Category Archives: Turn

Reading Hands, Turn

{Continued from the Flop, Part Three}

[NN] “The turn is the street of hope”, Nate the Natural asserted. “You hope you’re ahead in the hand. If not, you hope you can bluff your opponent out. But if all else fails, you still have hope that you’ll be able to hit one of your outs.”

[FF] “I’m usually hoping I won’t mess up the hand…”, Figaro the Fish added, “if I haven’t already.”

[DD] “Roderick’s usually hoping his opponent isn’t about to suck out on him”, Deb the Duchess noted.

[NN] “Which is why if you’re ahead, you need to figure out which cards you’re worried about and charge accordingly. Suppose we’re in position as before, on a wet board of K♥Q♥T♣, when the 4♦ hits, and our opponent checks again.

After he check-called the flop, we put him on:

	AA, JJ
	ATs+, K8s+, QTs+, Ah9h-Ah2h+, Jh9h+, Th9h
	AKo, KJo+, QJo-JTo

So he has a lot of draws in his range where he currently has less than top pair:

	JJ (8 outs for straight and 2 outs for set)
	JhTh (17 outs for flush or straight)
	AhTh-Ah2h, Th9h (9 outs for flush)
	KJo, QJo-JTo (8 outs for straight)

That’s half of his hands. Except for the J♥T♥, you can give your opponent the wrong odds to call with a half-pot or larger bet. Assuming of course you don’t pay off on the river if a scare card hits.

If you have a King yourself, your opponent is even more likely to be on a draw, so a bet here is basically required.”

[FF] “What if my he check-raises me?”

[NN] “That’s very unlikely around here, but if it happens, just fold and silently congratulate your opponent on a nice play.”

[DD] “I’ll have to try that with my next drawing hand!”

[NN] “On the other hand, if you have a King on the dry board of K♥7♣2♠ and the 4♦ hits, you need to know how often your opponent would have called your flop continuation bet with a weaker King or an underpair. The looser you’re perceived and the tighter he plays, the more reason you have to check behind to avoid the check-raise or check-call he was planning.

As the saying goes, ‘Big hands want to play big pots…’, and you have just top pair here, so keep the pot small. Your opponent most likely has at most five outs,1 so the free card isn’t much of an issue.”

{To be continued…}

Footnotes:

  1. An underpair has five outs to make a set or two pairs, a weaker King has four kicker outs, and an Ace has three outs for an overpair.
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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.

{ The Hold ‘Em at Home blog is brought to you by THETA Poker Pro, the strongest, fastest, and most configurable Texas Hold ‘Em game for iPhone, iPad, iPod touch, and Apple TV. }

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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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Basic Player Reading – Levels of Thinking

[BB] “Like in chess, better poker players generally think deeper than weaker players”, Benny the Book compared. “It’s not a clean correlation because it doesn’t help to think incredibly deeply about the wrong things. Chess computers could think many more plies than humans for decades before they could beat the strongest grandmasters.”

[BB] “Poker is a funny game in this respect. You want to think deeper than your opponent, but don’t want to think too much deeper than your opponent. A seasoned pro and a complete neophyte might make the same exact bet for completely different reasons, so you need to determine how deeply your opponent thinks to know how to play against them. You want to be thinking exactly one level deeper than your opponent.”

[BB] “Beginners think at the first level: what cards do I have and how good is my hand? Some beginners are able to adjust for the texture of the board, but most won’t think to devalue their nut flush if the board is paired.”

[BB] “Intermediate players think at the second level: what does my opponent seem to have? But they only have a vague notion of your hand strength: strong, fair, or weak. Sometimes intermediate players prematurely and inaccurately try to place you a specific hand, but that usually doesn’t work out well for them.”

[BB] “Advanced players also think at the second level, but they start you on a fairly wide hand range and try to narrow it as the hand progresses. When they’ve deduced correctly, they can cause you major problems.”

[BB] “Expert players think at the third level: what does my opponent think I have? That’s as far as anyone goes in the games we’re playing in. Which means that if you can go to the fourth level (what does my opponent think that I think they have), you’ll dominate here.”

[BB] “Do you remember the battle of wits scene in The Princess Bride?”1

[JJ] “Of course!” Joey the Juvenile confirmed.

[BB] “Westley, then referred to anonymously as the man in black, shows Vizzini some poisonous iocane powder, turns his back for a moment, then says, ‘All right. Where is the poison? The battle of wits has begun. It ends when you decide, and we both drink and find out who is right… and who is dead.'”

