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Tuesday, February 2, 2010

Mathematics of NLHE Ep 2

Covered in this episode:

  • Pot Odds
  • Implied odds
  • Fold Equity
  • Hand Combinations
  • G-Bucks

Pot Odds

  • Definition: Ratio between the value of the pot and the value that must be called to continue.
    • Ex. The pot has $40 in the middle and there is a $10 bet. It's your turn to act, what are your pot odds?
      • $40 +$10 = pot = $50
      • To continue with the hand we would have to call the $10 bet to win $50 so the ratio is 50:10 or 5:1 which also means we have to win the hand 1 in 6 times in order to call
  • To correctly continue in a hand, in terms of Pot Odds, your price to continue in the hand must be better than the odds against you making your hand (or having the best hand).
    • Ex. If you are a 2:1 dog, as you know that means you'll lose twice foe each time you win. To continue you'd need at least 2:1 on your money to break even. Any better than that and you'll make money over the long haul. Any worse, you'll lose.
  • If we know our odds of winning the hand, we can determine what odds we need to call a bet profitably
    • Ex. 1 - Let's say you have 6h8h and the board is 9723r and you're playing against a tight player who bets $40 into a $50 pot on the turn. You suspect, at worst, this player has a pair of nines and the only way you can win is to hit one of your straight outs. Do you have pot odds to call?
      • OESD has 8 outs to improve
      • We are roughly a 5:1 dog (4.7:1 exactly) to improve on the river
      • He bet $40 into $50 pot so total = $90 and it costs us $40 to win it
      • We are risking 40 to win 90, getting 90:40 or 2.25:1 odds on our call
      • We need 4.7:1 to make the call, we can't profitably continue in this hand based on pot odds alone.
    • Ex. 1 continued - Let's say we've got the same 6h8h again and the same 972r board but this time we're on the flop and we're playing against a loose passive player who goes too far with his hands and calls to much but generally won't put money in postflop without top pair or better. The pot is $100 and we've got %950 behind in our stack. He makes a pot bet, $100 into $100 pot. Based on our pot odds, call or fold?
      • We've got 8 outs to the straight, that means we need 2.2:1 on a call based on pot odds alone. We're getting 200:100 or 2:1 on a call so we fold right?
      • WRONG! Our flop odds assume we get to see both the turn and the river but often in NLHE that's not the case. Therefore it is better to calculate based on the odds you'll hit on the next street.
      • So do we still fold because we need 4.7:1 and we are getting 2:1?
      • No. Why?
      • We have a read that this player calls much too liberally postflop and that he rarely puts money in the pot without at least top pair. He's willing to pay us off whenever he has a decent pair and he usually has a hand when he bets so he'll have something to pay us off with often. Another piece of good news is that the stacks are deep. We have $950 left to play with and we have reason to believe he will put a lot of $ in with marginal holdings.

Implied Odds

  • Ratio of the total amount of money to be won from the pot + from stacks behind to value of the current bet
  • Ex. 1 continued - What are our implied odds?
    • We are calling $100 bet to win the 4200 in the pot plus potetinally the $950 left to play with.
    • We could be looking at as much as 950+200 = 1150:100 or 11.5:1. Since we need roughly 5:1, we are getting over double what is required.

