Mathematical formula collection.

    • z4tz
      z4tz
      Bronze
      Joined: 24.05.2008 Posts: 2,043
      I see questions quite often for the quite simple formulas so i thought i'd gather them in one post. Hopefully some of you can have some use for it. Also posted some examples for them to see how they are used. If there is something missing or any misstakes, please let me know.

      Mathematic formulary for poker.

      Odds:
      (unknown cards - outs) : outs

      Pot odds:
      (Pot+Bet):Investment

      Implied odds
      Draw odds - pot odds = implied odds needed

      Equity to call bet:
      EQ=Bet/(Pot+2*Bet)

      Pot odds <=> Equity
      x:1 => 1/(x+1)=EQ
      1/EQ-1=x => x:1


      EV-Calculations
      ( EV= P(1)*$(P(1))+ P(2)*$(P(2))+…+ P(N)*$(P(N)) )

      Calling a bet:
      EV= P_win*(Pot+Bet) - P_lose*Bet

      Fold equity:
      EV= P_fold*Pot + (1 – P_fold) * (EQ * Total pot – Investment)

      Purebluff FE, EQ=0:
      EV= P_fold*Pot - (1 - P_fold)*Investment


      Tournament maths
      M-factor
      M=Stacksize/(bb+sb+Ante*Players) *

      Effective M-factor
      EM = M * (players/10)

      Effective Big Blind
      EBB=2*(bb+sb+Ante*Players)/3 *

      *A little trick is to replace (bb+sb+Ante*Players) with (Pot before any action preflop) for calculations on the table.
  • 3 replies
    • z4tz
      z4tz
      Bronze
      Joined: 24.05.2008 Posts: 2,043
      Examples

      Odds:
      Flush draw from flop to turn gives 9 outs. 52-5= 47 unknown cards.
      47-9:9
      38:9
      4.2:1

      Pot odds:
      To call a bet of $8 into a $10 pot.
      (10+8):8
      18:8
      18/8:8/8
      2.25:1

      Implied odds:
      To call a bet of $5 into a $5 pot with a flush draw on the flop.
      Pot odds (5+5):5 => 2:1
      Odds with flush draw 4.2:1
      4.2:1 – 2.2:1 = 2.2:1
      $5*2.2 = $11 needed to break even on later streets.

      Equity to call bet:
      To call a 10$ bet in a 15$ pot.
      EQ=10/(15+2*10)
      EQ=0.286 => 28.6%

      Pot odds <=> Equity
      Pot odds of 3:1 gives:
      1/(3+1)=0.25 => 25%
      Equity of 40% gives (EQ=0.4):
      1/0.4-1=x
      2.5-1=1.5
      Pot odds 1.5:1




      EV-Calculations

      Calling a bet:
      Expected value of calling a 5$ bet in a 20$ pot with 19% equity.
      EV=0.19*(20+5) - 0.81*5
      EV=4.75 - 4.05
      EV= +$0.7

      Fold equity:
      Ammount of fold equity needed to make a 3-bet push preflop profitable. The blinds are 1/2 and there is a raise to $8. We have a stack of $25 and expect to have 27% equity if called.
      To find break even set EV=0. (Both players are outside the blinds)
      0=P_fold*(8+1+2)+(1-P_fold)*(0.27*(1+2+25+25)-25)
      0=P_fold*11+(1-P_fold)*(-10.69)
      0=P_fold*11 - 10.69 +10.69*P_fold
      10.69=21.69*P_fold
      P_fold=0.493 =>49.3% of the times he needs to fold.

      Purebluff FE:
      Raising a cBet with 0% equity if called. Pot is $10 and villain cbets $7. We raise his bet to $20
      To find break even set EV=0.
      0=P_fold*(10+7) - (1 - P_fold)*20
      0=P_fold*17 - 20 + P_fold*20
      20=P_fold*37
      P_fold=0.54 =>54% of the times he needs to fold.

      Tournament maths

      M-factor:
      M on a table with 9 players 25/50 and 5 ante with 2200 chips.
      M=2200/(50+25+5*9)=18.33

      Effective M-factor:
      5 Players left on the table with 75/150 and 20 ante with 2200 chips.
      M=2200/(150+75+20*5)=6.78
      EM= 6.78*(5/10)=3.39

      Effective bb:
      9 Players on a table with blinds of 250/500 and 50 ante.
      Ebb=2*(500+250+50*9)/3
      Ebb=800
    • doctorkgb
      doctorkgb
      Silver
      Joined: 01.04.2009 Posts: 1,263
      Sticky imo. Thnx
    • Wolfjustice
      Wolfjustice
      Bronze
      Joined: 09.01.2012 Posts: 38
      great post