Implied Probability Calculator

How do you use the Implied Probability Calculator?

This UK-focused tool converts your odds into implied probabilities, shows the built-in bookmaker margin (overround), and can normalise prices to “no‑vig” (fair) probabilities in one click. It accepts decimal (UK default), fractional (e.g., 6/4), and American (+150, −120) formats. Use it to sanity‑check football markets, compare sportsbooks, and plan staking.

Step-by-step: enter decimal, fractional, or American odds

Follow this quick workflow to get accurate implied probabilities and optional “no‑vig” prices.

• Select your odds format (Decimal, Fractional, or American).

Enter the price(s):

• Decimal: e.g., 1.80, 2.10, 3.40
• Fractional: a/b, e.g., 4/5, 6/4, 9/2
• American: +150, −110, −200

Choose market type:

• Two‑way (e.g., tennis match winner)
• Three‑way (e.g., football 1X2: Home/Draw/Away)

Review outputs:

• Implied probability for each outcome
• Total overround (bookmaker margin)
• Optional “no‑vig” (fair) probabilities and fair odds

• (Optional) Toggle “Remove margin (vig)” to normalise probabilities so they sum to 100% and display fair odds.

Formulas used by the calculator:

• Decimal to probability: P = 1 / Decimal
• Fractional (a/b) to probability: P = b / (a + b)

American:

• For positive (e.g., +150): P = 100 / (US + 100)
• For negative (e.g., −120): P = −US / (−US + 100)

Worked examples:

Input odds	Format	Implied probability
1.80	Decimal	55.56%
4/5	Fractional	55.56%
−125	American	55.56%
2.50	Decimal	40.00%
6/4	Fractional	40.00%
+150	American	40.00%

Can I remove the bookmaker margin (vig) in one click?

Yes. The calculator can normalise a market’s implied probabilities to remove the overround so they sum to 100%. This reveals the “fair” (no‑vig) probabilities and fair odds implied by the posted prices.

Two‑way example (tennis match):

• Posted decimal odds: Player A 1.80, Player B 2.10

Implied probabilities (with vig):

• A: 1/1.80 = 55.5556%
• B: 1/2.10 = 47.6190%
• Sum = 103.1746% (overround ≈ 3.1746%)

No‑vig normalisation (divide each by the sum 103.1746%):

• A fair probability: 55.5556% / 103.1746% = 53.8462% → fair odds 1 / 0.538462 = 1.8571
• B fair probability: 47.6190% / 103.1746% = 46.1538% → fair odds 1 / 0.461538 = 2.1667

Three‑way example (football 1X2):

• Posted decimal odds: Home 2.20, Draw 3.40, Away 3.30

Implied probabilities:

• Home: 1/2.20 = 45.4545%
• Draw: 1/3.40 = 29.4118%
• Away: 1/3.30 = 30.3030%
• Sum = 105.1693% (overround ≈ 5.1693%)

No‑vig normalisation (divide each by 105.1693%):

• Home fair probability ≈ 43.25% → fair odds ≈ 2.31
• Draw fair probability ≈ 27.98% → fair odds ≈ 3.57
• Away fair probability ≈ 28.77% → fair odds ≈ 3.47

Notes:

• Normalisation preserves the relative prices while removing margin.
• For multi‑outcome markets (totals with multiple lines, correct score, etc.), the same method applies: compute each implied probability, sum them, then divide each by the sum.

What is an implied probability chart and how do I read it?

An implied probability chart maps odds to their equivalent chances of occurring. It helps you quickly see whether a price suggests a 60% or a 25% chance, compare offers across bookmakers, and spot where “fair” prices would sit once the margin is removed. Read across a row to match Decimal, Fractional, and American odds to the same probability.

Decimal	Fractional (UK)	American	Implied probability
1.20	1/5	−500	83.33%
1.50	1/2	−200	66.67%
1.80	4/5	−125	55.56%
2.00	1/1 (Evens)	+100	50.00%
2.50	6/4	+150	40.00%
3.00	2/1	+200	33.33%
4.00	3/1	+300	25.00%
5.00	4/1	+400	20.00%
10.00	9/1	+900	10.00%
101.00	100/1	+10000	0.99%

How to interpret the chart:

• Lower decimal odds mean higher implied chance; higher odds mean lower chance.
• “Evens” (1/1, 2.00, +100) always equals 50% implied probability.
• Use the chart to sanity‑check value: if your model says a team wins 45% but the market implies 40% (2.50), the posted price may be favourable before considering vig.
• Remember: implied probability reflects the price, not the true chance; margins and market biases can shift these numbers.