Fold Equity

  • FE is essentially the estimated % chance that a bet or raise will induce our opponent to fold.
  • FE is often used in estimating how successful semi-bluffing (bluffing with outs to improve) is in a specific situation, but it applies to pure bluffs as well.
  • FE is measured by which hands in our opponent's hand range he/she will fold to a bet or a raise.
  • Ex. We have AsTs and the flop is 8s9s7h with $100 in the pot. The stacks are $950. Opponent bets $100 into $100. Call, fold or raise?
    • In this hand if our opponent has something like K9, we could have a ton of outs: 9 spades, 3 Jacks, and 3 sixes (remember not to double count the Js and 6s), 3 tens and 3 aces = 21 outs.
    • With our big draw, we are actually a 68% favorite over top pair so folding is obviously out of the question. Raise or call?
    • We are getting 2:1 on a call si if we were guaranteed to see both cards, we'd only need 33% equity. We have 68% equity with 2 cards to come, but even with only 1 card to come we have over 45% equity so calling would be ok.
    • Raising is the best option as we have a lot of FE and Pot Equity. We don't have much if any implied odds as the board is so coordinated nor do we have a made hand.
  • Some hands and situations get a big part of their profitability from the use of FE
  • Ex. 5/10 NL $1000 stacks. You have an aggressive image and the other players at the table are similarly aggressive. You open AKs from the CO to $35, the BTN calls and the SB who you've seen play small pairs very aggressively makes it $160 to go. Folds to you and you have a decision to make.
    • In this spot based on our read, we know the SB can be 3betting (or making a "squeeze play" as its known) with hands such as 22+, AQ+. That's the range we give him and we're a slight underdog to that range. So do we fold? Call and hope to hit taking our pot odds?
    • No, the best play is to 4bet him. The reason is FE. Our AK doesn't mind getting all in preflop against most of his range, but we'll force him to incorrectly fold hands that hes a slight favorite in.

Hand Combinations

  • Hand combinations refers to the number of ways a particular 2 card hand can be made given the known and unknown cards in the deck.
  • Knowing the number of combos possible for a given hand can help us determine the likelihood of that hand in a given range of possible holdings.
  • Ex. You hold AsAc and the board is Kc5d9h and your "gut" says our opponent either has a set or AK. How many hand combos are there of AK?
    • You hold AsAc so he cannot have those in his hand. The Kc is out on the flop so he cannot have that either. The possible cards left to make AK are: Ah, Ad, Kh, Kd, Ks. How many ways can we make AK from these cards?
      • There are 6 combos of AK. 2 Aces x 3 Kings = 6 combos
        • AhKh
        • AhKd
        • AhKs
        • AdKd
        • AdKs
        • AdKh
    • How many combos of sets? Let's start with a set of fives.
      • 5d is on the board so we need to make a 2 card hand out of: 5h, 5c, 5s
      • There are 3 combos of a set of fives.
        • 5h5c
        • 5c5s
        • 5h5s
      • Same for set of 9s
    • So if his range is really {AK, sets} then how likely is it that he has a set?
      • 6 combos of AK. 3+3+3 = 9 combos of sets. so of 15 combos, 9 are sets. 9/15 = 60% chance that he has a set if our hand range is correct.
  • Hand Combos to Memorize:
    • Any exact two card hand: ex AsKs - 1 combo
    • Specific suited starting hand: ex 98s - 4 suits = 4 combos
    • Unpaired hand with no other info: ex AK preflop - 4 aces * 4 kings = 16 combos
    • Pocket pair with no other info: ex QQ preflop - 4 suits, 2 cards = 6 combos
    • Pair + Kicker with no other info: ex AK on K95 - 4 aces * 3 kings = 12 combos (note: you do not know villain has AK, you only think he has a pair and kicker)
    • Two pair hand with no other info: ex AK on AK5 - 3 aces * 3 kings = 9 combos (note: same as above except you think he has two pair)
    • Set with no other info: ex 55 on K95 - 3 suits, 2 cards = 3 combos
    • When thinking about it at the table, realize you can use straight multiplication only when you're dealing with two separate cards such as aces matched with kings.


  • The term "G-Bucks" coined after Phil Galfond is the next level of complexity in EV calculations
  • G-Bucks calculations determine your expected value based not on our actual holding against your opponents holding, but instead your range of hands against his actual holding.
  • Over the long haul in a given situation, we're not playing the exact hand we have, we're playing a range of hands that our opponent has to react to. Therefore we see how his actual hand stacks up against our range.
  • We can determine our range based on:
    • the betting sequence
    • board texture
    • our overall read on how our opponent plays
    • and often what we believe his read is on us

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