[BB] “Vizzini tries to deduce which glass to pick, reasoning, ‘Now, a clever man would put the poison into his own goblet, because he would know that only a great fool would reach for what he was given. I’m not a great fool, so I can clearly not choose the wine in front of you. But you must have known I was not a great fool; you would have counted on it, so I can clearly not choose the wine in front of me.'”

[BB] “Vizzini is thinking at level three, considering what his opponent, the man in black would think he thinks.”

[BB] “Vizzini apparently solves the dilemma by switching the glasses while his enemy isn’t looking. The man in black doesn’t hesitate to drink from his own goblet, so it must have originally been the safe one. Because of the swap, Vizzini therefore has the drink that’s safe to imbibe, which he does.”

[JJ] “But Westley was thinking at level four!”

[BB] “Right. Although Vizzini did momentarily come close to the right answer, he spent most of his time considering the wrong data. Of course, a wise man wouldn’t ever consider drinking from a glass if there were a 50% chance that it would be fatal, not unlike a coin flip for your tournament life.”

[BB] “Wallace Shawn’s villain character flopped the nut flush draw and concluded that his Ace was good anyway. But he really had no outs as his opponent already had a full house. Both glasses were poisoned, and Cary Elwes’s hero character had spent years building up his immunity to iocane.”

[BB] “So make sure you use the appropriate level of thinking for your opponent. Against Figaro the Fish, there’s no need to go beyond level two. If you explain his actions with any deeper logic, you’ll end up making the wrong move. Against Vince the Veteran, level three is good. He’ll try to put you on a hand, so a little deception will go a long way. And against Deb the Duchess, level four would be right, although I still aspire to that.”

Footnotes:

  1. If you prefer, here is the transcript of the battle of wits.
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Semi-Bluffing

[BB] After a brief respite, Benny the Book has returned with another Hold ‘Em lesson for his son. “You, along with every other poker player, love to get away with a bluff. You win a pot that you shouldn’t have won. Maybe you flash your hole cards to show off what you just did (bad idea, by the way, unless you don’t plan to bluff again for a while).”

[BB] “But before you learn the art of the bluff, you should arm yourself with the safer flop and turn semi-bluff.”

[BB] “A seminar isn’t half a ‘nar’;1 a Seminole2 isn’t half a ‘Nole’; but a semi-bluff is exactly what it sounds like, half of a bluff. It’s partly a bluff because you don’t currently have the best hand but are hoping to win the pot by causing your opponents to fold. But it’s partly not a bluff because of your chance of improving to a better hand than your opponents.”

[BB] “The full range goes from total bluff (no pot equity) to mostly bluff to semi-bluff (up to 50% pot equity) to value bet (over 50% pot equity).”

[JJ] “What’s the distinction between a bluff and a semi-bluff?” Joey the Juvenile interrupted.

[BB] “The line between bluff and semi-bluff isn’t officially defined, but let’s say that a bet with anything less than 5 outs (e.g., an inside straight draw) is more bluff than semi-bluff. It’s not actually that important except to know that along the continuum you’re relying on fold equity more on the left and less on the right.”

[BB] “It’s also impossible to know where semi-bluffs end and value bets begin until you’ve seen all the hole cards. But none of us can read our opponents hands well enough that it matters whether you have 49% pot equity or 51%; as long as you know it’s closer to 50% than 20% or 80%, you’re doing great.”

[BB] “The less pot equity you have, the more you want to take down the pot immediately. The more pot equity you have, the more you don’t mind building the pot.”

[BB] “One issue that comes up with in-position semi-bluffs is when should you take the free card and when should you semi-bluff if nobody bets and you’re last to act?”

  • “The more outs you have, the more often you should check, since your pot equity is high but won’t be increased relatively by betting.”
  • “The more opponents you have, the more likely the flop was to hit your opponents’ hands, the more your opponents like to check-raise, and the more your opponents like to slowplay, the more you should check, since your fold equity is lower in each case.”
  • “The larger the pot is relative to your and your opponents’ stacks, the more you should check because you lower your implied odds by betting (i.e., there aren’t enough chips left to pay you off properly when you hit your draw). In the extreme case, never semi-bluff if you or one of your opponents is pot-committed.”
  • “Otherwise, semi-bluffing usually has a higher EV3 than checking.”

“When you are out of position, reduce the frequency of your semi-bluffs since your fold equity is lower (a player who has checked is more likely to fold than a player who hasn’t acted yet).”

