The step-by-step math behind every EV calculation, from odds conversion to final answer.
Calculators are faster. Nobody is arguing that. But if you don't understand the math behind the output, you're trusting a number you can't verify, can't adjust, and can't apply in situations where the inputs aren't straightforward.
Knowing how to calculate expected value manually gives you three things a calculator alone doesn't. First, you can sanity-check results. If a calculator tells you a -150 favorite at 60% win probability is +EV, you should be able to confirm that in your head within a few seconds. Second, you can adapt the formula to multi-leg bets, alternate lines, or situations where the standard inputs don't quite fit. Third, you develop intuition. After calculating EV a few dozen times by hand, you start recognizing +EV spots instinctively, before you plug anything in.
This guide walks through the full calculation from raw American odds to a final EV number. Every step is shown explicitly so you can follow along with your own bets.
American odds are the standard format at US sportsbooks, but they're awkward to calculate with directly. Converting to decimal odds first simplifies every calculation that follows.
Decimal = 1 + (150 / 100) = 1 + 1.50 = 2.50
This means a $1 bet returns $2.50 total ($1 stake + $1.50 profit).
Decimal = 1 + (100 / 110) = 1 + 0.909 = 1.909
A $1 bet returns $1.909 total ($1 stake + $0.909 profit).
Decimal = 1 + (100 / 200) = 1 + 0.50 = 1.50
A $1 bet returns $1.50 total ($1 stake + $0.50 profit).
Quick reference: -110 = 1.909. -150 = 1.667. +100 = 2.00. +200 = 3.00. +300 = 4.00. Memorizing a few common conversions speeds up the rest of the process.
Implied probability is what the sportsbook's odds are telling you about how likely they believe the outcome is. More precisely, it's the breakeven win rate at those odds, the point where the bet neither makes nor loses money over time.
Implied = 100 / (150 + 100) = 100 / 250 = 0.400 (40.0%)
You need to win this bet at least 40% of the time to break even at +150.
Implied = 110 / (110 + 100) = 110 / 210 = 0.524 (52.4%)
You need to win at least 52.4% of the time at -110 just to break even. That extra 2.4% above 50% is the sportsbook's margin.
Implied = 200 / (200 + 100) = 200 / 300 = 0.667 (66.7%)
At -200, you need to win two out of every three bets to break even. The heavier the favorite, the higher the breakeven bar.
Implied probability includes the vig. The sportsbook's margin is baked into the number. On a standard -110/-110 market, both sides imply 52.4%, totaling 104.8%. The extra 4.8% is the vig. This means implied probability slightly overstates the sportsbook's true estimate of each side's chances.
| American Odds | Decimal Odds | Implied Probability | Breakeven Win Rate |
|---|---|---|---|
| +300 | 4.00 | 25.0% | 1 in 4 |
| +200 | 3.00 | 33.3% | 1 in 3 |
| +150 | 2.50 | 40.0% | 2 in 5 |
| +100 | 2.00 | 50.0% | 1 in 2 |
| -110 | 1.909 | 52.4% | 11 in 21 |
| -150 | 1.667 | 60.0% | 3 in 5 |
| -200 | 1.50 | 66.7% | 2 in 3 |
| -300 | 1.333 | 75.0% | 3 in 4 |
Once you have your decimal odds and your win probability estimate, the EV calculation is a single equation.
Breaking that down into its components for a $1 stake:
Normalizing to $1 first makes the math clean. The EV per dollar is the same regardless of stake size. Once you have EV per dollar, multiply by your actual stake to get EV in dollars.
If you prefer a single-line calculation that goes straight from inputs to answer:
If EV per dollar is positive, the bet is +EV. If negative, the sportsbook has the edge. That's all there is to it.
Three complete examples covering the most common bet types. Each one walks through every step from raw American odds to a final EV number.
You've handicapped a game and believe your side covers 56% of the time. The sportsbook is offering standard -110 odds.
Step 1: Convert to decimal.
Decimal = 1 + (100 / 110) = 1.909Step 2: Find implied probability.
Implied = 110 / 210 = 52.4%Step 3: Compare your estimate to implied.
