a pegging outcome of 7-5 in favour of 2-0, 11-5 is something else entirely. It might be, and I assume that it is, that the bot will in fact grab a chance to peg 11 by allowing pone to peg 5, despite being 'defensive', and this might indeed feed into the W%, but again this comes back to understanding what the numbers actually are and what they mean. At any rate, I'd be looking at W%/L% as more meaningful than others.....
thelawnet says: Just to add, I've learned that when the data says something that is contradictory or makes no sense to us based on that which is logical, we should try to understand WHY that has happened. I.e. there is ALWAYS a logical explanation. Human error (us? someone else? the person who input the data? the computer programmer?), or the data aren't what they seem to be. For example, recently in the UK the number of young men dying appeared to be less than usual. Was it because of covid-19 meaning they stayed at home instead of getting into car crashes, committing suicide, and the other ways young men died? Well maybe! But in fact we don't know because the death statistics tell us only that fewer deaths have been REGISTERED in a given week, and with all those deaths requiring an inquest before registration, something which takes 6+ months, and presumably such inquests were stopped or reduced during covid-19. So what we were actually seeing is the deaths of young men MONTHS AGO being delayed a further few months, and we won't know statistically if there were more suicides, domestic murders, etc. for perhaps a year. Here my insight is only from noting 'hands B & C are given a higher win %, yet has a lower "sum"'. This seems illogical until we think about the missing data (our pegs, pone's hand) whereupon it makes complete sense without employing the other possible explanation (a different distribution of hand outcomes for both players). Beyond that I cannot really be too certain about what the bot does and how well it plays, since all I have access to is the output on here, which for all I know is not 'raw'....
thelawnet says: Jqt: Hi. I don't know what age my father taught me cribbage but maybe 10? But it has pleasant filial connotations for me as a result. I am not yet 40. At one time I was going to be a lawyer but I failed the interview to Oxford because I had no experience of the kind of thinking they wanted, and my pride was too much to go elsewhere, so I decided that the law was not for me after all, though the username circa 1998 has stuck with me for life! And I became a computer programmer. I have done quite a bit of online gambling which is something that I find very easily mathematically in that it's just numbers + randomness, so if you can crunch the right numbers the randomness will reward you! It is the total inability of most people to properly understand risk and numbers that makes the gambling business so lucrative for those that set the odds. Anyway, I've stepped away from that recently and am at 98 degrees East running (or not at the moment) a travel business. I must confess that I played many games of crib without quite getting to the point that it's all just numbers and probabilities. However that might not have been a bad thing in that it's a social, tactile ritual aside from win or lose, and perhaps it really is the taking part that counts!
JQT says: Yes, after playing Halscrib in say 1000 games, you can examine not just your win/loss ratio, but it also provides a 'Stats' page which shows your average Hand as Pone, and Hand and Crib as Dealer, along with pegging averages. It can be very humbling, and I have a few tens of thousands of saved games spread out among a few programs, many in the Halscrib family. My educated guess (for I did help to test some aspects of the program) is that it wouldn't take all of my fingers and toes to count the number of people on Planet Earth that could maintain 50% or better win-rate after 1000 games. And RAS is very clearly one of these players. You've got to understand that the 'snapshot' of what is shown here does not begin to scratch the surface of the calculations that produced the given choices. Like RAS, I have played this program for well over a decade, and I am still able to find new features. Sadly, it did not port well to Windows 10, and so you really do need to boot into Windows XP to run a version that will display the HELP files, and work as it was designed. I have hoped for years that a Win 10 upgrade would occur, but I think we may have to wait until the next mathematical professor and Cribbage Genius takes over when Hal left off. A Cribbage Program can never have its results compared with a 'perfect knowledge' game such as Chess, because in Cribbage, cards and thus luck is involved (it's probabilistic), but after a large enough sample of games, it will obliterate 99.5% or more of all human players. And even the very few who can hold their own or beat it cannot always understand some of its displayed logic, because it's sort of like trying to understand the Dirac Equation or Feynman Diagrams, something I often actually find easier to understand!
Ras2829 says: Hi thelawnet: In Halscrib, REX, and the Cribbage Prof, pegs are computed based on choice of strategy. If choosing optimal pegging strategy, pegs are net and can be + or -. If choosing an offense pegging strategy pegs will be + as they are the pegs gained by the player choosing offense. If choosing defense, the pegs will be shown with a minus and are the pegs expected to be gained by the opponent. So it is only when choosing an optimal pegging strategy that potential pegs are reduced by those given up to opponent. I do play 1,000 game blocks with alternate deals and even number of games in each session. I play a bit more aggressively than do any of the cribbots; so out peg the bots both as dealer and non-dealer by .1-.2. Also have slightly larger hand size by .1 of a point when non-dealer. Hand size as dealer and crib size tend to be identical after 1,000 games.
thelawnet says: Hi again JQT. Luck really isn't a big problem for computers, in that a computer can crush anyone at backgammon, for example. In general it's just a matter of playing sufficiently many games to determine whether one player is better than another. Cribbage is relatively low variance in that to prove that something like a slot machine is fair (i.e. as advertised, not that it pays out more than put in) you may need millions of spins (empirically), whereas a game of win/loss a much smaller sample size is needed.
JQT
4143 votes

Joined: October 2008
 
 
 
Sunday 3:33 AM
It's a toss-up between Toss (A T) and Toss (8 T) for me this morning.

Holding the RUN in *any* form seems toxic but if forced, I would perhaps consider Toss (A 2) to be the safest.

If I feel adventurous, I might try leading the 8 Card, and 'go fishing' for a Trey response.
JQT says: We see Eleven different discard decisions have been voted upon out of "6 Choose 2" equals 15 Permutations, or Fifteen Total Possible Choices (see the link below for an explanation of this Combinatorial Formula) all made before NOON Eastern Time, and this can only mean one thing: Another Excellent Puzzle by Cribbage Master, RAS. https://www.mathway.com/popular-problems/Finite%20Math/600082
JQT says: Toss (A T) seems about a half of a point 'safer' in the Opponent's Crib than Toss (8 T), even with some Negative Delta due to our 9 Card; and also, I don't think this hand I've chosen should peg that much worse than a hand with three 'small' cards, as the only way to retain a three-card "Magic Eleven" would be to Toss (4 T) and begin with ZERO points!
Rosemarie44
2052 votes

Joined: March 2016
 
 
 
Sunday 3:38 AM
First hand as pone is usually offense strategy. Start with 3 points and a hand that has more cuts for improvement over 2-4-8-9 (second choice).
dec
6351 votes

Joined: April 2008
 
 
 
Sunday 3:38 AM
This to me the most offensive keep with an eleven count in the hand that might influence the pegging. Eight lead here for me. dec
james500
3916 votes

Joined: June 2013
 
 
 
Sunday 4:33 AM
9 lead?

Maybe: 9(9)-5(14)-A(15/2)...
Or: 9(9)-X(19)-4(23)-5(28)-2(30)-go-A(31/2).
mfetchCT425
1394 votes

Joined: February 2009
 
 
 
Sunday 5:58 AM
Great puzzle by Ras. Quite a few options to consider today with this subpar hand. I decided to keep the ‘5’ in my hand (A-4) along with the touching 9-10. Only the 3 or 7 cut do not help.
Jazzselke
2583 votes

Joined: March 2009
 
 
 
Sunday 6:07 AM
Tough call today. Do not like 48910, a lead of the 4 can get caught in a 3- card run with the 3 middle cards. Will go with a toss of 28 instead of A2, A4 or 24.
Jazzselke says: Choice of A4910.
Jazzselke says: Choice of A4910.
Gougie00
5723 votes

Joined: March 2008
 
 
 
Sunday 7:03 AM
Kept the 8-9-10, hoping for help. Tossed the A-2 because it probably won't go *Boom*.
horus93
1281 votes

Joined: December 2017
 
 
 
Sunday 9:41 AM
Really interesting puzzle today.

I saw 8-T as a "goldilocks" between a-2 or 2-4 (probably significantly more likely than 8-t to give up a huge crib) on the one hand and a-t (really bad hand) on the other.

