Poker math: formulas, odds and free calculators

Cards and probability

A deck in numbers: 52 cards, 1,326 starting hands

C(52, 2) = 52 × 51 ÷ 2 = 1,326 starting hands, where C(n, k) counts the ways to choose k cards from n. One kind: pair 4 × 3 ÷ 2 = 6, suited 4, offsuit 4 × 3 = 12

How often you are dealt a pair, aces or a suited hand

Chance of a hand = its combinations ÷ 1,326: aces 6 ÷ 1,326 = 1 in 221, any pair 78 ÷ 1,326 = 1 in 17

Combinations of one hand: 6, 4 and 12

Pair: n × (n − 1) ÷ 2 with n unseen cards of its rank: 4 → 6, 3 → 3, 2 → 1. Unpaired: unseen of one rank × unseen of the other: 3 × 4 = 12

Poker odds and percentages: how to convert odds against

a to b against = b ÷ (a + b): 4 to 1 = 1 ÷ 5 = 20%. Odds against = misses ÷ hits: 25% misses 75%, 75 ÷ 25 = 3 to 1

At least once: the complement rule

P(at least once in n tries) = 1 − (1 − p)ⁿ: aces in 100 hands, 1 − (220/221)¹⁰⁰ ≈ 1 − 0.635 = 36.5%

What the flop brings: sets, flush draws and pairs

Flops = C(50, 3) = 50 × 49 × 48 ÷ 6 = 19,600. At least one hit = 1 − misses ÷ 19,600: a set with a pocket pair 1 − 17,296 ÷ 19,600 = 11.8%

The ten hand ranks and how often they happen

One hand in how many = all hands ÷ hands of the rank: a royal flush is 2,598,960 ÷ 4 = 649,740 with five cards, 133,784,560 ÷ 4,324 = 30,940 with seven

Pot odds and outs

Pot odds and implied odds

Required equity = call ÷ (pot + call), where pot includes the bet you face. Extra winnings needed = call ÷ equity − (pot + call)

Fold, call or raise: choosing the action

Price = call ÷ (pot with his bet + call): 10 ÷ (30 + 10) = 25%. Call when you win more often than the price

Counting outs: clean and tainted

One card: outs ÷ 47 on the flop, outs ÷ 46 on the turn. Two cards: 1 − (47 − outs) ÷ 47 × (46 − outs) ÷ 46. Quick rules: outs × 2% or outs × 4%

Drawing out: hands that change

One card: outs ÷ 47. Two cards: 1 − (47 − outs)/47 × (46 − outs)/46. Rule of 2 and 4: ≈ 2% or 4% per out

Free cards and protection

Break-even bet against a draw on the turn = pot × outs ÷ (46 − 2 × outs): 20 × 9 ÷ 28 = 6.4 bb

Pocket pairs and set mining

Needed from each set = 7.5 × call; needed after the flop = 7.5 × call − other players' chips in the pot: 7.5 × 3 − 4.5 = 18 bb

Equity and hand ranges

Domination: why A9 loses to AK

Equity = (wins + splits ÷ 2) ÷ boards: (407,694 + 79,513 ÷ 2) ÷ 1,712,304 = 26.1%

Suited connectors and broadway cards

Suited bonus = suited equity − offsuit equity: 45.4% − 42.3% = 3.1 points

Reading hands as distributions

Pairs: C(4,2) = 6 combinations; unpaired hands: 4 × 4 = 16 (4 suited, 12 offsuit). One card of a rank visible: pairs 3, unpaired 3 × 4 = 12

Your hand against a distribution

Equity vs range = Σ combos(h) × equity vs h ÷ Σ combos(h)

Playing accurately with the cards face up

EV(call) = equity × (pot + call) − call, with his all-in in the pot; call when equity > call ÷ (pot + call)

Expected value, betting and game theory

Poker expected value (EV): how to evaluate a call

EV = Σ probability × result. Calling: EV = equity × pot − (1 − equity) × call = equity × (pot + call) − call, with his bet in the pot

Why we bet: value and bluffs

A bluff must work bet ÷ (pot + bet) of the time: 10 ÷ (20 + 10) = 1 in 3 = 33.3%

Bet sizes as a share of the pot

Caller's price = bet ÷ (pot + 2 × bet). A bluff must work bet ÷ (pot + bet). Half the pot: 15 ÷ 60 = 25% and 15 ÷ 45 = 33.3%

Half-street games and the bluffing ratio

α = s ÷ (1 + s). Bluffs per value bet = α; share of the betting range that is bluffs = α ÷ (1 + α); minimum defense = 1 − α = 1 ÷ (1 + s)

Poker game theory: optimal and exploitative strategy

Equilibrium: each strategy is a best reply to the other. Pot-sized bet: one bluff for every two value bets, so bluffs are bet ÷ (pot before the bet + 2 × bet) = 1 ÷ 3 of the bets

The AKQ game: poker game theory with three cards

Pot P (more than 1 bet), bet 1: bluff the Q with frequency 1 ÷ (P + 1); call with the K with frequency (P − 1) ÷ (P + 1)

The [0,1] game: continuous hands

α = 1 ÷ (P + 1). Caller calls with the best c = 2(1 − α) ÷ ((1 + α)(2 − α)); bettor value-bets the best c ÷ 2 and bluffs the worst α × c ÷ 2

The road to poker: several streets

Same fraction f each street: pot × (1 + 2f)^n = pot + 2 × stack

Short stacks, tournaments and bankroll

Heads-up with high blinds: jam or fold

EV of jamming compared with folding, stacks S bb, blinds 0.5 and 1: P(fold) × 1.5 + P(call) × [equity × 2S − (S − 0.5)]

Tournaments: chips aren't cash

ICM: P(1st) = stack ÷ total chips, then the same for the remaining places; equity = Σ P(place) × prize. Required equity to call (ICM) = loss ÷ (loss + gain)

Poker variance: why short-term results fluctuate

After N hands: mean = win rate × N/100; spread = 1.96 × SD × √(N/100); measured win rate ± 1.96 × SD ÷ √(N/100)

Poker risk of ruin: bankroll and variance

Risk of ruin ≈ e^(−2 w B ÷ σ²); bankroll for a target risk r: B = −σ² × ln(r) ÷ (2w)

Kelly criterion in poker: sizing your bankroll risk

Kelly fraction f* = (b × p − q) ÷ b for odds b to 1, win chance p, lose chance q = 1 − p. Growth = p ln(1 + b f) + q ln(1 − f)

Bankroll rules: buy-ins, moving up and moving down

Buy-ins = bankroll ÷ (100 × big blind): 1,000 ÷ (100 × 0.5) = 20. Move down below rule × 100 × big blind