“You can also semi-bluff raise in position and semi-bluff check-raise out of position, but these are both riskier plays with less fold equity.”

“Lastly, I’d add that a side benefit of semi-bluffs is that they help balance4 your flop and turn betting ranges. You’d be too easy to read if you always had a good hand when you bet. By adding semi-bluffs to your arsenal, your strong bets for value will get called more often as your opponents learn that you could also be semi-bluffing.”

“How big your semi-bluffs should be is too opponent- and hand-dependent to say with any certainty. Against unobservant opponents, make your semi-bluffs as small as possible while still inducing folds. Against better competition, the closer your semi-bluff sizing matches your value bet sizing, the harder you will be to read. After we talk about bluff sizing, which is simpler because of the lack of pot equity, we’ll return to the math of this subject.”

Footnotes:

  1. “Seminar”, “seminate”, and “seminal” all come from the root “semen-” (seed) rather than “semi-” (half).
  2. “Seminole” comes from the Creek Indian Simano, meaning “wild, untamed, runaway”.
  3. See last month’s article on “Expected Value”.
  4. Sorry, that’s a future article that I haven’t written yet.
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The Nuts

[BB] “A good exercise for beginners to develop card sense is to figure out what the nuts are when you see the flop, and see if they change on the turn and the river. For example, what’s the nuts on a K♦8♠5♣ flop?”, Benny the Book quizzed his son, Joey the Juvenile.

[JJ] “That’s a totally dry flop. No pair and no flush or straight possible, so the answer must be three of a kind”, Joey responded after a moment’s thought.

[BB] “Right, specifically three Kings. What if the 7♣ hits on the turn?”

[JJ] “Now a straight is possible.”

[BB] “Correct again. A 9-high straight. And let’s try two different river cards, first the 8♥.”

[JJ] “That paired the board, so a full house is possible. No, wait… four Eights.”

[BB] “And what if the river is the 6♣ instead?”

[JJ] “9-high Straight flush!”

[BB] “Okay, this is already too easy for you. In another minute, you’ll realize that it comes down to just five simple rules:

  1. If three cards to a straight flush are on the board (e.g., 8♦7♦4♦), then a straight flush is the nuts (the highest one if more than one are possible).
  2. Otherwise, if the board is paired or more, then a four of a kind is the nuts (the higher denomination if two pairs or a full house are on the board).
  3. Otherwise, if the board has three of one suit, then an Ace-high flush is the nuts (add the highest missing two cards of the flush suit to the board).
  4. Otherwise, if three cards to a straight are on the board, then a straight is the nuts (the highest one if more than one are possible).
  5. Otherwise, three of a kind is the nuts (three of the highest card on the board).”1

[BB] “The two cards you hold can prevent certain nut possibilities; for example, you could hold a card that blocks the straight flush, or you could hold a card that blocks quads.”

[BB] “One oddity is that if you have a full house, it’s never the nuts if you have a pocket pair. It’s the nuts only if the board’s highest denomination is paired, and you have one of those and one of the highest other denomination on the board and no straight flush is possible. E.g., TT on T8842 board and KT on a KTT84 board are not the nuts, while KT on a KKT84 board is the nuts. Although the Ten in the second hand makes quads impossible, KK still wins (although the player with the KK doesn’t know quads are impossible).”

[BB] “On the river, the five cards on the table can only be the nuts if they form a royal flush, four of a kind with the nut kicker, or a royal straight with no flush possible.”

Footnotes:

  1. The weakest hand that can be the nuts is QQ on a Q8732 board with no flush possible. Three Kings and three Queens are the only sets that can be the nuts as any other three of a kind makes a straight possible.

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Expected Value

[BB] “Every action you take in a poker game has an expected value, which is the average amount of chips or money that action will win or lose in the long run”, Benny the Book explained to his son.

EV of Folding

[BB] “Folding has an expected value, or EV, of 0, since you aren’t gaining or losing any more chips. Remember that the chips in the pot are no longer yours, so they don’t factor in when you fold. Whether there are 10 chips or 10,000 chips in the pot, folding will change your chip stack by the same amount, zero.”

EV of Calling

[BB] “Following up on our discussion of pot odds, the EV of calling is calculated as the odds of winning times the pot size minus the odds of losing times the call size. For example, suppose the pot is P and you call a half-pot all-in bet on the turn when you’re sure you need to hit a flush on the river. Since 9 outs is about 20%1, the EV of the call is 0.2 * 1.5P – 0.8 * 0.5P = -0.1P2. You’ll lose just over a tenth of the original pot size on average by making this bad call. If you also had a straight draw, you have 15 outs, about 32.5%, which we can call 1/3, and (1/3)*1.5P – (2/3)*0.5P = 1/6P3. You’ll win almost a sixth of a pot on average.”