Which odds formats are supported and what are the formulas?

Our calculator supports Decimal (UK default), UK Fractional (e.g., 4/5, 7/2), and American (+150, −110) odds. All three describe the same thing in different notations: the price and its implied probability. Below are the exact formulas, worked examples, and the correct way to adjust for bookmaker margin (overround) to obtain “no‑vig” fair probabilities and fair odds.

Decimal odds to percentage: with and without margin

Decimal odds include your stake. To convert a single Decimal price D to implied probability P, use P = 1 / D. In a market with multiple outcomes, the sum of all implied probabilities typically exceeds 100% due to margin (overround). To remove the margin, normalise each probability by the total sum.

• Single outcome conversion: P = 1 / D (e.g., D = 1.80 → P = 0.5556 = 55.56%).
• Market overround: Overround = Σ(1 / Dᵢ). Example two‑way market D = {1.91, 1.91} → ΣP = 1/1.91 + 1/1.91 = 1.0471 (104.71%).
• No‑vig normalisation: Pᵢ′ = Pᵢ / ΣP, so probabilities sum to 100%. Fair decimal odds Dᵢ′ = 1 / Pᵢ′.

• Example (two‑way): D = {1.80, 2.10}. Raw P = {55.5556%, 47.6190%}; ΣP = 103.1746%.

No‑vig P′:

• Outcome A: 55.5556 / 103.1746 = 53.8462% → fair odds 1 / 0.538462 = 1.8571
• Outcome B: 47.6190 / 103.1746 = 46.1538% → fair odds 1 / 0.461538 = 2.1667

How do you convert UK fractional odds to implied probability?

Fractional odds are written a/b. They show net profit a for every stake b. The implied probability is P = b / (a + b). To compare across markets or remove margin, first convert to Decimal using D = 1 + (a / b), compute each probability as 1 / D, sum them, and normalise.

• Formula: P = b / (a + b); Decimal equivalent: D = 1 + a / b.

Examples:

• 4/5 → P = 5 / (4 + 5) = 0.5556 = 55.56%; D = 1 + 4/5 = 1.80
• 7/2 → P = 2 / (7 + 2) = 0.2222 = 22.22%; D = 1 + 7/2 = 4.50
• 10/11 → P = 11 / (10 + 11) ≈ 0.5238 = 52.38%; D = 1 + 10/11 ≈ 1.9091

• Notes: “Evens” is 1/1 (P = 50%, D = 2.00). “Odds‑on” means a < b (P > 50%).

American odds (+/−): what’s the formula and common traps?

American odds express profit on a £100 stake when positive, or the stake needed to win £100 when negative. Convert to implied probability using different formulas for positive and negative lines.

Formulas:

• US > 0: P = 100 / (US + 100); Decimal D = 1 + (US / 100)
• US < 0: P = |US| / (|US| + 100); Decimal D = 1 + (100 / |US|)

Examples:

• +150 → P = 100 / 250 = 0.40 = 40.00%; D = 1 + 1.50 = 2.50
• −110 → P = 110 / 210 ≈ 0.5238 = 52.38%; D = 1 + 100/110 ≈ 1.9091
• −200 → P = 200 / 300 = 0.6667 = 66.67%; D = 1 + 100/200 = 1.50

Common traps:

• Sign confusion: the minus sign (favourite) does not mean “negative” probability; it usually implies P > 50%.
• Forgetting absolute values: for negatives, use |US| in both numerator and denominator.
• Mixing stake and profit: American odds are quoted on a £100 reference; Decimal always includes returned stake.
• Rounding drift: round only at the end to avoid small errors compounding across legs or normalisation.

The table below cross‑checks the three formats using consistent formulas. Probabilities are rounded to two decimal places, and Decimal equivalents to four where needed.

