Category Archives: Preflop

Reading Hands, Preflop: Part Two

[FF] “I think I understand the idea behind hand ranges now,” Figaro the Fish began, “but I’m pretty clueless as to what ranges to put people on. It just seems so complicated!”

[NN] “It is complicated, so don’t feel bad”, Nate the Natural consoled.

[DD] “Maybe you could give us some more common pre-flop examples?” Deb the Duchess inquired.

[NN] “Sure. Especially among casual players like we have here, you often get an early limper that encourages the rest of the table to limp in because we love to see flops.”

[NN] “The early position limper has the tightest range of the limp chain, anywhere from 20% of hands to 40% of hands, not including the biggest hands like Aces or Kings which don’t want to be playing against a lot of opponents. This is what it might look like for 30%:

	QQ-22
	A2s+, K5s+, Q7s+, J8s+, 98s-54s
	A8o+, K9o+, Q9o+, J9o+, T9o

Subsequent limpers might be playing 40% of hands:

	QQ-22
	A2s+, K2s+, Q7s+, J8s+, 98s-54s
	A2o+, K9o+, Q9o+, J9o+, T9o-65o

But there’s always a guy who loves suited cards and limps along with almost 55% of hands:

	QQ-22
	XXs
	A2o+, K9o+, Q9o+, J9o+, T9o-54o

And don’t forget that the big blind gets to play for free, so he could have any two cards with up to about two limpers in front, and all but Aces or Kings with more (that’s 99%).”

[NN] “If a middle position player is known to raise a loose 30%, say,

	AA-22
	A2s+, K9s+, Q8s+, J8s+, T8s+, 98s-54s
	A8o+, K9o+, Q9o+, J9o+, T9o, 98o

then a player in late position could call with the same 30% with the positional advantage.”

[NN] “Another common occurrence is when a short-stacked player moves all-in pre-flop. You can use a formula1 to estimate how weak the player’s range is, but most people here just use their gut instinct. Depending on the player’s patience, the shoving range tends to widen with each hand that gets folded. In a rebuy tournament, the range is significantly wider early and tightens up tremendously after the rebuy period ends.”

[LL] “My range widens considerably once the side game has started”, Leroy the Lion admitted. “I don’t want to bust out and have to wait around doing nothing.”

[NN] “The button vs. small blind vs. big blind (BSB) battle is a special scenario that happens more with better players that it does here, but it’s still important. Some players will raise 100% of the time if folded to on the button. Other players, especially weaker ones, don’t value position that highly and are likely to play the same cards from the button as they will from the cutoff or hijack.”

[NN] “A limp from the button here is interesting, since it tends to deny a stronger hand. One player might raise 40% of hands,

	AA-22
	A2s+, K9s+, Q9s+, J9s+, T9s-54s
	A2o+, K2o+, Q9o+, J9o+, T9o

limp 40%,

	K8s-K2s, Q8s-Q2s, J8s-J2s, T8s-T2s, 97s-92s, 86s-82s, 75s-73s, 64s-62s, 53s-52s, 43s
	Q8o-Q2o, J8o-J3o, T8o-T5o, 95o+, 85o+, 75o+, 65o, 54o

and fold the remaining 20%:

	72s, 42s-32s
	J2o, T4o-T2o, 94o-92o, 84o-82o, 74o-72o, 64o-62o, 53o-52o, 43o-42o, 32o

while a raise-or-fold player could raise 50%:

	AA-22
	A2s+, K2s+, Q2s+, J6s+, T6s+, 96s+, 86s+, 76s, 65s
	A2o+, K5o+, Q7o+, J7o+, T7o+, 98o

and fold the rest.”

[NN] “If the blinds are known to be tight, the stealing range from the button could be 70% or more.

	AA-22
	A2s+, K2s+, Q2s+, J2s+, T2s+, 93s+, 84s+, 74s, 63s, 53s+, 43s
	A2o+, K2o+, Q3o+, J5o+, T6o+, 96o+, 86o+, 76o"

[NN] “The big blind might then try to resteal with just the top 20% of hands:

	66+
	A4s+, K8s+, Q9s+, J9s+, T9s
	A9o+, KTo+, QTo+, JTo"

[LL] “A wider range would probably be better.”

