Chapter 9 of Essential Poker Math by Alton Hardin: Expected Value (EV) Calculations

Poker is a game of incomplete information and variance. While luck plays a role in short-term results, skilled players make profitable decisions based on Expected Value (EV) calculations. Understanding EV allows players to assess whether a particular play will be profitable in the long run, even if it doesn’t always result in an immediate win.

In Chapter 9 of Essential Poker Math, Alton Hardin explains that EV is a mathematical measure of how much a player expects to win or lose, on average, over time based on specific poker situations. Rather than focusing on short-term outcomes, poker players must evaluate the long-term profitability of their decisions using EV calculations.

By mastering EV, players can consistently make mathematically sound choices, leading to increased profitability and a greater understanding of poker strategy.

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What is Expected Value (EV) in Poker?

Expected Value (EV) represents the average outcome of a decision over a large number of repetitions. Every action in poker—betting, calling, folding, bluffing—has an associated EV, which can be either:

• Positive (+EV): A profitable decision in the long run.

• Negative (-EV): An unprofitable decision in the long run.

• Neutral (EV = 0): A break-even decision.

EV Formula

EV = (\text{Win %} \times \text{Win Amount}) – (\text{Lose %} \times \text{Loss Amount})

This equation accounts for:

1. The probability of winning (Win%) and how much is won when successful.

2. The probability of losing (Lose%) and how much is lost when unsuccessful.

The key takeaway is that an individual hand does not matter—what matters is the expected profit over thousands of hands.

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Components of an EV Calculation

To calculate EV, we need three key pieces of information:

1. Probability of Winning and Losing

   • The likelihood that your hand will win or lose.

   • Example: If you hold A♠ K♠ against Q♦ J♦, your preflop equity is 65% to win.

2. Amount Won When Successful

   • The total amount of money you win if your hand holds up.

3. Amount Lost When Unsuccessful

   • The amount of money you lose when your opponent wins.

By plugging these values into the EV formula, we can determine whether a play is profitable.

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Example 1: EV Calculation for a Call

Scenario: Flush Draw on the Turn

You are on the turn holding A♠ 10♠, and the board is K♠ 7♠ 2♣ J♦. Your opponent bets $50 into a $100 pot, making the total pot $150.

Question: Should you call the $50 bet with a flush draw?

Step 1: Determine Probabilities

• You have 9 outs to complete your flush.

• There are 46 unknown cards (deck minus your 2 cards and 4 board cards).

• Your probability of hitting a flush on the river:

  • Flush Completes: 946\frac{9}{46}469​ = 19.6% (~20%)

  • Flush Misses: 3746\frac{37}{46}4637​ = 80%

Step 2: Calculate Win and Loss Amounts

• If you win, you collect the $150 pot.

• If you lose, you lose the $50 call.

Step 3: Apply the EV Formula

$$EV=(Win\%\times WinAmount)-(Lose\%\times LossAmount)$$ $$EV=(0.20\times 150)-(0.80\times 50)$$ $$EV=30-40$$ $$EV=-10$$

Conclusion: This call is -EV (unprofitable). Over many repetitions, you will lose $10 on average by making this call. Unless you have implied odds (potential future winnings if you hit your flush), this is a mathematically incorrect call.

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Example 2: EV Calculation for a Bluff

Scenario: Bluffing the River

You are heads-up on the river with 8♠ 9♠ on a board of K♠ J♣ 5♦ 4♠ 2♥. The pot is $200, and you decide to bluff $100.

Your opponent will fold 40% of the time and call 60% of the time.

Question: Is this bluff profitable?

Step 1: Determine Probabilities

• Opponent folds (wins pot) = 40%

• Opponent calls (loses bluff) = 60%

Step 2: Calculate Win and Loss Amounts

• If the bluff succeeds, you win the $200 pot.

• If the bluff fails, you lose your $100 bet.

Step 3: Apply the EV Formula

$$EV=(Win\%\times WinAmount)-(Lose\%\times LossAmount)$$ $$EV=(0.40\times 200)-(0.60\times 100)$$ $$EV=80-60$$ $$EV=+20$$

Conclusion: This bluff is +EV (profitable). Over time, you will make an average of $20 per bluff in this spot.

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The Importance of EV in Poker Decision-Making

Understanding EV allows players to focus on long-term profitability rather than short-term results. Even if a decision loses in one hand, what matters is whether it is mathematically profitable over many hands.

Key Takeaways:

1. +EV decisions win money over time, even if they don’t always succeed.

2. -EV decisions lose money over time, even if they sometimes win in the short run.

3. Variance (short-term luck) does not change the correctness of an EV calculation.

By making consistent +EV decisions, a player ensures long-term success in poker.

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Applying EV to Different Poker Situations

EV calculations apply to all aspects of poker strategy. Below are some common situations where EV can be used.

1. EV of Calling an All-In

When facing an all-in bet, you should call if your equity justifies the risk.

Example:

• Pot: $300

• Opponent all-in bet: $100

• Your call: $100

• Required win rate:

100300+100=25%\frac{100}{300+100} = 25\%300+100100​=25%

If your hand equity is greater than 25%, calling is profitable.

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2. EV of Raising Instead of Calling

If you have a draw, sometimes raising is more +EV than calling, especially if it generates fold equity.

Example:

You hold J♠ 10♠ on a board of K♠ Q♣ 3♦.

• Calling gives you a 31% chance to improve.

• Raising forces folds 40% of the time, increasing profitability.

By applying EV, you can determine the optimal aggressive play.

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Common Mistakes in EV Calculations

1. Ignoring Future Betting Rounds

EV calculations should account for implied odds—additional money that can be won in future streets.

2. Focusing Only on Short-Term Results

Many players judge decisions based on a single outcome. EV calculations focus on long-term profitability.

3. Miscalculating Pot Odds and Equity

Accurate EV calculations require correct probability estimates. Using tools like Equilab or PokerStove helps refine hand equity calculations.

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Conclusion: Mastering EV for Long-Term Poker Success

Expected Value (EV) is one of the most important mathematical concepts in poker. By consistently making +EV decisions, poker players ensure long-term profitability, regardless of short-term variance.

Key Takeaways

✔ Use the EV formula to evaluate decisions.
✔ Make +EV plays consistently for long-term success.
✔ Don’t focus on short-term results—EV matters over time.
✔ Adjust for opponent tendencies, pot odds, and implied odds.

By applying EV-based thinking, poker players can shift their focus from individual hand outcomes to making mathematically sound choices, leading to higher win rates and sustained success in poker.