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Implied Probability

Q: What's the formula for implied probability?

Decimal odds: implied = 1 / decimal. (Decimal 2.50 → 1/2.50 = 40%). American positive (+150): implied = 100 / (odds + 100). (+150 → 100/250 = 40%). American negative (-150): implied = |odds| / (|odds| + 100). (-150 → 150/250 = 60%). Fractional (5/2): implied = denominator / (numerator + denominator). (5/2 → 2/7 = 28.57%). Memorize the decimal formula; it's the universal one. All other odds formats are display conversions of decimal.

Q: Is implied probability the same as 'fair' probability?

No. Implied probability = what the price says before vig removal. Fair / no-vig probability = what the book estimates after vig removal. True probability = what would actually happen in nature (unknowable). Example: Lakers -150 implies 60%. After stripping 3.5% vig: 57.97% fair. The book believes Lakers win 58% of the time; the bettor's model might say 55%. Sharp bettors trade against the gap between fair probability and their model probability — that gap, multiplied by edge math, is expected value.

Q: How do I use implied probability to find +EV bets?

Four-step workflow: ① Strip vig from the market to get fair probabilities (multiplicative method works for two-way); ② Compare to your model — if you have a defensible probability estimate, the gap is your edge; ③ Convert edge to EV : EV% = (model_p × payout) − ((1 − model_p) × stake); ④ Filter for size : only act when edge > 1% AND you have a sample-size justification. Most bettors fail at step 2 — they don't have a real model, so they're comparing the book's estimate to their gut. The gap is then noise, not edge.

/ɪmˈplaɪd ˌprɒbəˈbɪlɪti/ · raw probability

Dice and probability — implied probability is the gateway to every sports betting model — Image: Pixabay Content License

From price to probability — one equation

Implied probability is the most fundamental transformation in sports betting. Every odds quote — American, decimal, fractional, Hong Kong, Malay, Indonesian — is a probability dressed up in display formatting. The formula is the same regardless of format:

# Universal formula
implied_probability = 1 / decimal_odds

# Examples
decimal 2.00  →  50.00%
decimal 1.50  →  66.67%
decimal 3.50  →  28.57%
decimal 1.10  →  90.91%
decimal 10.0  →  10.00%

Format conversion cheat sheet


Format	Formula → implied	Example → implied
Decimal	1 / decimal	2.50 → 40.00%
American (+)	100 / (odds + 100)	+150 → 40.00%
American (−)	\|odds\| / (\|odds\| + 100)	-150 → 60.00%
Fractional	denom / (num + denom)	5/2 → 28.57%
Hong Kong	1 / (HK + 1)	1.50 → 40.00%
Malay (+)	1 / (M + 1)	0.50 → 66.67%
Malay (−)	\|M\| / (\|M\| + 1)	-0.50 → 33.33%
Indonesian (+)	1 / (I + 1)	2.50 → 28.57%

Every professional bettor stores these conversions in a spreadsheet or function — there is no excuse for eyeballing.

The overround problem

Chalkboard with mathematical formulas — implied probability is the start, not the end — Image: Pixabay Content License

Raw implied probabilities always sum above 100% in a real sportsbook market. This is the structural fingerprint of vig.

# NFL spread: Cowboys -3 (-110) vs Eagles +3 (-110)
implied_DAL = 110/210 = 52.38%
implied_PHI = 110/210 = 52.38%
sum         = 104.76%   # overround

# Fair (no-vig) probability
fair_DAL = 52.38 / 1.0476 = 50.00%
fair_PHI = 52.38 / 1.0476 = 50.00%

The 4.76% overround is the book's commission. Until you remove it, you cannot honestly say what the book believes. Raw implied probabilities are inflated estimates — useful for quick orientation, dangerous for actual modeling.

Why this matters — the bettor's workflow

Read the price. Lakers -150, Celtics +130.
Compute implied probabilities. 60% and 43.48%. Sum = 103.48%.
Strip vig. Fair: 57.98% Lakers, 42.02% Celtics.
Compare to your model. If your model says Lakers win 55%, the book thinks they're 3 points stronger than your view — the bet is -EV.
Compute EV if betting Celtics. If your model says Celtics 45%: EV = 0.45 × 2.30 − 1 = +3.5%.