[DD] “We’re only talking about what people do, not what they should do.”

[NN] “One last example… stealing from the small blind in a blind vs. blind battle is tough because the player is out of position. A player might raise 30% of the time (like the loose middle position raise) against a tight big blind:

	AA-22
	A2s+, K9s+, Q8s+, J8s+, T8s+, 98s-54s
	A8o+, K9o+, Q9o+, J9o+, T9o, 98o"

Footnotes:

  1. For example, the SAGE (Sit And Go Endgame) formula can be used to determine whether to move all in or fold.
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Reading Hands, Preflop: Part One

[FF] “I’m really trying, but I just can’t figure out what cards people are holding. There’s just too much to pay attention to, and I seem to notice all the wrong things. I have a general idea that Roderick the Rock plays much tighter than Carlos the Crazy, but the only thing I really noticed was that the pizza stain on Rod’s sweatshirt was exactly the same shape and size as the mole on Carlos’s neck.”

[NN] “But a different color, I hope”, Nate the Natural suggested.

[LL] “Paying attention has never been your strong point”, Leroy the Lion suggested. “I know you pay up promptly when you lose bets. You pay compliments to pay your respect when you pay off after someone value bets you out of your chips. You even pay at the pump, but we don’t expect you to pay attention. At least not to the right things.”

[NN] “Pay Leroy no mind, Figaro. You know that if you pay heed to our advice it will pay dividends, especially if you pay your dues and work on your game.”

[DD] “At least you’re looking at your opponents”, Deb the Duchess commented. “I think all you need is a system — some straightforward step-by-step recipe you can follow.”

[FF] “That would be great; after twelve years, I can make mac and cheese now without even looking at the side of the box.”

[DD] “Maybe Nate or Leroy can explain help you here.”

[NN] “Sure. Let’s start before the flop. Where’s the button?”

[FF] “Leroy is twirling it between his fingers. The tournament hasn’t started yet.”

[NN] “No, I mean, you always have to know where the button is. The dealer usually has it here, but not when Elias the Eagle or someone else is permanent dealer, like in a casino. Hence the plastic button that says ‘DEALER’ on it.”

[NN] “Assuming you’re already familiar with how each player plays, before each hand you want to make sure you know:1

  1. The location of the button, so you can know what position each player is in.
  2. The number of players at the table. Expect tighter play with more and looser player with fewer.
  3. The size of the blinds and antes relative to each player’s stack (or M) and the average stack. Rough estimates will do.
  4. In tournaments, when and how much the next blind increase is. Is the rebuy period ending then? How far away is the bubble?
  5. The size of each player’s chip stack relative to each other, especially the smallest stacks who may move all-in preflop or soon thereafter.
  6. Any specific player traits that are relevant to the current situation. E.g., the cutoff likes to steal the blinds or the button doesn’t loosen his range much despite his position.
  7. Other random factors… Is someone on tilt because of a bad beat? Did someone just leave or join the table (and what effect will that have on table dynamics)? In our particular case, did a side game just start up so the short stack might suddenly loosen up his requirements for shoving? Did the sporting event on TV just end so some people will now be focusing better?

With each bet, call, or raise, take into account:2

  1. The position of the player: earlier implies a stronger range, while later means weaker (possibly as weak as any two cards on the button).
  2. The tightness of the player: tighter means stronger; looser means weaker.
  3. The aggressiveness of the player: passive means stronger; aggressive means weaker.
  4. The size of the bet relative to the pot: larger usually means stronger; smaller means weaker.
  5. The size of the bet relative to the stack size: larger usually means stronger; smaller means weaker. An all-in is usually weaker (but beware players who may shove strong because they hope you think that).

With their first action in a hand, place each player on an initial hand range. Looser players will have wider hand ranges, while tighter players will have narrower ones. Adjust for how much each player likes being suited, connected, and paired. Keep in mind stack sizes, as speculative hands need more chips behind to be playable.

If the betting loops around preflop (and on subsequent streets), narrow down each player’s range.