Format	Example odds	Implied probability	Decimal equivalent
Decimal	1.91	52.36%	1.9100
Fractional (UK)	5/4	44.44%	2.2500
American	−110	52.38%	1.9091
American	+150	40.00%	2.5000
Fractional (UK)	4/6	60.00%	1.6667
Decimal	3.20	31.25%	3.2000

How do you calculate parlay/acca implied probability?

In UK betting, a “parlay” is commonly called an “accumulator” (acca). The combined decimal odds are the product of the legs, and the acca’s implied probability equals the product of the legs’ implied probabilities—provided the legs are independent. You can also de‑vig each leg first (normalise within its market) to estimate a “fair” acca probability and fair odds. The steps and worked example below show both approaches.

Parlay implied probability calculator: quick workflow

Use this workflow to compute an acca’s implied probability, with and without bookmaker margin (vig):

• Enter each leg’s odds in your chosen format (Decimal, Fractional, or American).

Convert each leg to implied probability Pᵢ:

• Decimal D → P = 1 / D
• Fractional a/b → P = b / (a + b)
• American: +X → P = 100 / (X + 100); −X → P = X / (X + 100)

(Optional but recommended) For each leg’s market, remove margin by normalising:

• Compute all outcomes’ implied probabilities in that market and sum ΣP.
• Fair probability for your selection: P′ = P / ΣP; fair odds D′ = 1 / P′.

Calculate the acca:

• With vig: P_acca = Π Pᵢ; Decimal_acca = Π Dᵢ
• No‑vig (fair): P′_acca = Π Pᵢ′; Decimal′_acca = 1 / P′_acca

• Returns (settled at posted odds): Return = Stake × Π Dᵢ. A void/push usually reduces the acca by one leg.

Worked leg‑by‑leg inputs and outputs (both “with vig” and “no‑vig”) are shown in the table; acca‑level totals are summarised beneath it.

Leg	Market (decimal odds pair)	Selected odds (decimal)	Raw implied probability	Fair probability (no‑vig)
1	1.80 vs 2.10	1.80	55.56%	53.85%
2	1.91 vs 1.91	1.91	52.36%	50.00%
3	1.50 vs 2.80	1.50	66.67%	65.12%

Acca results from the table above (assuming independence of legs): With vig: P_acca = 0.5556 × 0.5236 × 0.6667 ≈ 19.39%; Decimal_acca = 1.80 × 1.91 × 1.50 = 5.157. No‑vig: P′_acca ≈ 0.5385 × 0.5000 × 0.6512 ≈ 17.53%; Decimal′_acca ≈ 5.704.

Accumulator vs parlay in the UK: any difference?

The mechanics are effectively the same; differences are mostly terminology and some house rules.

• Naming: “Accumulator” or “acca” (UK) vs “parlay” (US/elsewhere). Both multiply leg odds.
• Settlement: A void/push leg typically reduces the acca by one leg (e.g., a four‑fold becomes a treble).
• Returns: Return = Stake × product of decimal odds; profit = Return − Stake.
• Bonuses/insurance: UK firms often offer acca bonuses or “acca insurance” on eligible markets; terms vary by operator.
• Each‑way accas: Common in UK racing/football props; win and place parts are settled separately at place terms.
• Minimum odds/eligible markets: Books may set per‑leg minimums and exclude certain props; always check rules.

What about correlated legs and same‑game parlays?

Standard accas assume legs are independent. Correlated selections (e.g., Home Win and Over 2.5 Goals in the same match, or Player to Score and Team to Win) break that assumption. Most books either prohibit such “related contingencies” in a standard acca or price them via a dedicated Same‑Game Parlay (Bet Builder) engine that accounts for correlation.

• Do not multiply singles for SGPs: the displayed SGP price already includes correlation; use P = 1 / D directly for the combined SGP, not Π of singles.
• Positive correlation inflates the naïve product: multiplying singles will overstate the true chance and misprice the acca.
• Negative correlation deflates the naïve product: the true chance can be higher than Π of singles.
• Availability: Some markets remain blocked in builders; offerings and rules differ by operator.
• Best practice: For multi‑leg same‑match combos, rely on the book’s builder price and treat its implied probability as 1/decimal; avoid DIY multiplication unless legs are demonstrably independent.

