1. Pot Odds & Required Breakeven Equity (%)
Required Equity (%) = Call Amount / (Pot Before Call + Bet + Call Amount)
Table Shortcut: Against A Half-Pot Bet (50%), Required Equity = 0.5 / 2.0 = 25.0%. Against A Full-Pot Bet (100%), Required Equity = 1.0 / 3.0 = 33.33%.
Worked Example: The Pot Is $100 And Your Opponent Bets $50 (Making The Pot $150 Before Your Action). You Must Call $50 To Win A Total Final Pot Of $200. Required Equity = $50 / $200 = 25.0%. If Your Hand Wins At Least 25% Of The Time, Calling Is +EV.
2. The Rule Of 4 And 2 (Flop & Turn Draw Equity)
Flop Equity (2 Cards To Come) ≈ Outs x 4 | Turn Equity (1 Card To Come) ≈ Outs x 2
Table Shortcut: A 9-Out Flush Draw Holds ~35% Equity On The Flop (9 x 4 - 1) And 19.6% On The Turn (9 / 46). An 8-Out Open-Ended Straight Draw Holds ~31.5% On The Flop And 17.4% On The Turn.
Worked Example: You Hold As Ks On A Qs 7s 2d Flop (9 Remaining Spades Out Of 47 Unseen Cards). Using The Rule Of 4: 9 x 4 = 36% Approximate Equity (Exact Combinatoric Equity Is 34.97%). On The Turn With One Card Left: 9 x 2 = 18% (Exact Is 19.57%).
3. Expected Value (EV) Of Any Call, Bet Or Shove
EV = (% Win x $ Won When Winning) - (% Lose x $ Lost When Losing)
Table Shortcut: Whenever EV > $0.00, The Action Generates Long-Term Profit Regardless Of Single-Hand Variance.
Worked Example: Your Opponent Shoves $100 Into A $100 Pot On The River (Total Pot $200, Costs $100 To Call). You Estimate Your Bluff-Catcher Wins 40% Of The Time And Loses 60% Of The Time: EV = (0.40 x $200) - (0.60 x $100) = $80 - $60 = +$20.00 Per Call.
4. Semi-Bluff All-In EV With Fold Equity
Total EV = (Fold % x Current Pot) + (Call % x [(Eq x Final Pot) - Risked Stack])
Table Shortcut: Combining 35% Fold Equity With 35% Draw Equity When Called Turns Almost Any Semi-Bluff Shove Into A Massive +EV Play.
Worked Example: There Is $90 In The Pot And You Shove $100 With A Flush Draw (35% Equity When Called). If Villain Folds 45% Of The Time: EV = (0.45 x $90) + 0.55 x [(0.35 x $290) - $100] = $40.50 + 0.55 x ($1.50) = +$41.33.
5. Alpha (α): Pure Bluff Breakeven Percentage
Alpha (α) = Bet Size / (Pot Size + Bet Size)
Table Shortcut: A 33% Pot Bluff Needs 25.0% Folds; A 50% Pot Bluff Needs 33.3% Folds; A 75% Pot Bluff Needs 42.9% Folds; A 100% Pot Bluff Needs 50.0% Folds.
Worked Example: You Fire A $75 Pure Bluff With 0% Equity Into A $100 River Pot. Alpha = $75 / ($100 + $75) = 75 / 175 = 42.86%. If Your Opponent Folds More Than 42.86% Of The Time, Your Bluff Prints Immediate Profit.
6. Minimum Defense Frequency (MDF)
MDF (%) = 1 - Alpha = Pot Size / (Pot Size + Bet Size)
Table Shortcut: Against A 33% Pot Bet, Defend 75.0%; Against A 50% Pot Bet, Defend 66.7%; Against A 100% Pot Bet, Defend 50.0%; Against A 150% Overbet, Defend 40.0%.
Worked Example: Your Opponent Bets $66 Into A $100 River Pot. MDF = $100 / ($100 + $66) = 60.24%. To Prevent Your Opponent From Profitably Bluffing Any Two Cards, You Must Call Or Raise With At Least 60.24% Of Your Range.
7. Optimal Polarized River Bluff-To-Value Ratio
Bluff Share Of Betting Range = Bet Size / (Pot Size + 2 x Bet Size)
Table Shortcut: At 50% Pot Sizing: 25% Bluffs / 75% Value (1:3 Ratio). At 75% Pot Sizing: 30% Bluffs / 70% Value. At 100% Pot Sizing: 33.3% Bluffs / 66.7% Value (1:2 Ratio).