For example, a tight early position raise at a full table might represent the top 10% of hands: 77+, A9s+, KTs+, QTs+, AJo+, KQo,3

while a loose open raise from the hijack might be 50% of all hands: 22+, A2s+, K2s+, Q2s+, J5s+, T6s+, 96s+, 86s+, 76s, 65s, A2o+, K5o+, Q7o+, J7o+, T7o+, 98o.

Some players will only three-bet with Aces or Kings, while others will do so with a pair or any two big cards in position. That reraise will fold out the weaker part of the loose raiser’s range, so a call may be a top 20% hand (66+, A2s+, K8s+, Q9s+, J9s+, A9o+, KJo+, QTo+, JTo), while a rereraise represents the goods (QQ+, AKs, and maybe AKo).”

{From across the room…}

[RR] “Shuffle up and deal!”

[NN] “Sorry, Fig, looks like we’ll have to continue this some other time… Are you following so far?”

[FF] “I’m picturing hand ranges as arrows pointing up and to the left. Sometimes they’re short and sometimes they’re big, and they shrink with each extra bet.”

[NN] “They’re also slightly lopsided, but it sounds like you get the point.”

Footnotes:

  1. Dan Harrington’s Harrington on Hold ’em Volume I: Strategic Play (page 18) lists 11 Elements of a Hand, the first six of which are:
    1. What’s the status of the tournament?
    2. How many players are at your table?
    3. Who are the players at your table?
    4. How does your stack compare to the blinds and antes?
    5. How big are the other stacks at your table?
    6. Where do you sit in relation to the aggressive and passive players?
  2. The last five of Harrington’s Elements of a Hand are:
    1. What bets have been made in front of you?
    2. How many active players are left after you act?
    3. What are the pot odds?
    4. What is your position at the table after the flop?
    5. What are your cards?
  3. Hand ranges are from Equilab with minor adjustments (e.g., most players treat a pair of Threes and a pair of Twos identically preflop).
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SAGE – Sit And Go Endgame

[SS] “Iggy, for heads-up, short-stack, push or fold decisions, I use SAGE”, Stan the Stat declared.

[II] “What’s that?” Iggy the Improver inquired.

[SS] “Sit And Go Endgame, Copyright ©2005 by James Kittock and Lee Jones. They produced a very simple formula based on your hole cards and your stack size, which I’ve adapted to use M1 instead of big blinds. It only applies when the lower of your two Ms (after deducting the blinds and antes) is 5 or less.”2

[II] “That’s pretty much Dan Harrington’s Red Zone.”

[SS] “Right. And his Dead Zone when you’re shoving any two cards.”

[SS] “To use SAGE, all you need to do is a simple calculation and a table lookup.

  • Step One: Calculate the Power Index of your hole cards:
    • Deuces through Kings are worth 2 through 13 respectively, and Aces are worth 15; your higher card counts double
    • Pairs are worth 22 points
    • Being suited is worth 2 points

    Power Index = HighCard*2 + LowCard + Pair*22 + Suited*2

  • Step Two: In the following modified table, look up the lower M and your situation (in the small blind first to act or in the big blind after the button has shoved). If your Power Index is at or above the value listed, move all-in, otherwise fold:
    M Jam (SB) Call (BB)
    <1 any any
    1 19 any
    2 22 24
    3 24 27
    4 25 29
    5 26 31

SAGE doesn’t specify what you should do in the big blind if the button limps, but you can start with the Jam column’s values and adjust up or down according to what you think the limp represents. The same adjustment applies to calling.”

[SS] “Some examples:

  • In the small blind with a lower M of 1, holding 5♦4♣: 5*2 + 4 + 0 + 0 = 14. Table value is 19, so fold.
  • In the big blind with a lower M of 2, holding Q♣J♣ after a shove: 12*2 + 11 + 0 + 2 = 37. Table value is 24, so call the all-in.
  • In the big blind with a lower M of 4, holding K♣2♥ after a shove: 13*2 + 2 + 0 + 0 = 28. Table value is 29, so fold (borderline case).
  • In the small blind with a lower M of 5, holding 4♠4♣: 4*2 + 4 + 22 + 0 = 34. Table value is 26, so move all-in.
  • In the big blind with a lower M of 5, holding A♠3♦ after a shove: 15*2 + 3 + 0 + 0 = 33. Table value is 31, so call the all-in.
  • In the big blind with a lower M of 7, holding A♥K♥ after a shove: not a Jam or Fold situation because M is too high. Pretty easy call here though.”

