🎯 Betting Calculator

How do I convert percentage to betting odds (decimal, American, fractional)?

In Canada (including Ottawa and Ontario’s regulated market), sportsbooks commonly present odds in decimal by default, with quick toggles to American and fractional. If you know the chance of an outcome as a percentage (its probability), you can convert that percentage into any odds format instantly. The key is understanding implied probability—the chance embedded in a price—and how to switch between formats accurately, with attention to rounding and bookmaker margin (vig).

What is implied probability and how is it derived from odds?

Implied probability is the likelihood of an outcome as suggested by a given set of odds, before accounting for the bookmaker’s margin. Converting between formats is deterministic:

• From decimal odds (D) to probability: p = 1 / D From American odds (A) to probability:

• If A ≥ +100: p = 100 / (A + 100)

• If A ≤ −100: p = |A| / (|A| + 100)

• From fractional odds (a/b) to probability: p = b / (a + b)

Once you have p, you can convert to any odds format:

• Probability to decimal: D = 1 / p Probability to American:

• If p < 0.5: A = +100 × (1 − p) / p

• If p ≥ 0.5: A = −100 × p / (1 − p)

• Probability to fractional: a/b = (1 − p) / p, then reduce to simplest ratio

Removing bookmaker margin (vig) for multi-outcome markets (e.g., 1X2): First compute raw implied probabilities for each outcome and sum them (S). Normalize each by dividing by S to get fair probabilities pfair,i = praw,i / S. Convert those to fair odds for true comparisons.

Rounding guide: sportsbooks typically round decimal odds to 2 decimals, American odds to whole numbers (often to the nearest 5 or 10), and fractional odds to simple ratios.

Decimal vs American vs fractional: which format should Canadians use?

All three formats encode the same information; the “best” format is about clarity and workflow in Canadian markets.

Decimal (e.g., 1.83):

• Most common display format in Canada; easiest mental link to probability (p ≈ 1/D).
• Total return per $1 stake is shown directly; quick for parlays and 1X2 comparisons.
• Recommended default for NHL, CFL, NBA, soccer 1X2, and futures browsing.

American (e.g., -120, +145):

• Favoured by U.S.-centric content, markets, and tools; good for line-shopping vs U.S. books.
• Immediate intuition for payout per $100 risked or won; handy near key prices (e.g., -110).
• Less intuitive for multi-outcome markets without conversion.

Fractional (e.g., 5/6, 12/5):

• Traditional in UK racing; concise for short prices.
• Requires conversion for probability-first workflows and parlays.

Recommendation for Canadians: Use decimal as your default for transparency and speed; switch to American when consuming U.S.-based analysis, and to fractional for UK racing content. Whatever you choose, always translate back to implied probability when evaluating value and comparing markets.

Quick formulas and step-by-step examples for local markets

Keep these formulas handy, then follow the worked Canadian-focused examples below.

• p from decimal: p = 1 / D
• p from American: A ≥ +100 ⇒ p = 100/(A+100); A ≤ −100 ⇒ p = |A|/(|A|+100)
• p from fractional: p = b / (a + b)
• Decimal from p: D = 1 / p
• American from p: p < 0.5 ⇒ +100×(1−p)/p; p ≥ 0.5 ⇒ −100×p/(1−p)
• Fractional from p: (1−p)/p, reduce to simplest terms

Turn a confidence percentage into odds (e.g., NHL side you rate at 58%).

• Probability: p = 0.58
• Decimal: D = 1 / 0.58 = 1.724… ⇒ 1.72
• American: −100 × 0.58 / 0.42 = −138.09 ⇒ −138
• Fractional: (1 − 0.58) / 0.58 = 0.724… ⇒ 21/29

Convert a common Canadian moneyline to probability (e.g., -120).

• Probability: p = 120 / (120 + 100) = 120 / 220 = 0.54545 ⇒ 54.545%
• Decimal: D = 1 / 0.54545 = 1.8333 ⇒ 1.83
• Fractional: (1 − 0.54545) / 0.54545 = 0.8333 ⇒ 5/6

Convert an underdog decimal price to other formats (e.g., 3.40 on CFL road team).

• Probability: p = 1 / 3.40 = 0.29412 ⇒ 29.412%
• American: +100 × (3.40 − 1) = +240
• Fractional: 3.40 − 1 = 2.40 ⇒ 12/5

Normalize an NHL 1X2 (three-way) market to remove vig.

• