Stan’s Lists – Hand Group Analysis

[SS] Stan the Stat was always happy to talk numbers, and he had a table full of captive but eager listeners who were waiting for the tournament to begin. “Suppose the tightest player you’ve ever met opens for three times the big blind under the gun. You know he only does this with the very best hands, Sklansky’s Group One and Group Two only: pairs from Aces to Tens, suited Aces down to Jacks, suited King-Queen, and offsuit Ace-King. You’re the only caller from the big blind with a pair of Nines. You didn’t spike a set on the flop, but did get three undercards. How often are you behind? In other words, how often does the nit have a bigger pair in the hole?”

[LL] “Five different pairs, six ways each”, Leroy the Lion calculated out loud. “Four suited combinations, four ways each. And one offsuit combination, twelve ways. So 30 over 30 plus 16 plus 12 equals 30/58, just over half the time”.

[SS] “Perfect. 51.7% of the time. The looser the player is the lower that percentage will be.”

Percentage of Pairs in Hand Groups

Hand Groups Pairs
Group 1 85.7%
Group 2+ 51.7%
Group 3+ 39.1%
Group 4+ 28.0%
Group 5+ 18.8%
Group 6+ 19.6%
Group 7+ 14.6%
Group 8+ 9.4%
Group 9+ 6.9%
All 5.9%

[SS] “Even for Group 3, you don’t want to have to work the math out at the table, so this chart is worth memorizing, at least approximately. You’ll need to adjust for how much the player likes to play pairs though.”

[RR] “Yeah, like the crazies who will call a 4-bet for a quarter of their stack with pocket twos to set-mine”, Roderick the Rock confirmed.

[SS] “Here are two related lists for situations like when an overpair flops against your pocket Queens:”

Percentage of Aces in Hand Groups

Group Aces
1 35.7%
2+ 51.7%
3+ 50.0%
4+ 52.0%
5+ 39.8%
6+ 25.6%
7+ 21.4%
8+ 23.9%
9+ 17.5%
All 14.9%

Percentage of Kings in Hand Groups

Group Kings
1 35.7%
2+ 44.8%
3+ 37.0%
4+ 30.7%
5+ 28.9%
6+ 25.6%
7+ 21.4%
8+ 18.1%
9+ 17.5%
All 14.9%

[SS] “Some players will find an excuse to play any Ace, so you’ll have to adjust for that. But otherwise, you don’t have to be that afraid you’re beat against a loose player.”

[LL] “And some players overvalue suited Kings, assuming that they’ll rarely get taken out by the nut flush.”

[RR] “Until the one time it costs them their stack.

[SS] “Lastly, here are all the denominations together, so you can figure out things like, ‘How likely is it that my opponent has a Eight for a straight draw on a Nine-Seven-Six flop?'”

Percentage of Each Denomination in Hand Groups

Group A K Q J 10 9 8 7 6 5 4 3 2
1 35.7% 35.7% 21.4% 21.4% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
2+ 51.7% 44.8% 24.1% 17.2% 10.3% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
3+ 50.0% 37.0% 32.6% 19.6% 15.2% 6.5% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
4+ 52.0% 30.7% 30.7% 22.7% 22.7% 6.7% 6.7% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
5+ 39.8% 28.9% 28.9% 28.9% 28.9% 10.2% 3.9% 3.9% 1.6% 1.6% 1.6% 1.6% 1.6%
6+ 25.6% 25.6% 19.6% 22.6% 22.6% 17.6% 12.6% 7.5% 6.5% 6.5% 5.5% 3.5% 3.5%
7+ 21.4% 21.4% 16.9% 19.9% 19.9% 20.6% 16.9% 11.6% 10.1% 8.6% 7.1% 4.1% 2.6%
8+ 23.9% 18.1% 15.2% 16.6% 16.6% 16.1% 16.6% 17.6% 14.7% 10.8% 10.4% 8.4% 6.5%
9+ 17.5% 17.5% 17.5% 14.3% 13.2% 14.3% 14.3% 15.3% 16.4% 16.4% 13.2% 12.2% 11.1%
All 14.9% 14.9% 14.9% 14.9% 14.9% 14.9% 14.9% 14.9% 14.9% 14.9% 14.9% 14.9% 14.9%
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Stan’s Lists – WSOP Main Event Winners


[RR] “So, how many lists do you have just for Texas Hold ‘Em?” Roderick the Rock wondered.

[SS] “Hmmm, I’ve never counted them”, Stan surprisingly claimed. “Dozens at least.”

[RR] “What are your favorites?”

[SS] “Oh, I don’t get too attached to any of them. My favorite is usually whatever I most recently created.1 I also prefer lists I’ve made over lists other people have made, but there are exceptions.”

[RR] “Such as?”

[SS] “Wikipedia’s list of World Series of Poker Main Event Champions is pretty cool.2 It’s even sortable, which is one of the reasons I used to maintain my own version of it (okay, I still do, but it isn’t any better than Wikipedia’s; just different).”

[RR] “Why don’t I believe that? There must be something extra in your list for you to bother.”

[SS] “Well, I added preflop hand groups (from another list, Dan Harrington’s heads up groups in 10% increments)3 because I suspected something unusual.”

[RR] “Well, I know that a pair of Aces has been the losing hand twice but never the winning hand. Hal Fowler cracked Bobby Hoff’s pocket rockets by hitting a gutshot with Seven-Six offsuit in 1979, and Carlos Mortensen hit an open ender with King-Queen suited to upend Dewey Tomko’s in 2001.”

[SS] “Which led me to think that the losing hands have been stronger preflop than the winning hands.”

[RR] “And?”

[SS] “Alas, it was true, but that changed in 2011 when Pius Heinz’s A♠K♣ (top 10%) held up against Martin Staszko’s T♣7♣ (top 50%). But the losing hands could easily leap back ahead again this year.”

[RR] “The fact that it’s even close says a lot and seems counterintuitively against the odds.”

[SS] “Not really. It’s easier to go broke with a good hand than a bad one, because you’re more likely to get all your chips in if you like your hand! On average, you’ll lose more with your bad hands, but most of that is from folding, which will never bust you.”

World Series of Poker Main Event Final Hands With Harrington’s Preflop Hand Groups

Year Winner Hand Tier Runner-Up Hand Tier
1970 Johnny Moss N/A4 N/A ? N/A N/A
1971 Johnny Moss ? ? Puggy Pearson ? ?
1972 Amarillo Slim Preston K♥J♦ 2 Puggy Pearson 66 1
1973 Puggy Pearson A♠7♠ 2 Johnny Moss K♥J♠ 2
1974 Johnny Moss 3♥3♠ 4 Crandell Addington A♣2♣ 3
1975 Sailor Roberts J♠J♥ 1 Bob Hooks J♣9♣ 3
1976 Doyle Brunson T♠2♠ 7 Jesse Alto A♠J♦ 1
1977 Doyle Brunson T♠2♥ 8 Gary Berland 8♥5♣ 8
1978 Bobby Baldwin Q♦Q♣ 1 Crandell Addington 9♦9♣ 1
1979 Hal Fowler 7♠6♦ 8 Bobby Hoff A♣A♥ 1
1980 Stu Ungar 5♠4♠ 8 Doyle Brunson A♥7♠ 2
1981 Stu Ungar A♥Q♥ 1 Perry Green T♣9♦ 5
1982 Jack Straus A♥T♠ 1 Dewey Tomko A♦4♦ 2
1983 Tom McEvoy Q♦Q♠ 1 Rod Peate K♦J♦ 1
1984 Jack Keller T♥T♠ 1 Byron Wolford 6♥4♥ 8
1985 Bill Smith 3♠3♥ 4 T. J. Cloutier A♦3♣ 3
1986 Berry Johnston A♠T♥ 1 Mike Harthcock A♦8♦ 1
1987 Johnny Chan A♠9♣ 2 Frank Henderson 4♦4♣ 3
1988 Johnny Chan J♣9♣ 3 Erik Seidel Q♣7♥ 5
1989 Phil Hellmuth, Jr. 9♠9♣ 1 Johnny Chan A♠7♠ 2
1990 Mansour Matloubi 6♥6♠ 1 Hans Lund 4♦4♣ 3
1991 Brad Daugherty K♠J♠ 1 Don Holt 7♥3♥ 10
1992 Hamid Dastmalchi 8♥4♣ 9 Tom Jacobs J♦7♠ 6
1993 Jim Bechtel J♣6♥ 6 Glenn Cozen 7♠4♦ 9
1994 Russ Hamilton K♠8♥ 3 Hugh Vincent 8♣5♥ 8
1995 Dan Harrington 9♦8♦ 5 Howard Goldfarb A♥7♣ 2
1996 Huck Seed 9♦8♦ 5 Bruce Van Horn K♣8♣ 2
1997 Stu Ungar A♥4♣ 3 John Strzemp A♠8♣ 2
1998 Scotty Nguyen J♦9♣ 4 Kevin McBride Q♥T♥ 3
1999 Noel Furlong 5♣5♦ 2 Alan Goehring 6♥6♣ 1
2000 Chris Ferguson A♠9♣ 2 T. J. Cloutier A♦Q♣ 1
2001 Juan Carlos Mortensen K♣Q♣ 1 Dewey Tomko A♠A♥ 1
2002 Robert Varkonyi Q♦T♠ 3 Julian Gardner J♣8♣ 4
2003 Chris Moneymaker 5♦4♠ 9 Sam Farha J♥T♦ 4
2004 Greg Raymer 8♠8♦ 1 David Williams A♥4♠ 3
2005 Joe Hachem 7♣3♠ 10 Steve Dannenmann A♦3♣ 3
2006 Jamie Gold Q♠9♣ 4 Paul Wasicka T♥T♠ 1
2007 Jerry Yang 8♦8♣ 1 Tuan Lam A♦Q♦ 1
2008 Peter Eastgate A♦5♠ 3 Ivan Demidov 4♥2♥ 10
2009 Joe Cada 9♦9♣ 1 Darvin Moon Q♦J♦ 2
2010 Jonathan Duhamel A♠J♥ 1 John Racener K♦8♦ 2
2011 Pius Heinz A♠K♣ 1 Martin Staszko T♣7♣ 5
2012 Greg Merson K♦5♦ 3 Jesse Sylvia Q♠J♠ 2
2013 Ryan Riess A♥K♥ 1 Jay Farber Q♠5♠ 4
20145 Martin Jacobson T♥T♦ 1 Felix Stephensen A♥9♥ 1
20155 Joe McKeehen A♥T♦ 1 Josh Beckley 4♦4♣ 3
20165 Qui Nguyen K♣T♣ 2 Gordon Vayo J♠T♠ 3

Footnotes:

  1. Last week’s post covered Stan’s Texas Hold ‘Em Odds From 1 to 52.
  2. Oddly, this table is better than the one on the dedicated page for Main Event champions, which doesn’t include the losing hole cards.
  3. Harrington on Hold ’em, Volume II: The Endgame, pages 365-366. Surprisingly, the best web link I could find was this rec.gambling.poker newsgroup message with Harrington’s Heads-Up hand rankings.
  4. Johnny Moss was named the winner after the players voted. The story goes that the first vote resulted in a seven-way tie, as each player voted for himself. They were then asked to vote for the second best player, and Moss won.
  5. Table updated with 2016 results on November 2, 2016, 2015 results on November 11, 2015, 2014 results on November 12, 2014, and 2013 results on November 6, 2013.

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Texas Hold ‘Em Odds from 1 to 52

[SS] Stan the Stat loved lists almost as much as he loved numbers. His favorite Go-Go’s song? “Girl of 100 Lists”.1 Slacker. Stan had created that many by the time he was seven years old. So it was no surprise when he proudly unveiled his latest list of numbers, Texas Hold ‘Em Odds from 1 to 52. “One for each card in the deck”, Stan boasted. “Of course, I had several choices for many of the odds, so I tried for variety. By coincidence, the last one stumped me the longest time.”

A♠ 1 to 1 Odds of finishing with a pair on the river with unpaired hole cards 1.08 to 1 48.15%
A♥ 2 to 1 Odds of improving from 3-of-a-kind to a full house or quads on the turn or river 1.99 to 1 33.40%
A♦ 3 to 1 Odds of being dealt suited cards 3.25 to 1 23.53%
A♣ 4 to 1 Odds of hitting a flush draw on the river 4.11 to 1 19.57%
K♠ 5 to 1 Odds of being dealt connectors 5.38 to 1 15.69%
K♥ 6 to 1 Odds of being dealt at least one Ace 5.70 to 1 14.93%
K♦ 7 to 1 Odds of hitting a 3-outer on the turn or river 7.01 to 1 12.49%
K♣ 8 to 1 Odds of flopping a flush draw with suited cards 8.14 to 1 10.94%
Q♠ 9 to 1 Odds of flopping an 8-out straight draw from max connectors (JT-54) 8.57 to 1 10.45%
Q♥ 10 to 1 Odds of being dealt two cards that are Jacks or higher 10.05 to 1 9.05%
Q♦ 11 to 1 Odds of filling an inside straight draw on the turn 10.75 to 1 8.51%
Q♣ 12 to 1 Odds of not flopping an overcard with pocket Sevens 11.73 to 1 7.86%
J♠ 13 to 1 Odds of being dealt 2-gappers 12.81 to 1 7.24%
J♥ 14 to 1 Odds of hitting a 3-outer on the river 14.33 to 1 6.52%
J♦ 15 to 1 Odds of completing a flush by the river with suited cards 14.63 to 1 6.40%
J♣ 16 to 1 Odds of being dealt a pocket pair 16.00 to 1 5.88%
10♠ 17 to 1 Odds of being dealt unsuited 2-gappers (e.g., 85o) 17.42 to 1 5.43%
10♥ 18 to 1 Odds of a monochromatic flop 18.32 to 1 5.18%
10♦ 19 to 1 Odds of beating KK with K2 offsuit (suit dominated, the worst all-in preflop matchup) 18.69 to 1 5.08%
10♣ 20 to 1 Odds of being dealt connected cards, 10 or higher 19.72 to 1 4.83%
9♠ 21 to 1 Odds of being dealt a pair of Fives or better 21.10 to 1 4.52%
9♥ 22 to 1 Odds of hitting a backdoor straight (e.g., from 876) 21.52 to 1 4.44%
9♦ 23 to 1 Odds of hitting a backdoor flush 23.02 to 1 4.16%
9♣ 24 to 1 Odds of a single opponent with random hole cards having quads on a 3-of-a-kind flop 24.00 to 1 4.00%
8♠ 25 to 1 Odds of being dealt any suited connectors 24.50 to 1 3.92%
8♥ 26 to 1 Odds of making a straight or better on the turn with random hole cards 26.15 to 1 3.68%
8♦ 27 to 1 Odds of making 3-of-a-kind by the turn with random hole cards 26.81 to 1 3.60%
8♣ 28 to 1 Odds of a 3-card straight flop 27.78 to 1 3.48%
7♠ 29 to 1 Odds of being dealt suited 2-gappers 29.14 to 1 3.32%
7♥ 30 to 1 Odds of the board having no overcards by the turn with pocket Sevens 30.48 to 1 3.18%
7♦ 31 to 1 Odds of the board having no overcards by the river with pocket Eights 31.21 to 1 3.10%
7♣ 32 to 1 Odds of being dealt suited cards Tens or higher 32.15 to 1 3.02%
6♠ 33 to 1 Odds of hitting a backdoor half-inside straight (e.g., 976) 32.78 to 1 2.96%
6♥ 34 to 1 Odds of hitting a backdoor flush to chop the pot when your opponent flops the worst flush (e.g., holding 32s) 34.36 to 1 2.83%
6♦ 35 to 1 Odds of making a full house or better on the river with random hole cards 34.71 to 1 2.80%
6♣ 36 to 1 Odds of nobody holding an Ace, King, or Queen at a 6-handed table 35.94 to 1 2.71%
5♠ 37 to 1 Odds of flopping an 8-out straight draw from 3-gappers 37.28 to 1 2.61%
5♥ 38 to 1 Odds of making a full house on the river with random hole cards 37.52 to 1 2.60%
5♦ 39 to 1 Odds of improving a pair to a full house on the turn and river 39.04 to 1 2.50%
5♣ 40 to 1 Odds of being dealt a weak suited Ace (A9s-A2s) 40.44 to 1 2.41%
4♠ 41 to 1 Odds of hitting a 1-outer on the river when three players are all-in (e.g., QQ vs. KK vs. AA on AKQ2 board) 41.00 to 1 2.38%
4♥ 42 to 1 Odds of making exactly Jack high on the turn with random hole cards 42.28 to 1 2.31%
4♦ 43 to 1 Odds of being dealt a pair of Tens or better 43.20 to 1 2.26%
4♣ 44 to 1 Odds of flopping a four flush holding unsuited cards 43.55 to 1 2.24%
3♠ 45 to 1 Odds of hitting an inside straight flush draw on the river 45.00 to 1 2.17%
3♥ 46 to 1 Odds of being dealt max stretch suited connectors (JT-54) 46.36 to 1 2.11%
3♦ 47 to 1 Odds of hitting a runner-runner 1-gap straight flush or a full house/quads missing three board outs (e.g., 8d8h vs. Ad5d + Jd9d2d [Jh, 9h, 2h mucked]) 46.83 to 1 2.09%
3♣ 48 to 1 Odds of flopping two pairs using both unpaired hole cards 48.49 to 1 2.02%
2♠ 49 to 1 Odds of at least one player holding 4-of-a-kind or better if 10 players make it to the river 49.21 to 1 1.99%
2♥ 50 to 1 Odds of an opponent holding a pair of Aces when you have an Ace at a 9-handed table 50.04 to 1 1.96%
2♦ 51 to 1 Odds of making a flush or better by the turn with random hole cards 51.43 to 1 1.91%
2♣ 52 to 1 Odds of hitting a runner-runner full house or quads missing one hole out (e.g., 88 vs. A7s vs. + QT2s [8 mucked])2 51.56 to 1 1.90%

[SS] “Plenty of Google hits for ’52 to 1′ too.”

[RR] “But they all really meant ‘1 in 52′, or ’51 to 1’?” Roderick the Rock surmised.

[SS] “Exactly right. I almost gave up and changed the list to go from ‘1 in 1’ to ‘1 in 52’, but I hated having the pointless ‘1 in 1’ (‘Odds of there being an error in this list’?). I ended up calculating dozens of runner-runner outs until I found one that worked.”

Footnotes:

  1. Track three on the Go-Go’s 1982 album Vacation was somehow never released as a single ;-). Jane Wiedlen’s lists included: “things I love”, “what shall I wear”, “who have I kissed”, and “things I must get done today”.
  2. Added missing 52 to 1 odds on July 7, 2014.
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Umberto in Omaholdem Land

The side game after the monthly tournament was usually Omaha High, sometimes a mix of Omaha and Hold ‘Em like tonight, and on rare occasions, Dealer’s Choice. This variety led to more than infrequent misdeals, but most were easily rectifiable problems such as a dealer stopping after two hole cards during an Omaha hand.

Vincent the Veteran had just lost his stack and was collecting his belongings before leaving the table as the next hand began. Figaro the Fish thought Vince was buying back in, so he dealt him in for the current hand of Hold ‘Em. Umberto the Unlucky, distracted saying goodnight to Vince, looked down to find four cards on the table and assumed the game was Omaha.

Because the table was short and Umberto was in the cutoff, he called Elias the Eagle’s small raise despite a mediocre hand: A♠10♣10♦10♥. Leroy tagged along from the big blind, and the three of them saw a flop of K♠9♠7♥. When both opponents checked, Umberto tried to steal with a three-quarter pot bet, but neither player was convinced.

The dangerous 8♠ hit the turn, and again Umberto got two checks. Since he now expected that at least one of his opponents wanted him to bet, he happily checked, intending to bluff a quiet river with his nearly useless hand.

The 6♠ river filled a ton of draws but still elicited another pair of checks. The Unlucky one had nothing but a pair of Tens,1 so the only way to win the pot was to bluff. With his A♠ blocking the nut flush, he ventured a one-quarter pot bet as a value bluff. The tactic failed as both opponents called, and all Umberto could do was sigh and reveal his horrible hand.

[FF] Figaro was paying enough attention to scream, “Why do you have four cards?”

[UU] “I thought we were playing Omaha!?”

[LL] “No matter, I have him beat”, Leroy the Lion interrupted, revealing the 5♥4♥ for a Nine-high straight.

[EE] “And I have you beat”, Elias the Eagle added, showing the A♣2♠ for the worst possible but nonetheless winning flush.

[UU] “Just my luck. With any one of my four hole cards, I could win this hand, yet having twice as many hole cards and six times as many two-card combinations as you, I can’t beat either of your Hold ‘Em hands.”

[LL] “That’s justice for playing such a horrible starting hand. I suppose you’d expect to do even better if you were dealt four-of-a-kind.”

[EE] “Seneca the Younger2 claimed, ‘Luck is what happens when preparation meets opportunity.’ But for Umberto, ‘Bad luck is what happens when aberration meets him with impunity.'”

Footnotes:

  1. In Omaha, you must use exactly two cards in your hand and three on the board.
  2. Roman philosopher, born as Lucius Annaeus Seneca around 4 B.C.
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An Unlucky Visit to the Pub

Umberto the Unlucky didn’t usually play in the pub game since it was too far from home, but because he was in the city on business that day, he joined Leroy the Lion and Carlos the Crazy for dinner before they hit the freeroll tournament. Not surprisingly, Umberto was the first player out of the tournament, having lost with a set of Jacks when his opponent filled an inside straight on the river.

His unluck continued in the side game. Leroy and Carlos had told him how loose and crazy the action was, so he resolved to play tighter than usual and see flops with the best hand most of the time. Two buy-ins later, his friends finally joined the table and asked how he was doing.

[UU] “What can you do when your under the gun eight big-blind preflop raise gets called by everyone at the table? I folded a pair of Queens when the small blind bet and the big blind raised the Seven-Six-Four flop”, Umberto whined by way of an answer. “What can you do when your opponents keep turning Eight-Five offsuit into nut straights, Queen-Two suited into flushes, and King-Three offsuit into two pairs over your Ace-King?”

[LL] “It’ll all work out in the long run”, Leroy noted sympathetically.

[CC] “Although the only way you’ll get ‘The Long Run’ tonight is by downloading the Eagles album to your iPhone”, Carlos quipped.

[UU] “I’d be happy to get just one buy-in back, but having you guys join us didn’t help my chances.”

[LL] “I’m on your left, and Carlos is on your right. Isn’t that exactly how you’d want it?”

[UU] “Well, at least now I see where Carlos honed his style the last few years.”

[LL] “Yeah, that’s why I try not to come here too often; I’ll need to unlearn everything I learn here. I’m Linus in a roomful of Lucys”

After a few relatively uneventful hands, Carlos lost his stack when he flopped set-under-set and didn’t hit his quads and left the table to get more chips. Umberto looked down in the big blind to find a lovely pair of red Aces. Almost every hand had been raised preflop, usually with at least half the table seeing the flop, but the Unlucky started to get antsy when Leroy started a chain of folds that circled the table rapidly. And sure enough, for the first time since he had joined the table, everyone folded to the big blind. With a dead small blind on the hand, Carlos failed to win even a single chip with his pocket rockets.

[LL] “Congrats. A short walk to begin your long run.”

[UU] “At least it was better than getting them cracked.”

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