Expected Value Calculator

What Is Expected Value?

Expected Value (EV) is the probability-weighted average of all possible outcomes of a decision. It tells you what you can expect to gain or lose "on average" if you repeated a decision many times. Crucially, it's not what will happen on any single trial - it's the mathematical long-run average.

The EV Formula:

EV = Sigma (P_i x V_i)

Where:

• P_i = Probability of outcome i
• V_i = Value (profit/loss) of outcome i
• Sum over all possible outcomes

Simple Example: A Fair Coin Flip Bet

You bet $10 on heads. Win: +$10, Lose: -$10

EV = (0.50 x $10) + (0.50 x -$10) = $5 - $5 = $0

This is a "fair" bet with zero expected value. Neither side has an edge.

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Positive vs. Negative EV

+EV (Positive Expected Value):

• Your expected profit is positive
• You have a mathematical edge
• You "should" take this bet/decision (ignoring other factors)
• Professional gamblers ONLY take +EV bets

-EV (Negative Expected Value):

• Your expected profit is negative
• The house/opponent has the edge
• Mathematically, you should decline this bet
• Every casino game is -EV for the player

Zero EV:

• Fair bet with no edge to either side
• Rare in real-world scenarios (someone usually has an edge)

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Why EV Matters More Than "Winning"

Consider two betting strategies:

Strategy A: Win 70% of the time, average win $100, average loss $300 EV = (0.70 x $100) + (0.30 x -$300) = $70 - $90 = -$20

Strategy B: Win 35% of the time, average win $400, average loss $100 EV = (0.35 x $400) + (0.65 x -$100) = $140 - $65 = +$75

Strategy A wins more often but is a losing strategy (-$20 EV). Strategy B loses more often but is a winning strategy (+$75 EV).

This is why professional gamblers focus on EV, not win rate.

Casino Games: The House Always Has +EV

Every casino game is designed to give the house positive expected value:

The only way to have +EV in a casino is through advantage play: card counting in blackjack, exploiting promotional offers, or finding rare game errors.

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Sports Betting: Finding +EV Lines

In sports betting, your EV depends on whether you're beating the implied probability:

Example: Betting on Team A at +150

Implied probability at +150 = 100 / (150 + 100) = 40%

If your analysis shows Team A has a 50% chance of winning:

• Win: 50% x $150 = $75
• Lose: 50% x -$100 = -$50
• EV = $75 - $50 = +$25 per $100 bet

This is a +EV bet. If your probability estimate is accurate, you'll profit long-term.

If Team A only has a 35% chance (below implied 40%):

• Win: 35% x $150 = $52.50
• Lose: 65% x -$100 = -$65
• EV = $52.50 - $65 = -$12.50 per $100 bet

This is -EV. The odds don't offer enough value.

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Poker: Every Decision Has an EV

In poker, calculating EV determines whether to bet, call, raise, or fold:

Example: Calling a River Bet

• Pot is $200, opponent bets $100
• You must call $100 to win $300
• You need to win 25% to break even (call $100 to win $400 total pot)

If you estimate a 35% chance your hand is best:

• Win: 35% x $300 = $105
• Lose: 65% x -$100 = -$65
• EV = $105 - $65 = +$40

Calling has +$40 EV - you should call even though you'll lose 65% of the time.

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The Gambling EV Framework

For any gambling decision:

1. Identify all outcomes (win, lose, push, partial win, etc.)
2. Calculate probability of each (from odds, pot odds, or your analysis)
3. Determine payout for each (including your stake)
4. Compute EV using the formula
5. Compare to alternatives (fold vs. call vs. raise)
6. Factor in variance (can you handle the swings?)

Investment Analysis: Risk-Adjusted Returns

When evaluating investments, EV helps quantify expected returns under different scenarios:

Example: Stock Investment Analysis

You're considering investing $10,000 in a growth stock:

• 30% chance: Bull market, stock gains 40% (+$4,000)
• 50% chance: Flat market, stock gains 10% (+$1,000)
• 20% chance: Bear market, stock loses 30% (-$3,000)

EV = (0.30 x $4,000) + (0.50 x $1,000) + (0.20 x -$3,000) EV = $1,200 + $500 - $600 = +$1,100

The expected return is $1,100 (11% on $10,000).