EV of Raising

[BB] “This is impossible to calculate without knowing how often your opponent is going to fold, call, or reraise, but we can lay out the general formula. For simplicity, let’s say we’re on the river, the pot is P, your opponent bets B, you raise R, and you have the better hand N% of the time (all percentages should be divided by 100 to give a number between 0.0 and 1.0). When your opponent folds, you win P+B. When your opponent calls you win P+B+R if your hand is better and lose B+R if your hand is worse. Let’s ignore the possibility of your opponent reraising for now. Your overall EV is (opp fold %) * (P+B) + (opp call %) * ((N * (P+B+R)) – ((1-N) * (B+R))).”

[JJ] “You expect me to crunch all that at the table?”, Joey the Juvenile objected.

[BB] “No, but it’s useful in postgame analysis. The important thing to note is that your raise wins some of the time by folding your opponent and some of the time when you have the best hand. Whenever you raise, you want to think about both of those possibilities to determine how much to bet. Against calling stations, you should expect fewer folds, so you should bet more for value. Against tight players who haven’t shown any strength, you can bluff more.”

[BB] Seeing his son smiling much too broadly, Benny regretfully wondered, “Why do I feel like I’m training a maniac?”

Footnotes:

  1. This was covered a few days ago in Counting Outs.
  2. The exact calculation is 9/46 * 1.5P – 37/46 * 0.5P equals about 0.29P – 0.40P = -0.11P.
  3. The exact calculation is 15/46 * 1.5P – 31/46 * 0.5P equals about 0.49 – 0.34 = +0.15P.
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Implied Odds

[BB] Benny the Book resumed explaining to his son, “Pot odds are pretty straightforward, but here’s the tricky part; implied odds are when you take into account additional chips that you could win if you hit your draw. If you or your opponent is all-in, the implied odds are the same as the pot odds. But if you both have more chips, you need to factor that in. There’s no clearcut formula for implied odds, unfortunately. You need to take into account the playing style of your opponent and even how they view you. Some players will frequently pay you off, while others will be very suspicious every time a possible draw fills.”

[JJ] “Are you implying that I need to use my excellent judgment here?”, Joey the Juvenile offered.

[BB] “Yes on the ‘judgment’ part; not sure about the ‘excellent’ part. Let’s say it’s early in a tournament so both you and your opponent have deep stacks. You’re facing a half-pot turn bet with nothing but a flush draw. Should you call?”

[JJ] “Well, the flush draw is 9 outs, which is 9 * 2 + 1.5 = 19.5%”, Joey responded then thought for a bit. “The half-pot bet means 25%, so my pot odds aren’t good enough. I can call if my opponent will call a small bet on a flush river card though, since my odds are only off by a little.”-

[BB] “Right, so it matters quite a bit whether you have suited or unsuited hole cards. In the latter case, that river put four cards to a flush on the board, and it doesn’t take Albert Einstein to find that fold. In the former case, it’s easier to believe that you don’t have the flush. The same thing is true with straight draws. The obvious straights are just as obvious as 4-flushes. If the board has King-Queen-Jack-Ten, everyone’s going to worry that you have the Ace, or even the Nine, for the straight. But the less obvious straight draws can be very well disguised. If the board has Queen-Ten-Three-Two, and an Eight hits on the river, will your opponent think you have the Jack-Nine? Your implied odds are barely better than your pot odds for obvious draws and significantly better for less likely draws.”

[JJ] “I’m usually on an unlikely draw.”

[BB] “Two undercards isn’t considered to be a draw.”

[BB] “Anyway, before you go too crazy and stick around even more because of your implied odds, keep in mind a couple caveats. First, if you’re not drawing to the nuts, you can still be beat. A classic case is when you have a suited King. The flop puts two of your suit and an Ace of another suit on the board. Every once in a while, your opponent has the nut flush draw, and you’re practically drawing dead.”

[BB] “Second, if you hit your draw on the turn, your opponent may have redraws to beat you on the river. This commonly happens when you hit your straight or flush but your opponent has three of a kind. They then have ten outs on the river; any card that puts a pair on the board gives them the boat or four of a kind. Another case is when you hit a straight but your opponent has a flush draw. Then they have the usual nine outs to win.”