[SS] “Here’s a version of the chart with hand ranges and percentages, but it’s really only so you can get a better feel for SAGE:”

Adjusted SAGE Jam/Call as Ranges

M Jam (SB) Call (BB)
<1 any
100.0%
any
100.0%
1 AA-22
Axs-8xs,76s-73s,65s
Axo-9xo,87o-83o,76o-75o
84.3%
any
100.0%
2 AA-22
Axs-9xs,87s-84s,76s
Axo-Txo,98o-94o,87o-86o
76.2%
AA-22
Axs-Txs,98s-94s,87s-86s
Axo-Jxo,T9o-T4o,98o-96o
69.2%
3 AA-22
Axs-Txs,98s-94s,87s-86s
Axo-Jxo,T9o-T4o,98o-96o
69.2%
AA-22
Axs-Qxs,JTs-J3s,T9s-T5s,98s-97s
Axo-Kxo,QJo-Q3o,JTo-J5o,T9o-T7o
57.5%
4 AA-22
Axs-Jxs,T9s-T3s,98s-95s,87s
Axo-Qxo,JTo-J3o,T9o-T5o,98o-97o
65.6%
AA-33
Axs-Kxs,QJs-Q3s,JTs-J5s,T9s-T7s
Axo,KQo-K3o,QJo-Q5o,JTo-J7o,T9o
48.6%
5 AA-22
Axs-Jxs,T9s-T4s,98s-96s
Axo-Qxo,JTo-J4o,T9o-T6o,98o
62.0%
AA-33
Axs,KQs-K3s,QJs-Q5s,JTs-J7s
Axo,KQo-K5o,QJo-Q7o,JTo-J9o
40.1%

[II] “Oh, that helps a lot. I can now see that you don’t even need the Ace or pair part of the formula, since you’re always shoving with those.”

[SS] “There are two borderline exceptions with a pair of Twos, but yes, close enough. That simplifies the Power Index calculation to your low card plus double your high card plus two if suited.”

Footnotes:

  1. See M and Q if you aren’t familiar with M.
  2. Kittock and Jones use multiples of the big blind, but that unnecessarily ignores antes.

Related Links:

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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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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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Preflop Odds Heads Up 4

[SS] “Okay, one last preflop matchup trivia question”, Stan the Stat promised. “You have almost no chance on this one, so I’ll give you the answer after you each take just one guess… What nontrivial preflop all-in matchup is the closest to a coin flip?”

[FF] “I always thought a pair of Queens against Ace-King was close”, volunteered Figaro the Fish.

[HH] “I’d go with something near what you mentioned earlier”, Harriet the Hazy suggested. “Like Jack-Ten offsuit against a pair of Twos.”

[RR] “Or the Queen-Jack suited against Twos”, Roderick the Rock recommended.

[LL] “Maybe something lower like Eight-Seven suited against the same”, Leroy the Lion offered.

[EE] “Yeah, it’s probably something like that”, Elias the Eagle amended. “I’ll guess the same but against Threes.”

[TT] “Since I get to guess for free / How ’bout the suited Four-Three / Against the ultimate poo / The offsuit Seven and Two?” Tyrone the Telephone attempted.

[SS] “All excellent guesses… well, maybe not Figaro’s, which is 54% suited. Tyrone was very close to the sixth best matchup: 62o vs. 54o where the Two matches a suit (the high card or kicker value of the Six almost exactly balances the straight potential of the 54). Fifth is 73s vs. 22 in different suits, and fourth is 97o vs. 22 with four suits. Third is what you all danced around, QJo vs. a pair of double suit-dominated Threes. Roderick just missed that one. Second is T9o vs. 55 with one matching suit. All of these are 50.01% for the better hand, as is number one, which wins by a smidge: ATs vs. 33 with one matching suit, which is so close to a coin flip that if you played it out 7,075 times, you’d only expect to win one more time than your opponent (50.0071%)! In all the cases with pairs, the pair wins with more sets and boats but loses to more straights and, perhaps surprisingly, pairs (and even two pairs sometimes because of the dreaded three-pair hands).”