Without step 3 (vig removal), your edge calculation is biased by the overround. A bettor comparing model 45% to raw implied 43.48% would think the Celtics bet has 1.5% edge; in reality the fair-line edge is closer to 3%.

Three-way markets — overround stacks differently

Soccer 1X2 (home win / draw / away win), tennis Grand Slam outright (128 players), Formula 1 race winner — markets with more than two outcomes have stacked vig and implied probabilities that can sum to 105-130%.

# EPL match — Liverpool / Draw / Arsenal
LIV: 2.10  →  47.62%
DRW: 3.40  →  29.41%
ARS: 3.80  →  26.32%
sum:                  103.35%

# Power method (logit) for three-way fair
fair_LIV ≈ 46.5%
fair_DRW ≈ 28.2%
fair_ARS ≈ 25.3%

For three-way markets, the power method (logit transformation) produces a more accurate fair probability than the simple multiplicative method, because it accounts for favorite-longshot bias asymmetrically.

Favorite-longshot bias

Percentage chart — implied probability vs realized frequency — Image: Pixabay Content License

Decades of academic research (Snowberg & Wolfers 2010; Sauer 1998) document a persistent gap between implied probability and true probability across betting markets: longshots are systematically overpriced, favorites are systematically underpriced.

Horse racing: a horse priced at 100/1 (implied 1%) wins about 0.6% of races.
NFL futures: Super Bowl longshots at +10000 (implied 1%) historically deliver below 0.5% win rate.
Soccer first goalscorer: 25/1 longshots historically priced at 4% implied actually score about 2.5% of the time.

The bias is small in major two-way markets (NFL spreads, NBA totals) and large in futures and exotics. Practical implication: implied probability on longshots should be discounted 30-50% to estimate true probability.

The "no-arbitrage" interpretation

If the implied probabilities across all outcomes summed to exactly 100%, the market would be a fair-coin lottery — no profit for the book. The overround is what keeps the market structurally profitable. The corollary: a sportsbook's theoretical hold = overround percentage, assuming balanced action across outcomes.

This is why books cap exposure on each outcome. If 90% of bettors take the Cowboys at -110, the book's implied-probability-balanced 52.38%/52.38% becomes a real liability of 90%/10%. Vig only protects the book when action is balanced, so books move lines to attract opposing money — even if the line move doesn't reflect changing probability beliefs.

Implied probability and the closing line

The closing line — the final price before kickoff — is the market's consensus probability estimate. Closing implied probability is the cleanest single estimate of true probability in financial markets (Levitt 2004). Sharp bettors are measured by their ability to beat the closing implied probability, not the opening line — CLV (Closing Line Value) is the gold standard.

A bet placed at +140 (41.67% implied) on a market that closes at +120 (45.45% implied) represents a 3.78% probability swing in the bettor's favor — strong CLV evidence the bettor identified a mispriced market early.

Common implied-probability mistakes

Treating raw implied as fair. Most popular betting calculators show raw implied probability without vig removal. A +EV calculation using raw implied is biased low.
Assuming overround is evenly distributed. Books frequently load vig onto the favorite or the longshot asymmetrically. The Shin method handles this; the multiplicative does not.
Ignoring three-way markets. Bettors apply the two-way multiplicative formula to soccer 1X2 and get fair probabilities that don't actually sum to 100%.
Confusing implied with confidence. An implied probability of 60% is not "the book is 60% sure" — it's "the book's price reflects a fair probability of about 58% after vig removal."

Implied probability in arbitrage

An arbitrage opportunity exists when implied probabilities across two books on opposing sides sum to less than 100%. Example:

# DraftKings: Lakers +145 → implied 40.82%
# FanDuel:   Celtics -135 → implied 57.45%
40.82% + 57.45% = 98.27%

# Arb margin = 100% − 98.27% = 1.73%
# Bet sized inversely to implied probability locks profit
stake_LAL = $408.20
stake_BOS = $574.50
total stake = $982.70
guaranteed payout = $1,000
profit = $17.30  ≈ 1.76%

Pure arbitrages on US-licensed books rarely exceed 1-3% and rarely last more than minutes. Offshore + exchange combinations occasionally show 4-6% windows during sharp price disagreements.