Football focus: where implied probability helps in UK markets

In UK football markets (Premier League, EFL, domestic cups, and Europe), converting odds to implied probability clarifies how likely a book is pricing outcomes such as Match Odds (1X2), Both Teams To Score (BTTS), and Over/Under lines. Reading prices as percentages helps you compare firms, spot margin (overround), and decide whether your own estimate justifies a bet, a hedge, or a pass.

Match odds, BTTS, and Over/Under: typical ranges and tips

Think in probability first, then in prices. For any single selection, implied probability equals 1/decimal odds. In multi‑outcome markets (e.g., 1X2), the sum of implied probabilities exceeds 100% due to margin; remove it by normalising each outcome by the total. Use the breakpoints and practical tips below to anchor your intuition.

Key breakpoints:

• 2.00 (Evens) → 50.00%
• 1.80 → 55.56% (solid favourite); 2.50 → 40.00% (underdog)
• 3.00 → 33.33% (outsider); 4.00 → 25.00% (longer shot)

• 1X2 reading: Convert each of Home/Draw/Away to percentages, note the total (overround), then compare firms. A 2–3 percentage‑point difference on one side is meaningful.
• BTTS and Over/Under: Treat each as a two‑way market; compute 1/odds for both sides, sum to see margin, and normalise if you need a “fair” probability for modelling.

Sanity checks:

• If your model says Over 2.5 is 52% and the market is 1.95 (51.28%), the raw edge is small; de‑vig first to judge true value.
• Draw prices are sensitive to team styles; a mid‑20s implied percentage usually indicates a relatively draw‑heavy setup.

Decimal odds	Implied probability
1.40	71.43%
1.50	66.67%
1.67	59.88%
1.80	55.56%
1.91	52.36%
2.00	50.00%
2.50	40.00%
3.00	33.33%
3.50	28.57%
4.00	25.00%

How to use implied probability for value bets and hedging

For value, compare your estimated true probability p̂ with the market’s “no‑vig” probability p′ (normalised within the market). For hedging, use outcome probabilities to understand exposure, and if you want to lock a profit, size the opposing bet to equalise results.

• Find the break‑even probability: p_break‑even = 1 / D. Example: BTTS Yes at 1.95 → 51.28%.
• Remove margin (two‑way example): Over 2.5 at 1.95 (51.28%), Under 2.5 at 1.85 (54.05%). Sum = 105.33%. Fair Over 2.5 ≈ 51.28 / 105.33 = 48.69%; fair odds ≈ 2.055.

Value test:

• If your p̂ = 54% for BTTS Yes at 1.95: expected ROI ≈ p̂ × D − 1 = 0.54 × 1.95 − 1 = +5.30% (positive value).
• Prefer decisions against fair (no‑vig) p′ to avoid margin bias.

Hedging a two‑way market (equalising outcomes):

• Initial bet: Stake S on Over 2.5 at D_A = 2.20 (pre‑match).
• Live opposing price: Under 2.5 at D_B = 2.40.
• Hedge stake to equalise profit: H = S × D_A / D_B = 100 × 2.20 / 2.40 = £91.67.
• Net if Over wins: 100 × (2.20 − 1) − 91.67 = £28.33; if Under wins: 91.67 × (2.40 − 1) − 100 = £28.33.

Notes:

• Include commission/fees where relevant (e.g., exchanges) by adjusting effective odds.
• Equalising does not guarantee a positive result; if prices move against you, the locked outcome may be a small loss.

Implied probabilities help you quantify whether to hold, hedge partially, or fully lock in. Combine them with your live estimates and risk limits to avoid over‑trading small edges.

Should you tie staking to probability with Kelly criterion?

Kelly sizes stakes as a fraction of bankroll using your edge over the break‑even probability. For decimal odds D and your true probability p, the net odds are b = D − 1 and the full‑Kelly fraction is f* = (b × p − (1 − p)) / b = (D × p − 1) / (D − 1). Many practitioners use Fractional Kelly (e.g., 1/2 Kelly) to reduce drawdowns and estimation risk.