[RR] “Very cool, Stan”, Roderick the Rock acknowledged. “But now that you’ve given us all of these mostly non-nutritious snacks, what about the meat and potatoes of all-in heads-up matchups?”

[SS] “I was getting to that”, Stan the Stat claimed. “When I was first learning how to play Hold ‘Em, I set out to memorize all of the common odds. I thought preflop all-in percentages would be useful, but there were just way too many to remember1. Fortunately, grouping the matchups into just eight general categories with their approximate odds is quite sufficient.”

[SS] “If neither hand is paired, there are three groups of matchups:”

Opposing Hand Equity Example2
Unpaired Dominated 70%3 KQo vs. K8o (75%)
QTo vs. JTo (73%)
Two Undercards
or Alternating Ranks4
65%5 AJo vs. 63o (65%)
QTo vs. J9o (64%)
Tweeners 57%6 A9o vs. QJo (56%)

[SS] “Otherwise with a pair, there are five matchup groups:”

Opposing Hand Equity Example
Dominated With Undercard 90% KK vs. KQo (91%)
Unpaired Undercards 85%7 QQ vs. 94o (87%)
Lower Pocket Pair 81% AA vs. KK (82%)8
One Overcard,
Possibly Dominated
70% QQ vs. K8o (72%)
KK vs. AKo (70%)
Two Overcards 55%* 44 vs. A7o (55%)
44 vs. QJo (51%)
44 vs. QJs (49%)9

[SS] “* Within a given category (when relevant), being suited is worth a few percent for the flushes (just being able to make a winning flush is worth half a percent), and being connected is worth a few percent for the straight possibilities. In cases where the flush or straight is one of the few ways to win, the difference for the weaker hand can be up to five percent (e.g., AA vs. AKs is 5% better than AA vs. AKo, and AA vs. T9o is 5% better than AA vs. T5o).”

[RR] “Why isn’t it always five percent?”

[SS] “All the hands where the weaker hand hits a straight or flush but would have had a winning pair, two pairs, or three of a kind anyway don’t increase the percentages. The straight or flush is superfluous in those cases. Similarly, suited connectors don’t get the full gain for both the straight and flush possibilities, more like just seven percent in the best cases.”

[TT] “If you’re all-in, while nothing’s been fated / Know the odds, lest your hopes get inflated / It’s better not to be dominated / Or for ‘Next Bust’ you’ll be nominated”, Tyrone concluded.

Footnotes:

  1. The total number of possibilities is 812,175 (52-choose-2 * 50-choose-2 / 2), but ignoring suits, there are only 14,196 (13^2-choose-2) to memorize.
  2. These charts are based on the tables in John Vorhaus’s Killer Poker by the Numbers, pages 268-275, but have been modified with help from Mathematrucker. All odds are approximate, within a couple of percent except as noted in some of the following footnotes.
  3. The full range is fairly wide, going from about 65% to 78%, with higher cards tending toward the upper part of the range.
  4. Having alternating ranks (e.g, QTo vs. J9o) makes surprisingly little difference (the weaker hand mostly needs to pair up in either case, and having two undercards can give more straight possibilities).
  5. The full range is fairly wide, going from about 58% to 71%.
  6. The full range goes from about 51% to 60%.
  7. The full range is fairly wide, going from 76.93% (KK vs. 54s) to 90.10% (AA vs. K2o with matching suits).
  8. The full range only goes from 79.75% (33 vs. 22 of different suits) to 82.69% (TT vs. 99 with matching suits). Vorhaus listed this as 80%, probably from rounding, but it’s definitely closer to 81%.
  9. The classic race, QQ vs. AKo (57%) or vs. AKs (54%), would also go here as the AK is effectively unconnected with only two Queens left in the deck to make a royal straight.