Compare to a bond yielding 5% ($500 guaranteed):

• Stock EV: +$1,100 with variance
• Bond EV: +$500 guaranteed

The stock has higher EV but also higher risk.

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Project Evaluation: Go/No-Go Decisions

Example: Launching a New Product

Investment required: $500,000

Scenario analysis:

• 20% chance: Major success, net profit $2,000,000
• 35% chance: Moderate success, net profit $600,000
• 30% chance: Break even, net profit $0
• 15% chance: Failure, net loss $500,000

EV = (0.20 x $2M) + (0.35 x $600K) + (0.30 x $0) + (0.15 x -$500K) EV = $400K + $210K + $0 - $75K = +$535,000

Positive EV suggests proceeding, but the 15% chance of total loss must be weighed against company risk tolerance.

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Hiring Decisions

Example: Hiring a Senior Engineer

Two candidates with different risk profiles:

Candidate A (Safe choice):

• 80% chance: Good performance, adds $200K value
• 20% chance: Average performance, adds $100K value
• EV = (0.80 x $200K) + (0.20 x $100K) = $180K

Candidate B (High-potential):

• 40% chance: Star performer, adds $500K value
• 35% chance: Good performance, adds $200K value
• 25% chance: Poor fit, adds $50K value (after hiring costs)
• EV = (0.40 x $500K) + (0.35 x $200K) + (0.25 x $50K) = $282.5K

Candidate B has higher EV ($282.5K vs $180K) but also higher variance.

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Insurance and Risk Management

Should You Buy Extended Warranty?

$2,000 laptop, $300 extended warranty, covers 3 years:

• 5% chance: Major failure, $800 repair saved
• 10% chance: Minor issue, $200 repair saved
• 85% chance: No problems, $0 benefit

EV = (0.05 x $800) + (0.10 x $200) + (0.85 x $0) - $300 EV = $40 + $20 + $0 - $300 = -$240

The warranty has -$240 EV. From a pure EV standpoint, skip it. But if you can't afford an $800 repair, the -EV might be worth the peace of mind (utility consideration).

Understanding Variance

Variance measures how spread out the outcomes are from the expected value. Two bets can have the same EV but wildly different variance:

Low Variance Bet:

• 90% chance: Win $11
• 10% chance: Lose $99
• EV = (0.90 x $11) + (0.10 x -$99) = $9.90 - $9.90 = $0

High Variance Bet:

• 1% chance: Win $9,900
• 99% chance: Lose $100
• EV = (0.01 x $9,900) + (0.99 x -$100) = $99 - $99 = $0

Both have $0 EV, but the high variance bet is far riskier. You could lose 99 times before winning once.

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Calculating Variance and Standard Deviation

Variance Formula: Variance = Sigma [P_i x (V_i - EV)^2]

Standard Deviation: SD = Square Root of Variance

Standard deviation tells you how much typical outcomes deviate from EV.

Example:

• 50% chance: +$100
• 50% chance: -$100
• EV = $0

Variance = (0.50 x (100 - 0)^2) + (0.50 x (-100 - 0)^2) Variance = (0.50 x 10,000) + (0.50 x 10,000) = 10,000 SD = sqrt(10,000) = $100

This means outcomes typically deviate $100 from the EV.

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When to Accept Higher Variance

Accept high variance +EV when:

1. You have sufficient bankroll to survive losing streaks
2. Sample size is large (many repetitions to realize EV)
3. Individual outcomes don't ruin you (can afford worst case)
4. The EV edge justifies the risk (higher edge = more variance acceptable)

Avoid high variance when:

1. Bankroll is limited relative to potential losses
2. One-time decisions (single trial, variance dominates)
3. Loss aversion is high (psychological cost of losing)
4. Opportunity cost of ruin is high (can't recover from bankruptcy)

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Bankroll Management: Surviving Variance

Even with +EV, you can go broke through variance. The Kelly Criterion helps size bets:

Kelly Formula: Optimal Bet % = Edge / Odds

Or more precisely: f* = (p x b - q) / b

Where:

• f* = fraction of bankroll to bet
• p = probability of winning
• q = probability of losing (1 - p)
• b = odds received (profit per unit risked)

Example: +150 odds, 50% true probability

Edge = 50% - 40% (implied) = 10% b = 1.5 (profit of $1.50 per $1 risked) f* = (0.50 x 1.5 - 0.50) / 1.5 = 0.25 / 1.5 = 16.7%

Kelly says bet 16.7% of bankroll. Most professionals use "Half Kelly" or "Quarter Kelly" to reduce variance.