Worked Example: You Bet Full Pot ($100 Into $100) On The River With A Polarized Range. Because Villain Gets 2-To-1 Pot Odds (Needs 33.3% Equity), Your Betting Range Should Contain 1 Bluff Combination For Every 2 Value Combinations (33.3% Bluffs, 66.7% Value) To Make Villain Indifferent.
8. Stack-To-Pot Ratio (SPR)
SPR = Effective Remaining Stack / Flop Pot Size
Table Shortcut: Low SPR (1 To 3): Commit With Top Pair Or Overpairs. Medium SPR (4 To 8): Two Pair Or Strong Draws. High SPR (9+): Require Sets, Straights, Or Flushes To Stack Off.
Worked Example: In A 100BB ($300) Cash Game, A Player Opens To $12, Gets 3-Bet To $40, And Calls. The Flop Pot Is $84 And Both Players Have $260 Remaining Behind. SPR = $260 / $84 = 3.10, Making Overpairs (QQ+) Automatic Stack-Offs.
9. Geometric Bet Sizing (Multi-Street All-In Growth)
Geometric Bet Fraction (b) = [((2 x SPR + 1)^(1 / N)) - 1] / 2
Table Shortcut: At SPR 4 Across 3 Streets (N = 3): Bet 54.3% Pot On Flop, Turn, And River. At SPR 13 Across 3 Streets: Bet 100% Pot On Flop, Turn, And River.
Worked Example: You Reach The Turn With $100 In The Pot And $275 Effective Stacks Remaining (Turn SPR = 2.75) And Want To Bet The Exact Same Percentage On Turn And River (N = 2 Streets). Using The Formula: ((2 x 2.75 + 1)^0.5 - 1) / 2 = (2.5495 - 1) / 2 = 77.5% Pot On Both Turn And River.
10. Independent Chip Model (ICM Malmuth-Harville Equation)
P(Player i Finishes 1st) = Chips_i / Total_Chips | P(i = 2nd | j = 1st) = Chips_i / (Total_Chips - Chips_j)
Table Shortcut: In Tournaments, Doubling Your Chip Stack Always Less Than Doubles Your Real-Dollar ($EV) Equity Because First-Place Prize Money Is Capped.
Worked Example: Three Players Remain For Prizes Of $500 / $300 / $200 ($1,000 Total) With Chip Counts Of 5,000 (50%), 3,000 (30%), And 2,000 (20%). The Chip Leader Holds 50% Of The Chips, Yet His Exact ICM Equity Is $383.33 (38.3% Of The Prize Pool), Not $500.
11. Risk Of Ruin & Required Bankroll Equation
Risk Of Ruin (RoR) = exp( (-2 x Winrate x Bankroll) / (StdDev^2) )
Table Shortcut: Doubling Your Winrate Cuts Your Required Bankroll For A 1% Risk Of Ruin In Half; Doubling Your Standard Deviation Quadruples Your Required Bankroll.
Worked Example: A 6-Max Cash Player Wins +5 BB/100 With A Standard Deviation Of 85 BB/100. With A 2,500 BB Bankroll (25 Buy-Ins), Risk Of Ruin = exp((-2 x 5 x 2500) / (100 x 7225 / 100)) = exp(-3.46) = 3.14%. Increasing To 35 Buy-Ins Drops RoR Below 0.8%.
12. Hypergeometric Combinatorics & Card Removal
C(n, k) = n! / [k! x (n - k)!] | Total Hold'em Starting Combos = C(52, 2) = 1,326
Table Shortcut: Each Pocket Pair Has 6 Combos; Each Suited Hand Has 4 Combos; Each Offsuit Hand Has 12 Combos (16 Total For Any Unpaired Hand). Holding One Blocker Cuts Pocket Pair Combos By 50% (From 6 To 3).
Worked Example: Normally There Are C(4, 2) = 6 Combinations Of Pocket Aces (AA) And 4 x 4 = 16 Combinations Of Ace-King. If You Hold The Ace Of Spades (As) In Your Hand, Only C(3, 2) = 3 Combinations Of AA And 3 x 4 = 12 Combinations Of AK Remain Possible.