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Preflop Odds Heads Up 3

[SS] “Fig, I’ve got a bet for you”, Stan the Stat propositioned Figaro the Fish while retrieving a set of three dice from his left pocket.

[FF] “Okay…”, Figaro responded warily.

[SS] “I’ll bet you even-money on who can roll the higher number with one of these dice”, Stan offered. “And because I like you, we’ll roll as many times as you want, and you can pick whichever die you want each time.”

[FF] After examining the dice, the Fish decided, “Okay, I’ll take this one, the only one with nines on it.”

[SS] “Very well, I choose this one with nothing higher than a seven”, the Stat countered.

[LL] Leroy the Lion chuckled, “You’re not really going to con him are you?”

[SS] “No, just for fun; we’re all friends here.”

[LL] “Fig, it doesn’t matter which die you pick, these are nontransitive dice, where each die beats one of the other two and loses to the other one.”

[FF] “How is that possible?”

[SS] “It’s like Roshambo, where rock beats scissors beats paper beats rock. The six faces on each die add up to 30, but they’re set up so you win by a little and lose by a lot, allowing the second player to win five-ninths of the matchups. C > B > A > C:”

Side 1 Side 2 Side 3 Side 4 Side 5 Side 6
Die A 1 1 6 6 8 8
Die B 2 2 4 4 9 9
Die C 3 3 5 5 7 7

[SS] Reaching into his right pocket and pulling out some more dice, Stan continued, “Or consider this set of four, called Efron’s dice:”

Side 1 Side 2 Side 3 Side 4 Side 5 Side 6
Die D 0 0 4 4 4 4
Die E 1 1 1 5 5 5
Die F 2 2 2 2 6 6
Die G 3 3 3 3 3 3

[SS] “The edge is even bigger here as the dominating die wins two-thirds of the time. G > F > E > D > G.”

[FF] “Wow, that’s amazing. Can I borrow those for the next time I go to a bar?”

[LL] “Do you really like to get beat up?”

[SS] “What’s cool is that you can do the same thing with Texas Hold ‘Em hands, although your edge is much smaller.” Having rummaged through a deck of cards and pulled out six of them, Stan continued, “For example, consider these hands: A♣K♦, J♥10♥, and 2♠2♥. Which would you prefer, Figaro?”

[FF] “Definitely the Ace-King at a full table.”

[SS] “Nope, you’re heads up.”

[FF] “I’d still take it.”

[SS] “Well, then I’d select the Twos and have a 53.0% to 47.0% edge on you. Similarly, the Jack-Ten suited has 53.7% equity against the Twos, and the Ace-King offsuit does even better against the Jack-Ten at 58.8%.”

[LL] “The Ace-King is an obvious win over the Jack-Ten, while the other two matchups are classic races. Both connectors mainly need to pair up to beat the Twos, but the Jack-Ten makes enough straights and flushes to tip the scales, while the Ace-King doesn’t.”

[FF] “Pretty cool, indeed.”

Related Links:

  • While it’s not blindingly obvious how to use it, the table at the top right of the Mathematrucker Matchups page lets you see any family of heads-up matchups (by gap and suitedness). Just tap on the proper square in the grid. In the answer cells, a pink background means the hands on the left column are winning and a blue background means the hands on the top row are ahead! Tap on the percentage to display the winning hand type breakdown on the left.

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Preflop Odds Heads Up 2

[SS] “Again you’re all-in against a single opponent”, Stan the Stat posed. “But this time you can select both your own hole cards and your opponent’s. What should you pick?”

[RR] “I thought the best was something like Aces against Ace-Nine offsuit, with the Nine matching the suit of one of your Aces.”

[SS] “Close.”

[FF] “Ace-Eight?” Figaro the Fish ventured.

[SS] “Nope.”

[HH] “Ace-Seven?” Harriet the Hazy attempted.

[SS] “Nope.”

[LL] “Ace-Six?” Leroy the Lion guessed.

[SS] “And nope. Oddly, the Ace-Nine is the best of those because King-Queen-Jack-Ten is only a push, but the Ace-Six is next.”1

[EE] “So, it’s not a pair of Aces at all! Then it must be Kings vs. King-Two”, Elias the Eagle deduced.