Career Decisions: Job Offer Comparison

Example: Two Job Offers

Job A (Stable company):

• 95% chance: Stay employed, earn $120K/year
• 5% chance: Layoff after 6 months, earn $60K
• EV = (0.95 x $120K) + (0.05 x $60K) = $117K

Job B (Startup):

• 30% chance: Startup succeeds, earn $200K (salary + equity)
• 50% chance: Startup struggles, earn $100K
• 20% chance: Startup fails, earn $40K (6 months)
• EV = (0.30 x $200K) + (0.50 x $100K) + (0.20 x $40K) = $118K

Similar EV, but vastly different risk profiles. Choose based on your risk tolerance and financial situation.

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Education ROI

Example: MBA Program Analysis

Cost: $150,000 (tuition + opportunity cost)

Expected outcomes:

• 40% chance: Land high-paying job, +$50K/year salary increase
• 35% chance: Moderate improvement, +$25K/year increase
• 25% chance: No significant change, +$5K/year increase

Over 10 years (simplified, ignoring time value):

• 40%: $500K additional earnings - $150K cost = +$350K
• 35%: $250K additional - $150K cost = +$100K
• 25%: $50K additional - $150K cost = -$100K

EV = (0.40 x $350K) + (0.35 x $100K) + (0.25 x -$100K) EV = $140K + $35K - $25K = +$150K

The MBA has +$150K EV over 10 years, suggesting it's worth pursuing from a pure financial standpoint.

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Time Investment Decisions

Example: Learning a New Skill

Investing 200 hours to learn programming:

• 25% chance: Career pivot, earn $30K more annually
• 40% chance: Side income from freelancing, $10K/year
• 35% chance: Personal satisfaction only, $0 monetary benefit

Assume these benefits persist for 5 years:

• 25%: $150K additional earnings
• 40%: $50K additional earnings
• 35%: $0 additional earnings

EV = (0.25 x $150K) + (0.40 x $50K) + (0.35 x $0) = $37.5K + $20K = $57.5K

200 hours of learning has $57.5K expected value = $287.50/hour of study time.

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Negotiation and Offers

Example: Salary Negotiation

Current offer: $100,000

If you negotiate:

• 20% chance: Offer increased to $115,000
• 50% chance: Offer stays at $100,000
• 30% chance: Offer reduced to $95,000 or withdrawn

EV of negotiating = (0.20 x $115K) + (0.50 x $100K) + (0.30 x $95K) EV = $23K + $50K + $28.5K = $101.5K

EV of not negotiating = $100,000

Negotiating has +$1,500 EV. But if "withdrawn" is possible and devastating, the 30% downside might outweigh the math.

Mistake 1: Ignoring Probabilities (Wishful Thinking)

People often overweight desirable outcomes and underweight undesirable ones:

Reality check: "There's a chance I could win big!" is not analysis. If there's a 1% chance of winning $10,000 and 99% chance of losing $200:

EV = (0.01 x $10,000) + (0.99 x -$200) = $100 - $198 = -$98

That "chance to win big" has -$98 EV. The math doesn't care about hope.

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Mistake 2: Sunk Cost Fallacy

Previous investments should not affect future EV calculations.

Example: You've already lost $5,000 at the poker table.

Incorrect thinking: "I need to win back my losses, so I should play looser."

Correct thinking: Each new hand has its own independent EV. Your past losses don't change the math of future decisions. If a fold is +EV, fold regardless of what you've lost.

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Mistake 3: Gambler's Fallacy

Believing past outcomes affect future independent events.

Example: Roulette has landed on red 10 times in a row.

Incorrect: "Black is due! EV of betting black is now higher!"

Correct: Each spin is independent. The probability of black is still 18/37 (single zero) regardless of history. The EV hasn't changed.

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Mistake 4: Confusing EV with Certainty

+EV doesn't mean you'll win. -EV doesn't mean you'll lose.