• Only use Kelly on “fair” (no‑vig) edges derived from normalised probabilities.
• Kelly is sensitive to p errors; overestimating p inflates stake size and risk. Fractional Kelly or caps (e.g., max 2–3% per bet) are common safeguards.
• If f* ≤ 0, skip the bet; the price does not compensate for risk.

Decimal odds (D)	True probability (p)	Edge vs break‑even	Kelly stake f*	Comment
2.50	45%	+5.00 pp (45.00% − 40.00%)	8.33% of bankroll	Strong edge; many would use 1/2 Kelly ≈ 4.17%
1.91	55%	+2.64 pp (55.00% − 52.36%)	5.55% of bankroll	Moderate edge; fractional Kelly recommended
3.20	35%	+3.75 pp (35.00% − 31.25%)	5.45% of bankroll	Underdog with value; variance higher
2.10	44%	−3.62 pp (44.00% − 47.62%)	0.00%	No bet (negative edge)

Kelly converts probability and price into a disciplined stake size, but bankroll volatility can still be large. If in doubt, scale down (e.g., 1/4–1/2 Kelly), cap per‑bet risk, and continuously recalibrate your p estimates with fresh match data.

The other side of the coin: what’s the strongest argument against relying on implied probability?

Implied probability is derived from market odds, not from the true chance of an outcome. Book prices embed bookmaker margin (overround), behavioural biases, promo mechanics, and sometimes limited liquidity or stale numbers. If you read prices as pure probabilities without adjusting for these factors—or without cross‑checking high‑liquidity references like exchanges—you can misjudge value, risk, and staking.

Market prices embed margin, bias, and promotions

Odds reflect business constraints and customer behaviour as much as sporting estimates. Before treating an odds‑derived percentage as “truth,” separate pricing mechanics (overround, rounding, promos) from information content (trader models, market liquidity).

• Overround: The sum of implied probabilities in a market typically exceeds 100%. This is margin, not information. See overview in [Overround](https://en.wikipedia.org/wiki/Overround).
• Favourite–longshot bias: Historically, longshots are often overpriced and favourites underpriced relative to outcomes; documented in [Favorite–longshot bias](https://en.wikipedia.org/wiki/Favorite%E2%80%93longshot_bias).
• Promotions and boosts: Enhanced odds, free bets, and profit boosts distort price signals; the “implied probability” computed from a boosted price is not a neutral estimate of true chance.

Concept	Mechanism	Numerical illustration	Implication
Overround (1X2)	Prices include margin so P(Home)+P(Draw)+P(Away) > 100%	2.20 (45.45%) + 3.40 (29.41%) + 3.30 (30.30%) = 105.17%	De‑vig before comparing to your model or break‑even
Favourite–longshot bias	Behavioural demand tilts prices away from true odds	Documented in literature; sign varies by market/liquidity	Be wary of very big prices; model and sample size matter
Odds boost	Book increases payout; promo T&Cs may apply	Boost 2.00 → 2.10 changes break‑even from 50.00% to 47.62%	Great for EV if your p > 47.62%, but not a pure signal

When do models, sample size, and exchange prices matter more?

Where markets are thin, early, or promo‑distorted, model‑based estimates and high‑liquidity exchange prices can be better guides. Exchanges aggregate sharp order flow; near kick‑off with good liquidity, their odds often provide a stronger “consensus” anchor than a single book’s implied probability (after accounting for commission).

Prefer models/exchanges when:

• Liquidity is high and spread is tight (e.g., Premier League near KO).
• Markets are promo‑heavy or boosted in one direction.
• Props/SGPs are priced with templates and low limits.
• You have reliable priors and current‑season data (adequate sample).

• Adjust for exchange commission on net winnings when converting to probability.
• Reference: Betfair Exchange background at [Betfair](https://en.wikipedia.org/wiki/Betfair).

Commission‑adjusted implied probability for a back bet is computed from D_net = 1 + (D − 1) × (1 − c), where c is commission (e.g., 0.02, 0.05). Then p = 1 / D_net.