[SS] “Indeed, with nearly 95% equity.2 With an Ace, you can’t let your opponent have anything under a Five because that would permit too many straights. With the King, no such problem. The King-Deuce does tie a few more hands, but it wins many fewer because the Six through Nine all take part in five kinds of straights while the Two only makes two. Queens vs. Queen-Two is the next best.”3

Footnotes:

  1. Ace-Eight makes 12 more winning full houses than Ace-Seven, which in turn makes 12 more than Ace-Six.
  2. 94.92% from 1,612,287 wins and 26,192 ties out of 1,712,304 possibilities.
  3. 94.66%. Then Kings vs. King-Three (94.48%), Jacks vs. Jack-Two (94.35%), Queens vs. Queen-Three (94.22%), and finally Aces vs. Ace-Nine (94.08%), barely edging Kings vs. King-Four (94.04%) and Kings vs. King-Two with four suits (94.03%).

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Preflop Odds Heads Up

[SS] “If your only opponent is all-in with a pair of Aces, what’s the best hand you can hold?” Stan the Stat opened.

[FF] “The other two Aces”, Figaro the Fish observed.

[SS] “Besides that, of course.”

[RR] “I thought it was Ten-Nine suited, matching neither of the Ace’s suits”, Roderick the Rock opined.

[LL] “No, if that’s good then Nine-Eight suited has to be better. With Ten-Nine, a King-high straight is no good, since the Aces win with a royal straight”, Leroy the Lion objected.

[SS] “Both incorrect, but you’re getting warmer.”

[HH] “Eight-Seven suited”, Harriet the Hazy opted, like the next contestant on The Price is Right.

[SS] “Nope.”

[FF] “Seven-Six”, Figaro the Fish output.

[SS] “Still wrong.”

[TT] “Now it’s a cinch to derive / That it must be the Six-Five”, Tyrone the Telephone offered.

[SS] “Indeed. The Six-Five has just over 23% equity.1 But why is it slightly better than the Seven-Six?”

[TT] “The Seven-Six gets more hits / But the Six-Five gets more splits.”

[SS] “Indeed it does. Which ones?”

[RR] “Except for my initial guess, all these suited connectors can make the same number of winning straights and flushes, so that leaves full houses.”

[LL] “No, the Seven-Six makes more winning straights. They both win if there’s a wheel straight on the board, but a 65432 board is a win for the Seven-Six and only a push for the Six-Five.”

[EE] “So the 6-high straight is some of the Six-Five’s extra pushes. The rest are on the other side: the Jack-high and Ten-high straights, which don’t happen as often for the Seven-Six.”2

[SS] “And that’s most of the difference, although there are a few full houses. 77666 down to 77222 vs. 55444 down to 55222 accounts for a few more wins for the Seven-Six and a few more ties for the Five-Four. Lastly, JT98x wins more often for the Seven-Six than 432Ax does for the Six-Five.”

[SS] “The Six-Five has enough extra straight pushes to overcome the extra wins that the Seven-Six gets.”

[TT] “When facing Aces, your prospects are wee / Even with the best hole cards you could see / Out of thirteen tries, you’ll only win three.”3

Footnotes:

  1. The Six-Five suited has 23.056% equity, 0.0235% more than the Seven-Six suited (23.0325%).
  2. Out of 1,712,304 possible boards, the Six-Five suited wins 55 fewer times but ties 916 more times.
  3. This is an excellent estimate as three-thirteenths is the repeated decimal 0.230769.

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Basic Player Reading – The Next Step

[FF] Deb the Duchess was being anti-social, tapping away at a Texas Hold ‘Em game1 on her iPhone when Figaro the Fish and Nate the Natural approached. “Haven’t you beaten that game yet?” Figaro interrupted.

[DD] “I’m on the second to last tournament, the National Championship on the top difficulty”, Deb responded. “It’s going to be a long time before computers can beat the pros, but they’re still challenging to me.”

