Poker Hand Probability & Expected Value Calculator
Calculate poker hand odds, probabilities, and expected value for Texas Hold'em. Analyze pot odds and hand strength to improve decision-making.
Poker Odds & Probability Explained
Introduction to Poker Odds
Poker is a game of incomplete information played with mathematical foundations. Understanding hand probabilities, pot odds, and expected value separates winning players from losing ones. This calculator helps you analyze poker situations numerically.
Core Concepts
Outs
An "out" is any card remaining in the deck that would improve your hand to likely winner.
- Flush draw: 9 outs (13 cards of suit minus 4 already seen)
- Open-ended straight draw: 8 outs (4 cards high, 4 cards low)
- Pair to trips/set: 3 outs (remaining cards of that rank)
- Two unpaired cards: 6 outs (could hit either card)
Pot Odds Formula
Example: Pot is $100, bet is $20
Pot Odds = $20 / ($100 + $20) = 16.67%
This means you need 16.67% chance of winning to break even. If your hand has 20% equity, calling is profitable.
The Rule of 4 and 2 (Quick Approximation)
- Flop to River: Outs × 4 ≈ percentage chance of hitting
- Turn to River: Outs × 2 ≈ percentage chance of hitting
Example: 9-out flush draw on flop
Chance of hitting: 9 × 4 = 36% (close to actual 35%)
Worked Example: Flush Draw Decision
Scenario: Texas Hold'em Cash Game
Your Hand: K♥ 9♥ (flush draw)
Board: 2♥ 7♥ J♦ (flop, two hearts)
Situation: Opponent bets $20 into $100 pot
Analysis:
- Your outs: 9 (remaining hearts minus the 2 you see)
- Probability to hit by river: 9 × 4 = 36% (rule of 4)
- Pot odds required: $20 / $120 = 16.67%
- Your equity: 36% > 16.67% required
- Decision: CALL is profitable
Expected Value:
- If you call 100 times in this spot:
- You win 36 times: 36 × $120 = $4,320
- You lose 64 times: 64 × -$20 = -$1,280
- Net: $4,320 - $1,280 = $3,040
- Average per hand: $3,040 / 100 = $30.40 expected value
Poker Decision Rule: Call if your equity > pot odds. Fold if your equity < pot odds.
Common Hand Probabilities
| Draw Type | Outs | Flop-River % | Turn-River % | Odds Against |
|---|---|---|---|---|
| Flush Draw (4 cards) | 9 | 35% | 18% | 1.86-1 |
| Open-Ended Straight | 8 | 32% | 16% | 2.13-1 |
| Pair (to 2 pair/trips) | 5 | 20% | 10% | 4.0-1 |
| Gutshot Straight | 4 | 16% | 8% | 5.25-1 |
| Two Overcards | 6 | 24% | 12% | 3.16-1 |
Expected Value (EV)
Limitations and Assumptions
- Assumes fixed equity: Actual equity changes with player actions
- No position considered: Position affects decision-making
- Doesn't account for: Implied odds (potential future wins), fold equity, table dynamics
- Simplified EV: Real poker requires considering ranges, not just equity %
- Variance not modeled: Short-term results vary significantly
Conclusion
Profitable poker decisions rest on understanding odds, probabilities, and expected value. Call when pot odds justify it based on your hand's equity. Fold when odds don't align with your chances. This mathematical approach, combined with position and player dynamics, forms the foundation of winning poker strategy.