Quoted exchange decimal (D)	Commission (c)	Net decimal (D_net)	Implied probability (1/D_net)
3.50	5%	3.3750	29.63%
2.00	2%	1.9800	50.51%
1.80	5%	1.7600	56.82%

Notes:

• For lay bets, convert using lay price and liability; commission applies to net wins—check exchange rules.
• On books, also factor in each‑way terms, cash‑out fees, or boosts caps when translating to EV.

How did implied probability evolve from chalkboards to algorithms?

In the UK, off‑course betting shops opened in 1961 following the Betting and Gaming Act 1960, with prices chalked on boards and settled from fractional odds. Over decades, pricing moved from manual “tissues” to database‑driven models, live in‑play trading, and exchange‑informed algorithms. Today, risk teams blend internal forecasts, third‑party feeds, and market signals to set and update odds, while bettors use calculators to strip margin and compare to modelled chances.

Legal and market milestones:

• 1961: Betting shops legalised; fractional odds standardised on the high street.
• Late 1990s: Online sportsbooks broaden access; decimal odds proliferate across Europe.
• 2000: Launch of UK betting exchanges (e.g., Betfair) changes price discovery.
• 2010s–2020s: In‑play modelling, APIs, and same‑game parlay engines price correlation explicitly.

• See context: [Betting and Gaming Act 1960](https://en.wikipedia.org/wiki/Betting_and_Gaming_Act_1960).

Era	Pricing practice	Effect on “implied probability”
Pre‑1961	On‑course bookies, chalkboards, manual tissues	Heavily heuristic; probabilities implicit, not computed
1961–1990s	Retail shops, fractional odds, static coupons	Simple conversions; margin sizeable, slower updates
Late 1990s	Online books, early risk engines, decimal adoption	Faster repricing; conversions standardised
2000s	Exchanges launch, market‑making influenced by order flow	Consensus anchors emerge; need commission‑adjusted p
2010s–2020s	In‑play models, data feeds, SGP correlation pricing	Dynamic probabilities; de‑vig and correlation matter more

The bottom line: implied probability is a useful lens, but it is only a starting point. Adjust for margin, understand bias and promos, and where possible, benchmark against liquid exchange prices and your own validated models before making decisions.

Frequently Asked Questions

How do I convert a percentage probability back into decimal, fractional, and American odds?

Multiply nothing; just invert. Decimal odds D = 1/p, fractional odds are (D − 1) written as a simplified fraction a/b, and American odds are −100/(D − 1) when D

How do I remove vig on Asian Handicap or Draw No Bet markets where a push is possible?

Normalise only across the two quoted outcomes, then treat push as a separate refund event for settlement. If AH 0.0 is priced at 1.90 vs 1.90, raw implieds are 52.63% and 52.63% (sum 105.26%), no‑vig becomes 50.00% and 50.00%; the push probability is not inferable from these prices alone and requires a goal‑distribution model, but EV on a fair 50–50 line is unchanged by pushes because refunded stakes contribute 0 expected profit.

How do I convert UK each‑way terms into implied win and place probabilities?

Compute the place part as a standalone price using the terms, then invert to a probability. If the win decimal is 10.00 and the place terms are 1/5 for 3 places, the place decimal is 1 + (10.00 − 1) × 0.20 = 2.80, so the implied place probability is 1/2.80 = 35.71%, while the win implied probability is 1/10.00 = 10.00%; both figures include overround from their respective books and do not account for field size or dead‑heats.

What is the difference between overround, vigorish, and bookmaker margin, and how do I compute it on a two‑way line?

They are used interchangeably to describe the excess of implied probabilities over 100%. On a two‑way market with decimals D1 and D2, Overround = 1/D1 + 1/D2, so the margin is Overround − 1; for 1.91 vs 1.91, Overround = 1/1.91 + 1/1.91 = 1.0471, hence margin = 4.71%.

How many bets do I need before a 1–3 percentage‑point edge is statistically meaningful?

As a rule of thumb, 250 bets can make a 2.6 pp edge detectable at roughly 80% one‑sided confidence, while about 1,350 bets are needed for 95%. Example at 1.91 odds: break‑even p0 = 1/1.91 = 52.36%; if your true p = 55.00% (edge 2.64 pp), the standard error is √(p(1−p)/N); solving p − z·SE > p0 gives N ≈ 250 for z = 0.84 (80%) and N ≈ 1,358 for z = 1.96 (95%).