[BB] “Sufficiently warned?”

[JJ] “Sufficiently armed!”

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Pot Odds

[BB] “Do you know what pot odds are?”, Benny the Book asked his son.

[JJ] Joey the Juvenile jested, “Would that be the chance that the police discover that the pointy little plants in your garden are marijuana?”

[BB] With a scornful look, Benny chided his son, “In poker, the pot odds are the current odds that you are getting to make a call. If you don’t know what odds you’re facing, you can’t make an informed decision. If you aren’t making informed decisions, you’re making mistakes. If you’re making mistakes, you’re…”

[JJ] “Okay, okay. I get it”, Joey interrupted.

[BB] “If you’re ahead in a hand, it’s good to know what your opponent’s odds are of catching you, so you charge him enough for his draw. But let’s focus on the flip side.”

[BB] “If you’re behind, it’s even more important to know what your odds are. If you’re chasing draws with bad odds all day, you’re just asking to lose. We just covered what your odds are of hitting your draws. Next you need to look at what odds the pot is giving you. If there’s 1,000 in the pot, and you need 250 to call, then you’re getting 1,000-to-250, or 4-to-1 odds. So if you can hit your draw once for every four times you miss, which is 20%, then you’ll break even on pure pot odds.”

[JJ] “Why is 4-to-1 20% instead of 25%?”

[BB] “Oh, I should have explained that last time. 4-to-1 means 4-misses-to-1-hit, which is the same as 1-in-5 (1 hit in 5 attempts), which is 20%. Even I get 4-to-1 versus 1-in-5 confused occasionally, so I prefer to work in percentages all the time.”

[BB] “To summarize, your pot odds are the amount that’s in the pot to the amount you need to call. Reversed, it’s the amount that you need to call in the pot size, so the percentage is the amount you need to call divided by new total pot (times 100% unless you want to keep it as a decimal). Let’s stick with this last equation from here on out.”

[BB] “If the pot is 250 and your opponent bets 500 on the turn, what are your pot odds for calling?”

[JJ] “That’d be 500 divided by (250 + 500 + 500) equals 40%. You’d need like 19 outs to make that a good call.”

[BB] “Exactly. Overbetting the pot makes almost all draws unprofitable for one street.”

[BB] “If the pot is 2,000 and your opponent moves all-in on the flop for 1,500, how many outs do you need to make it a good call if he never bluffs?”

[JJ] “1,500 divided by 5,000 is 30%. You’d need about 14 outs… no, that’s one street… you’d need 7 or 8 outs.”

[BB] “Excellent! I think you’ve already got it.”

[JJ] “Thanks. And you know those pointy green things in my bedroom? They’re buckeye leaves ;-).”1

Footnotes:

  1. To be honest, they do look a little alike.
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Counting Outs

[BB] Joey the Juvenile has been calling too much, and not on the phone. His dad, Benny the Book explained, “When you’re behind in a hand, you can’t just call every bet because you might improve to a better hand. You need to know what your chances are, and the first thing is to count how many cards will improve your hand enough to beat your opponent’s. These are called your outs:”

Draw Outs Notes
Flush Draw 9 Outs 13 – 4 = 9
Straight Draw 8 Outs 2 * 4 = 8
2 Overcards 6 Outs 2 * 3 = 6
Inside Straight Draw 4 Outs
1 Overcard 3 Outs

[BB] “If you have a combination draw, such as a straight flush draw, add the outs then subtract any duplicated outs. For example, if you’re pretty sure your opponent has a pair of Aces1 and you’re holding Q♥J♥ on a 10♥9♥2♣ board, you have 9 + 8 – 2 = 15 outs (the K♥ and 8♥, which actually give you a straight flush, being the two duplicated outs).”

[BB] “Once you know how many outs you have, you can convert to a percentage or odds using this chart:”