[NN] “Don’t be fooled. Computers will be better than the pros at Hold ‘Em sooner than you think. If IBM can conquer Jeopardy! with Watson by building a large enough information database and crafting a smart enough language parser, other programmers can certainly keep improving their Hold ‘Em algorithms until they outclass even the top poker pros. Paraphrasing The Simpsons,2 Ken Jennings conceded on his Final Jeopardy answer screen, ‘I, for one, welcome our new computer overlords.'”

[DD] “How soon do you think that’s going to happen?”

[NN] “I think Black Friday set the timetable back a couple years, since the breakthroughs will probably come outside the U.S. now, but certainly within a decade.”

[NN] “Computers can do the math perfectly. They can calculate odds almost instantaneously with massive lookup tables and powerful processors. They can analyze your hand history to see how you play. All that’s really left are a few advancements in the algorithms that decide what to do with all this information. It essentially comes down to hand reading, and I don’t mean palmistry.”

[DD] “That’s what separates the pros from the amateurs.”

[NN] “The reason hand reading is so daunting to most humans is that there’s so much to keep track of on just a single hand, let alone over a session of hands with the same players. Computers don’t have the slightest problem with it. Which is why most good online players use HUDs (heads-up displays).”

[NN] “To put your opponent on a hand range even before the flop, you need to take into account his stack size; the blinds and antes; his position; the action ahead of him, including who bet what from where; the amount he called, bet, or raised; how he’s been playing recently; and his playing style, which alone can be broken down into dozens of smaller areas; and more.”

[FF] “But how am I supposed to keep track of all that?”

[NN] “To start as simply as possible, the most important element is a player’s style. In a home game, you’ll get to know the regulars quite well without even trying. You two certainly know how I play, and I know how you play. In an unfamiliar ring game or any larger tournament, you’ll have no history with your opponents, but every hand adds to your database of information about them.

  • How many hands do they play (i.e., are they loose or tight preflop)?
  • How often do they 3-bet? 4-bet? 5-bet? (i.e., how tight a range does each of those represent)?
  • Do they correctly value position (e.g., do they play many more hands in late position than early and can you discount their bets in position vs. out of position)?
  • Do they tend to call or raise (i.e., are they too passive or too aggressive)?
  • Do they bluff too often or too infrequently (i.e., can you discount the strength of their bets or should you take them as real)?
  • Do they call too much or can you bluff them out of pots (i.e., should you value bet them or steal from them)?
  • Can they make big folds? (i.e., should you try to make a big river bluff or all-in bluff)?
  • Do they like to chase draws? (i.e., should you charge them more for their draws and/or not try to bluff if you think they have a draw)?
  • Will they bet if they have just a draw (i.e., could your third pair be the best hand despite their bet)?
  • Do they tend to underbet the turn and river (i.e., will you get a good price to hunt for your draws)?
  • Do they overbet the flop when the board is scary (e.g., can you put them on a hand like top pair or an overpair)?
  • Do they adjust to their opponents (i.e., do you need to take into account what they think of your style)?
  • Is their style static or can it change (i.e., once you’ve figured out how they’re playing, can you count on that always)?”
  • Do they play differently when they’re shortstacked? (e.g., can you devalue a shove from them once they’re short enough on chips)?”
  • Do they like to steal the blinds from the button? The cutoff? The hijack? (e.g., can you devalue all of those raises)?
  • How often do they check-raise, if at all? (e.g., if they check to you, can you bluff knowing that you won’t get raised)?”

[FF] “Yikes!”

[NN] “If the event is a tournament, a few more questions are relevant:

  • Are they very afraid of busting out? And do they have any rebuys left if it’s a rebuy tournament?
  • Does their style change as the blinds go up?
  • Are they happy just to cash or are they trying to win it all?
  • Do they play more tightly near a bubble or will they attack if their stack is healthy?”

[FF] “My head is spinning.”

[NN] “I know it’s overwhelming at first, so start by just watching the player on your immediate left or right or the most active player at the table, and note only a few of those pieces of information. As you get comfortable, add more information and more players. You can do it. You just have to try.”

Footnotes:

  1. See “A New Game in Town – THETA Poker Pro”.
  2. You can watch the segment from the “Deep Space Homer” episode.
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