How do I compute lay‑side break‑even probability on exchanges after commission?

For a lay at decimal D with commission c on net wins, the layer’s break‑even event probability is p* = (1 − c) / ((D − 1) + (1 − c)). If D = 3.50 and c = 5%, p* = 0.95 / (2.50 + 0.95) = 27.54%; you must estimate the true event probability below 27.54% to have positive EV when laying.

How can I estimate the push probability on totals or Asian lines without a full model?

Use adjacent half‑goal lines after removing vig and difference their probabilities. If the de‑vigged P(Over 2.5) = 52.0% and P(Over 3.5) = 40.0%, then the mass at exactly 3 goals is about 52.0% − 40.0% = 12.0%; that 12.0% approximates the push chance on Over/Under 3.0 or AH ±0.0 tied outcomes.

How can I back out implied correlation between two legs from a Same‑Game Parlay price?

Compare the SGP implied probability to the product of the singles and solve ρ from a Bernoulli correlation approximation. With singles pA = 40% and pB = 35%, the naive joint is 0.14; if the SGP decimal is 5.00 (pAB = 20%), then ρ ≈ (0.20 − 0.14) / √(0.4·0.6·0.35·0.65) = 0.06 / 0.2337 ≈ +0.26, indicating material positive correlation priced into the builder.

How do I fairly compare prices across bookmakers when opposing lines differ?

De‑vig each book within its own market, then compare the resulting fair probabilities or fair odds. If Book A has Over 2.5 at 1.95 and Under 2.5 at 1.85, the de‑vigged Over is 51.28% / (51.28% + 54.05%) = 48.69% (fair odds 2.055); if Book B posts 2.02 and 1.87, Over de‑vig = 1/2.02 ÷ (1/2.02 + 1/1.87) = 0.4951 ÷ 1.0298 = 48.08% (fair odds 2.079). Book A’s Over is the stronger fair quote here.

How do boosted odds and profit boosts change implied probability?

Treat a boost as a new decimal and invert. A direct boost from D to D′ sets break‑even probability to 1/D′, so moving 2.50 to 2.875 lowers break‑even from 40.00% to 34.78%; for a profit boost of x% on winnings, D′ = 1 + (D − 1) × (1 + x), and then p_break‑even = 1/D′.

What rounding precision should I use to avoid drift in multi‑leg calculations?

Keep at least 4 decimal places on decimals and 4–5 on probabilities to limit compound error below 1% across many legs. For a leg at 55.5556%, rounding to 55.6% introduces a 0.00044 absolute error; across 10 independent legs this can accumulate to roughly 0.8% relative error in the product, whereas retaining 5 dp caps the same scenario near 0.2%.

How do I compute the probability of at least one winner across several singles?

Use the complement rule. For independent legs with probabilities p1, p2, …, pn, P(at least one) = 1 − ∏(1 − pi); for p = {0.556, 0.524, 0.667}, P(at least one) = 1 − (0.444 × 0.476 × 0.333) = 1 − 0.0705 = 92.95%, which is very different from an acca’s all‑legs‑win probability of 0.556 × 0.524 × 0.667 = 19.39%.

How do UK Rule 4 deductions affect implied probability and fair odds in racing?

Apply the deduction to winnings, recompute the net decimal, then invert. With a 20p Rule 4 and a quoted 5.00, the net decimal is D_net = 1 + (5.00 − 1) × 0.80 = 4.20, so the implied probability consistent with the reduced payout becomes 1/4.20 = 23.81%.

How do I estimate a fair acca probability when two legs are weakly correlated but I only know a correlation coefficient?

Adjust the naive product using a Bernoulli correlation term. For two legs A and B with pA and pB and correlation ρ, use P(A ∩ B) ≈ pA pB + ρ √(pA(1 − pA)pB(1 − pB)); with pA = 55%, pB = 60%, and ρ = 0.20, the joint becomes 0.55 × 0.60 + 0.20 × √(0.2475 × 0.24) = 0.33 + 0.20 × 0.2435 = 37.87%, a 14.6% relative increase over the naive 33.0%.