Category Archives: River

Reading Hands, River

{Continued from Reading Hands, Turn}

[NN] “The river is the exciting conclusion to a four-act play”, Nate the Natural continued. “With a good read, you can pick off a bluff by a missed draw, like naming the murderer in a whodunit.”

[FF] “I don’t get killed that often, but they’re certainly always stealing my chips”, Figaro the Fish amended.

[NN] “Okay then, just a bit of larceny to discover… or commit. If you think you’re behind, can you try to steal the pot? If you’re ahead, how much value can you get from the final street of betting? If you’re in position or do you fear a check-raise? If you’re out of position, are you better off betting or hoping to pull off your own check-raise?”

[NN] “So, if a blank hits on a draw-heavy board after your opponent has check-called you the whole way, you’re not going to get paid off much. You have to hope the draw included a weak pair. In tournaments, you may not want to risk a small value bet if your opponent is a known check-raiser (unless of course you think he check-raise bluffs too often).”

[NN] “Likewise, if a money card hits on the river but your opponent still checks, there isn’t much point in betting.”

[NN] “The interesting case is when a draw comes in and your opponent leads out.”

[FF] “Easy fold.”

[NN] “Against most of the players here, probably. But what about against someone crafty like Elias the Eagle?”

[DD] “I try not to get into hands with him in the first place.”

[NN] “True, but you have top pair, and you never even had a chance to fold, since he never bet or raised. So here you are now with a board that shows K♥Q♥T♣4♦2♥. Elias bets half the pot. What are the odds he actually has the flush?”

[DD] “I have to fold or else he justifies his odds for chasing his draw.”

[NN] “The Birdman chases a lot of draws, because his implied odds are higher than ours are. When we didn’t bet him off on the turn, he called with pretty much 100% of his holdings, so he still has:

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

Count up the hands, and you’ll discover that he was on a straight draw more than three times as often as a flush draw. But by representing the flush, he more than doubled his ‘outs’, got us to fold, and stole our chips!”

[DD] “So, the only way to apprehend the criminal is to catch him in the act by calling his river bluff.”

[NN] “Right. Now suppose the board was dry instead: K♥7♣2♠4♦2♥, and your opponent is Roderick the Rock instead of Elias.”

[FF] “No draws there, so he has a real hand.”

[NN] “Yet he’s only been check-calling us.”

[DD] “He has top pair but doesn’t like his kicker.”

[FF] “Maybe a pocket pair lower than Kings?”

[NN] “It depends on who you are. If his opponent is Carlos the Crazy, Rod would have no problem calling with a pair of Tens. If it’s Mildred the Mouse, he’s folded all but his best Kings.”

[DD] “So, not only does he have a King, but it almost has to be King-Queen. Because he would have raised with Ace-King preflop.”

[NN] “Very good. So if we’re Mildred, and we actually hold pocket Sevens for a set, how much should we bet to extract the maximum value?”

[FF] “I’d probably pay off anything up to half a pot.”

[DD] “He’s tighter than you are. I don’t think he’s paying off much at all. I might try a quarter pot or even smaller.”

[NN] “I agree. That’s all you’re likely to get. He shouldn’t call anything, but we all hate to get bluffed, and we’re all curious to see what our opponents have.”

[DD] “Mildred isn’t ever bluffing here.”

[NN] “What if the opponent was Elias with an unknown hand instead of Mildred? If he bets a quarter pot, should Rod call? A half pot? Pot?”

[DD] “Roderick would probably call the first two but fold to a pot bet.”

[FF] “Unless Elias had been bullying Rod out of a bunch of pots recently.”

[DD] “Precisely when Elias is most likely to show up with the goods.”

[NN] “Maybe. But if you do a good job of putting him on a hand range, he won’t be able to fool you nearly as often as he does now.”

[DD] “Thanks, Nate. You could write a great book about reading hands.”