Outs Turn Odds River Odds Turn+River Odds
22 Outs 1.1-to-1 (46.81%) 1.1-to-1 (47.83%) 0.38-to-1 (72.25%)
21 Outs 1.2-to-1 (44.68%) 1.2-to-1 (45.65%) 0.43-to-1 (69.94%)
20 Outs 1.3-to-1 (42.55%) 1.3-to-1 (43.48%) 0.48-to-1 (67.53%)
19 Outs 1.5-to-1 (40.43%) 1.4-to-1 (41.30%) 0.54-to-1 (65.03%)
18 Outs 1.6-to-1 (38.30%) 1.6-to-1 (39.13%) 0.60-to-1 (62.44%)
17 Outs 1.8-to-1 (36.17%) 1.7-to-1 (36.96%) 0.67-to-1 (59.76%)
16 Outs 1.9-to-1 (34.04%) 1.9-to-1 (34.78%) 0.75-to-1 (56.98%)
15 Outs 2.1-to-1 (31.91%) 2.1-to-1 (32.61%) 0.85-to-1 (54.12%)
14 Outs 2.4-to-1 (29.79%) 2.3-to-1 (30.43%) 0.95-to-1 (51.16%)
13 Outs 2.6-to-1 (27.66%) 2.5-to-1 (28.26%) 1.1-to-1 (48.10%)
12 Outs 2.9-to-1 (25.53%) 2.8-to-1 (26.09%) 1.2-to-1 (44.96%)
11 Outs 3.3-to-1 (23.40%) 3.2-to-1 (23.91%) 1.4-to-1 (41.72%)
10 Outs 3.7-to-1 (21.28%) 3.6-to-1 (21.74%) 1.6-to-1 (38.39%)
9 Outs 4.2-to-1 (19.15%) 4.1-to-1 (19.57%) 1.9-to-1 (34.97%)
8 Outs 4.9-to-1 (17.02%) 4.8-to-1 (17.39%) 2.2-to-1 (31.45%)
7 Outs 5.7-to-1 (14.89%) 5.6-to-1 (15.22%) 2.6-to-1 (27.84%)
6 Outs 6.8-to-1 (12.77%) 6.7-to-1 (13.04%) 3.2-to-1 (24.14%)
5 Outs 8.4-to-1 (10.64%) 8.2-to-1 (10.87%) 3.9-to-1 (20.35%)
4 Outs 10.8-to-1 (8.51%) 10.5-to-1 (8.70%) 5.1-to-1 (16.47%)
3 Outs 14.7-to-1 (6.38%) 14.3-to-1 (6.52%) 7.0-to-1 (12.49)%
2 Outs 22.5-to-1 (4.26%) 22.0-to-1 (4.35%) 10.9-to-1 (8.42%)
1 Out 45.9-to-1 (2.13%) 45.1-to-1 (2.17%) 22.3-to-1 (4.26%)

[JJ] “I can’t memorize all those numbers!”, Joey the Juvenile complained.

[BB] “I knew you’d say that. Now you know why should learn a formula to calculate the percentage. There are several formulas you can use for just the turn, just the river, and combined, so I’ll let you pick whichever ones you’re comfortable with:”

Accuracy Turn River Turn+River Comments
Excellent Outs times 2.13% Outs times 2.17% Turn odds plus River odds minus Turn odds times River odds2 More accurate than you’ll ever need
Very Good Outs times 2-1/8% Outs times 2-1/6% Outs times 4%, minus (Outs – 8) if 9 or more Outs3 As accurate as you’ll ever need
Good Outs times 2%, plus 1% if 5+ Outs, plus 2% if 13+ Outs Same as Turn Outs times 4%, minus (Outs – 8) if 9 or more Outs3 The easiest reasonable approximation
Bad Outs times 2% plus 1% Same as Turn Outs times 4% Not good for low or high numbers of outs

[JJ] “Which formulas do you use?”

[BB] “I like the ‘Very Good’ approximations. Say I have 13 outs. It’s easy enough to say that’s 13 * 2 + 2 = 28% on the turn (actually three-eighths of a percent less, but I always round), the same on the river (28-1/6% if you don’t round), and 13 * 4 – (13 – 8) = 47% combined. The turn and river numbers are very close, while the combined number is off by only a bit over a percent, so they’re more than good enough.”

[BB] “One additional note, you can count a backdoor flush or a backdoor straight as 1 out for the turn and river combined, since they’re both around 4%.”

[BB] “Once you know the odds, you can compare them to the pot odds or implied odds to see if calling makes sense, which is an explanation for another day…”

Footnotes:

  1. The suits, being unknown, are ignored. Yes, one of them could be a heart, so you really have only 8 flush outs, but all cards that you can’t see are treated equally (any card that’s in another player’s hand or in the muck could still be in the deck).
  2. For example, using the Good approximation, if you have 9 Outs, then you have a 19% chance on the turn, a 19% chance on the river, and 38% minus 19% of 19% (close to 20% of 20%, or 4%) equals a 34% chance on the turn and river combined.
  3. From David Solomon, mentioned in Harrington on Hold ’em: Volume 2.
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