Edge = 56.0% - 52.4% = +3.6%Step 4: Calculate EV per $1.
EV = (0.56 × 0.909) - (0.44 × 1.00) EV = 0.509 - 0.440 = +$0.069Result: +$0.069 per dollar wagered. On a $100 bet, you gain $6.90 on average. This is a clear +EV spot with a 3.6% edge.
You believe an underdog wins 40% of the time. The book is offering +180.
Step 1: Convert to decimal.
Decimal = 1 + (180 / 100) = 2.80Step 2: Find implied probability.
Implied = 100 / 280 = 35.7%Step 3: Compare.
Edge = 40.0% - 35.7% = +4.3%Step 4: Calculate EV per $1.
EV = (0.40 × 1.80) - (0.60 × 1.00) EV = 0.720 - 0.600 = +$0.120Result: +$0.12 per dollar wagered. On a $50 bet, you gain $6.00 on average. This bet loses more often than it wins, but the payout structure creates positive EV. This is why win rate alone is misleading.
You like the over on a hockey game. The book has it at -120.
Step 1: Convert to decimal.
Decimal = 1 + (100 / 120) = 1.833Step 2: Find implied probability.
Implied = 120 / 220 = 54.5%Step 3: Compare.
Edge = 58.0% - 54.5% = +3.5%Step 4: Calculate EV per $1.
EV = (0.58 × 0.833) - (0.42 × 1.00) EV = 0.483 - 0.420 = +$0.063Result: +$0.063 per dollar wagered. On a $100 bet, you gain $6.30 on average. A 3.5% edge on a juiced line.
A heavy favorite looks like a lock. The book has them at -250. You think they win 68% of the time.
Step 1: Convert to decimal.
Decimal = 1 + (100 / 250) = 1.40Step 2: Find implied probability.
Implied = 250 / 350 = 71.4%Step 3: Compare.
Edge = 68.0% - 71.4% = -3.4%Step 4: Calculate EV per $1.
EV = (0.68 × 0.40) - (0.32 × 1.00) EV = 0.272 - 0.320 = -$0.048Result: -$0.048 per dollar wagered. You lose about $4.80 per $100 bet on average. The team probably wins, but the price demands they win more often than you believe they do. This is the most common trap in sports betting: backing a likely winner at the wrong price.
If a $100 bet at +150 wins, you receive $250 total. Your profit is $150, not $250. The EV formula uses profit (what you gain above your stake), not total payout. Using total payout inflates your EV calculation and makes every bet look better than it is.
The conversion formulas for positive and negative American odds are different. Applying the positive formula to negative odds (or vice versa) will produce nonsense. If you get an implied probability above 100% or below 0%, you've used the wrong formula.
Implied probability tells you the breakeven point at those odds. It doesn't tell you how likely the outcome actually is. The entire EV framework depends on comparing your probability (your estimate of reality) against implied probability (the sportsbook's breakeven). If you use implied probability as your win estimate, EV will always come out to zero or negative because of the vig.
Small rounding errors compound. If you round decimal odds from 1.909 to 1.91, and round implied from 52.38% to 52%, those tiny differences can flip a marginal +EV assessment to breakeven or slightly negative. Carry at least three decimal places through the calculation and only round the final answer.
The hardest part isn't the math. The formula is arithmetic. The hard part is generating an accurate win probability. If your probability estimates are systematically wrong, every EV calculation built on them will be wrong too. The formula is only as good as the inputs.
Use a calculator when you're evaluating multiple bets quickly, when you want instant results during live betting windows, or when you need to compare several odds across different sportsbooks in a short time. Speed matters in those situations, and a calculator eliminates arithmetic errors.
Calculate by hand when you're learning the concept, when you want to build intuition for what different edges and payouts feel like, or when you're working through a non-standard situation (alternate lines, conditional bets, correlated parlays) where you need to modify the formula. The manual process forces you to understand every component instead of trusting a black box.
Most experienced bettors eventually do both. They calculate by hand enough to develop reliable instincts, then switch to a calculator for day-to-day use because it's faster. The hand calculation stays in reserve for when something doesn't look right and they need to verify the output.
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