Sources & further reading

Snowberg, Erik & Wolfers, Justin. "Explaining the Favorite-Longshot Bias: Is It Risk-Love or Misperceptions?" Journal of Political Economy, 2010.
Levitt, Steven D. "Why are gambling markets organised so differently from financial markets?" Economic Journal, 2004.
Štrumbelj, Erik. "On determining probability forecasts from betting odds." International Journal of Forecasting, 2014.
Sauer, Raymond D. "The Economics of Wagering Markets." Journal of Economic Literature, 1998.
Pinnacle Betting Resources — "How to calculate true probability and remove margin" (open documentation).

FAQ

What's the formula for implied probability?

Decimal odds: implied = 1 / decimal. (Decimal 2.50 → 1/2.50 = 40%). American positive (+150): implied = 100 / (odds + 100). (+150 → 100/250 = 40%). American negative (-150): implied = |odds| / (|odds| + 100). (-150 → 150/250 = 60%). Fractional (5/2): implied = denominator / (numerator + denominator). (5/2 → 2/7 = 28.57%). Memorize the decimal formula; it's the universal one. All other odds formats are display conversions of decimal.

Why do implied probabilities of both sides add to more than 100%?

Because the sportsbook embeds vig (commission) in the price. A fair coin flip would be priced at decimal 2.00 / 2.00 (50% / 50% summing to 100%). A sportsbook prices it at 1.909 / 1.909, summing to 104.76%. The 4.76% excess is the overround — the book's profit margin. To recover the book's actual probability estimate, divide each implied probability by the overround. This 'no-vig' or 'fair' probability is what you compare to your model. Three-way markets (soccer 1X2) have larger overrounds because vig stacks across three outcomes.

Is implied probability the same as 'fair' probability?

No. Implied probability = what the price says before vig removal. Fair / no-vig probability = what the book estimates after vig removal. True probability = what would actually happen in nature (unknowable). Example: Lakers -150 implies 60%. After stripping 3.5% vig: 57.97% fair. The book believes Lakers win 58% of the time; the bettor's model might say 55%. Sharp bettors trade against the gap between fair probability and their model probability — that gap, multiplied by edge math, is expected value.

What does implied probability tell me about a longshot?

Mostly it tells you the bookmaker's overround is wide and there's favorite-longshot bias. Empirical finance research (Snowberg & Wolfers 2010, 'Explaining the Favorite-Longshot Bias') shows that betting market longshots are systematically overvalued in implied probability vs. true probability. A horse priced +1500 (6.25% implied) actually wins ~5% of races on average — the book's overround compresses the longshot's price further. Practical takeaway: implied probability on a 50/1 longshot is usually inflated 25-50% above reality. Longshots are statistically the worst bets in sports betting.

How do I use implied probability to find +EV bets?

Four-step workflow: ① Strip vig from the market to get fair probabilities (multiplicative method works for two-way); ② Compare to your model — if you have a defensible probability estimate, the gap is your edge; ③ Convert edge to EV: EV% = (model_p × payout) − ((1 − model_p) × stake); ④ Filter for size: only act when edge > 1% AND you have a sample-size justification. Most bettors fail at step 2 — they don't have a real model, so they're comparing the book's estimate to their gut. The gap is then noise, not edge.

What's the difference between implied probability and inferred probability?

They're often used interchangeably, but precise usage differs. Implied probability is the direct conversion of odds: 1/decimal. Inferred probability is what you infer about the underlying event after considering overround, market depth, and time decay. A late-money line move from 1.90 to 1.85 changes implied probability from 52.6% to 54.1%, but the inferred probability could be higher still if you know that the move was driven by sharp action rather than public action. Inferred probability is implied probability filtered through market microstructure intuition. Pros track both.

// published 2026-05-23 · updated 2026-05-23 · OddsCipher Desk