Advanced Poker Math: EV Calculation & Game Matrices

Poker is fundamentally a game of expected value. Advanced mathematical tools let you analyze situations beyond simple pot odds.

Expected Value (EV)

**EV Formula**: EV = (Probability of Winning × Amount Won) − (Probability of Losing × Amount Lost)

**Fold Equity**: EV of betting includes the chance the opponent folds. EV(bet) = (Fold% × Pot) + (Call% × EV when called).

**Bluff EV**: A pure bluff needs Fold% > Bet / (Pot + Bet) to be +EV. A half-pot bluff needs folds > 33%.

**Value EV**: When betting for value, you need >50% equity when called against the continuing range.

**Hand combinations**: Unpaired = 16 combos (4 suited + 12 offsuit), Paired = 6 combos, Suited only = 4 combos

**Card removal**: Holding the A♠ removes 25% of AA combos (from 6 to 3) and blocks AK suited combos

**Range counting**: A 15% opening range is roughly 200 combos. Count removed combos to assess how often they have it.

**Payoff Matrix**: A grid showing payoffs for each player's strategy pair. Useful for simplified river decisions.

**Frequency analysis**: If you value bet 30 combos and bluff 15 combos, you are betting 45 combos with a 2:1 value-to-bluff ratio.

**Optimal ratio**: On the river, a pot-sized bet requires 2 value combos for every 1 bluff combo to make the opponent indifferent.

**Break-even fold %** = Bet / (Pot + Bet) → half-pot needs 33% folds; full pot needs 50%

**Required equity vs a bet** = Call / (Pot + Bet + Call) → facing half-pot, need 25% equity

**Implied odds**: Add future expected winnings to pot odds when you can win more on later streets

**Reverse implied odds**: Subtract future expected losses when you hit your hand and still lose

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