A +$50 EV bet that loses 60% of the time will show losses most of the time in small samples. Only over many repetitions does EV manifest. Professional gamblers understand they'll have losing days, weeks, even months - but positive EV compounds over thousands of decisions.

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Mistake 5: Ignoring Variance When It Matters

Scenario: You have $10,000 and need $15,000 for a medical procedure.

Option A: 50% chance of doubling to $20,000, 50% chance of losing to $5,000 Option B: 100% chance of keeping $10,000

EV of Option A = $12,500 EV of Option B = $10,000

Option A has higher EV, but if you lose, you're ruined. Here, the certainty of Option B (or finding a lower-variance path to $15,000) might be correct despite lower EV. Utility and survival matter.

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Mistake 6: Not Accounting for All Outcomes

Every EV calculation must include ALL possible outcomes, including unlikely ones:

Incomplete analysis: "If I bet $100 on this +200 underdog with 40% true probability, my EV is +$20!"

Complete analysis: What if the game gets cancelled? What if there's a push? What if your probability estimate is wrong? Robust EV calculations account for all scenarios, including model uncertainty.

Decision Trees: Multi-Stage EV

Real decisions often have multiple stages with conditional probabilities:

Example: Startup Fundraising

Stage 1: Pitch to VCs

• 30% get term sheet, proceed to Stage 2
• 70% no term sheet, company value = $0

Stage 2: Due diligence

• If Stage 1 success: 60% close funding, 40% deal falls through

Stage 3: With funding

• 20% major exit ($10M founder value)
• 40% moderate exit ($2M founder value)
• 40% failure ($0 value)

Calculating EV:

Path to major exit: 0.30 x 0.60 x 0.20 = 3.6% probability, $10M value Path to moderate exit: 0.30 x 0.60 x 0.40 = 7.2% probability, $2M value All other paths: 89.2% probability, $0 value

EV = (0.036 x $10M) + (0.072 x $2M) + (0.892 x $0) EV = $360K + $144K = $504K

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Conditional EV: Bayesian Updating

Your EV estimates should update as new information arrives:

Example: Sports Betting with Injury News

Initial assessment: Team A has 55% chance to win at +120 (implied 45%) Initial EV = (0.55 x $120) - (0.45 x $100) = $66 - $45 = +$21

Breaking news: Star player doubtful with injury.

Updated assessment: Team A now has 40% chance to win. Updated EV = (0.40 x $120) - (0.60 x $100) = $48 - $60 = -$12

The same bet went from +EV to -EV with new information.

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Game Theory: EV Against Thinking Opponents

Against opponents who adapt, EV depends on their strategy:

Example: Poker Bluffing

If you never bluff:

• Opponents always fold when you bet big
• You only get value from weak hands calling
• EV of betting decreases

If you always bluff:

• Opponents always call
• Your bluffs always lose
• EV of betting decreases

Optimal strategy: Mix bluffs and value bets so opponents can't exploit you. Game theory optimal (GTO) play maximizes EV against perfect opponents.

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Monte Carlo Simulation: Complex EV Estimation

When outcomes are too complex for closed-form solutions, simulate thousands of scenarios:

Example: Options Trading

• Stock can move anywhere from -30% to +50%
• You hold a call option struck at current price
• Many variables: volatility, time decay, interest rates

Process:

1. Generate 10,000 random price paths
2. Calculate option payoff for each path
3. Average all payoffs = Expected Value

This Monte Carlo approach handles complexity that formulas can't.

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Expected Value vs. Expected Utility

For personal decisions, "utility" (subjective value) often differs from dollar value:

• $1 million to a billionaire vs. someone in poverty
• Winning $100 vs. losing $100 (loss aversion)
• Certainty vs. equivalent gamble (risk aversion)

Example: Would you accept a bet with 50% chance of winning $200,000 and 50% chance of losing $100,000?

EV = (0.50 x $200K) + (0.50 x -$100K) = +$50K

Most people reject this +EV bet because losing $100K hurts more than gaining $200K helps. This is rational behavior based on utility, not EV.

Professional gamblers/traders train themselves to be "utility-neutral" - to make decisions purely on EV regardless of emotional impact.