[FF] “I don’t know about palmistry, but you sure could write a good poker book.”

[NN] “Thanks, but Ed Miller already has. How To Read Hands At No-Limit Hold’em is expensive1 but worth the price. You can easily win that outlay back in a single cash game or tournament.”


  1. Currently still selling for its original list price of $49.99 at Amazon. The book deserves a full review, but I’m not qualified to write it. Maybe in a couple years.

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.”


  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. }


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.”


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

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.”


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

Zeebo’s Theorem

[YY] “Have you guys heard of Zeebo’s Theorem?” Yuri the Young Gun opened.

[FF] “Named for the youngest of the Marx Brothers?” Figaro the Fish speculated.

[VV] “No, that was Zeppo”, corrected Vincent the Veteran.

[JJ] “I thought it was an instant messaging client for your browser”, Joey the Juvenile offered.

[LL] “No, that was Meebo”, Leroy the Lion declined. “Zeebo was a cheap Brazilian game console.”

[HH] Harriet the Hazy wondered, “Wasn’t he a Greek philosopher who said that you can never get to your destination because you have to get halfway there an infinite number of times?”

[VV] “No, that’s Zeno’s Paradox.”1

[YY] Yuri took the floor back. “Zeebo’s Theorem, named not for the ‘To Kill a Mockingbird’ character but for poker pro Greg Lavery who goes by Captain Zeebo2 online, states that ‘No player is capable of folding a full house on any betting round, regardless of the size of the bet.'”

[LL] “Seems pretty accurate to me. Ever so rarely I’ve seen some top poker pros make big laydowns, but I’ve never seen any of us do it.”

[LL] “Just last week at the pub game, Patrick the Pickled busted out of both the tournament and the side game when he crashed his boat into a four-of-a-kind iceberg twice. On the first hand, he had Nines over Jacks, but Kieran the Keeper had overbet all-in on the river with quad Jacks. On the second one, his opponent shoved his big stack and was practically begging him to fold, taunting, ‘Don’t call. You’re gonna regret it if you call. I have you crushed!'”

[LL] “I don’t know if the double reverse psychology ploy had any effect, but after a full five minutes, Pat called with Nines full of Sevens and got stacked by four Sevens.”

[YY] “Both of the players with quads followed Zeebo’s Theorem perfectly. If you think your opponent has a full house, get all the chips in and expect to be called.”

[TT] “Hold ‘Em players please take note / When your foe begins to gloat / You may have a leaky boat / If you call, that’s all she wrote”, Tyrone the Telephone concluded.


  1. Zeno of Elea actually stated a number of paradoxes; this is his Dichotomy paradox (see the second version at the end of the section).
  2. Lavery’s handle is captZEEbo.

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.”


  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.

Related Links:


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?”


  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.

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


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

Seven-Card Hands

[FF] “Stan, what’s the average hand in Hold ‘Em on the river?” queried Figaro the Fish.

[SS] “Well, the average hand that sees the river depends a lot on the players in the game, but if you mean the average hand, including all the mucked hands, then it’s just a pair”, responded Stan the Stat.

[SS] “Since you asked about the flop hands last time, I was ready for this followup. About three-sevenths (43.82%) of all 7-card hands are just pairs. Two pairs comes next at 23.50% then no pairs at 17.41%. The rest of the hands appear less than one-sixth of the time total (Three of a Kind: 4.83%, Straight: 4.62%, Flush: 3.03%, Full House: 2.60%, Four of a Kind: 0.17%, and Straight Flush: 0.031%).”

[SS] “But so much depends on the betting that this information is almost useless, except for one important point. Beginners have a bad habit of getting over-enamored of their overpairs. Even without any clues from the betting, an overpair is beaten by almost two-fifths of all hands. Top pair is even worse than that. So if someone who isn’t totally crazy keeps betting (or even just calling your bets) with no obvious draws, you can lay down your overpair on the river with no regrets.”

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