But checking myself on discardpro and reading the comments I don't think I was right. I often overestimate the danger of a-2 it seems.
horus93 says: Seems I'm in good company after all. Even when I'm right I'm wrong!
dgergens
938 votes

Joined: January 2018
 
 
 
Sunday 10:03 AM
May not be the popular vote, but the small company is good company.
cribbagepogo
3249 votes

Joined: October 2007
 
 
 
Sunday 11:09 AM
Agree with Mike F. Not too dangerous and I don't have to type War and Peace to explain.
joekayak
1873 votes

Joined: May 2016
 
 
 
Sunday 11:44 AM
kept the three points.
Coeurdelion
5589 votes

Joined: October 2007
 
 
 
Sunday 3:02 PM
For a low-scoring hand such as this there are still several options - runs with 4-8-9-10 (A-2), A-8-9-10 (2-4), the pegger A-2-4-9 (8-10) and the defensive discard 2-4-8-9 (A-10):

4-8-9-10: 3pts - 5pts (Schell: 5.07) = -2pts

A-8-9-10: 3pts - 5¼pts (Schell: 5.33) = -2¼pts

A-2-4-9: 2pts - 5pts (Schell: 5.10) = -3pts

2-4-8-9: 2pts - 4½pts (Schell: 4.48) = -2½pts

Potential:

4-8-9-10: Improves with AAA, 222, 3333, 444, 5555, 6666, 7777, 888, 999, 101010, JJJJ = 38 cuts = 82.6% up to 6/8pts with 7777, 888, 999, 101010 = 13 cuts.

A-8-9-10: Improves with AAA, 444, 5555, 6666, 7777, 888, 999, 101010, JJJJ = 31 cuts = 31/46 = 67.4% up to 6/7/8pts with 5555, 6666, 7777, 888, 999, 101010 = 21 cuts.

A-2-4-9: Improves with AAA, 222, 3333, 444, 5555, 6666, 888, 999 + 15xXs = 42 cuts = 91.3% up to 6/8pts with AAA, 222, 3333, 444, 999 = 16 cuts.

2-4-8-9: Improves with AAA, 222, 3333, 444, 5555, 6666, 7777, 888, 999, 101010 = 34 cuts = 34/46 = 73.9% up to 5/6/7pts with 222, 3333, 444, 7777, 999, 101010 = 20 cuts.

Position:

As First Pone I'll play Offense and try to reach 18pts which is unlikely with this hand!

Pegging:

I think A-2-4-9 will peg best with three low cards and a high card while 2-4-8-9 should be next best.

Summary:

A-2-4-9 has the lowest starting value but has the most cuts for improvement and 16 cuts for 6/8pts. It also should peg best. So I'll throw the 8-10.
HalscribCLX
5312 votes

Joined: February 2008
 
 
 
Sunday 3:45 PM
At 0-0* playing an Offense strategy for the pegging the dynamic expected averages and Win/Loss %s are:

__________________Our
Offense__Hand_Pegs__Crib___Total___W9 %____W10 %
2-4-8-9___4.46+1.70+(-4.26)=1.90____18.7____19.2
A-2-4-9___4.70+1.93+(-4.75)=1.88____20.6____19.7
4-8-9-10__5.24+1.30+(-4.84)=1.70____19.7____19.4
A-8-9-10__5.28+1.35+(-5.05)=1.58____19.9____19.2

Offense_____L9 %____L10 %
2-4-8-9______36.9____53.7
A-2-4-9______38.5____54.1
4-8-9-10_____38.0____54.0
A-8-9-10_____38.4____54.3

2-4-8-9 is very slightly best for expected averages by 0.02pt over A-2-4-9. However A-2-4-9 is slightly better for Win %s but slightly worse for Loss %s. As we're playing Offense I'll select the more Offense oriented keep and select 8-10 to discard.

After the 4 cut I'll lead the 4 and play Offense:

Lead_____________Our Pegging Pts.
4______________________2.22
A______________________2.03
9______________________1.88
2______________________1.33
RubyTuesday
908 votes

Joined: January 2019
 
 
 
Sunday 3:50 PM
I wanted to keep the run, looked at the discard tables and threw A 2
Ras2829
5146 votes

Joined: November 2008
 
 
 
Sunday 5:55 PM
This hand was from a game with REX and have seen it a few times since in electronic and live play. The three consecutive small cards with high pegging value put holding the A-2-4-9 up just a hair. If you lack confidence in your pegging game, play the 2-4-8-9, which has the higher combined value. You can come real close to sorting out pegging value using this simple approach: A-5 have high pegging value, 6-7-8-9 have moderate pegging value, Jack has low pegging value, and 10,Q,K have none. Appreciate the positive comments relative to the quality of the puzzle. It's off., off., off., and lead the 4 keeping A-2 intact.
Ras2829 says: Of course if primarily concerned with potential hand value, holding 2-4-8-9 with the gapper at the lower end and the touching cards at the upper end is the way to go.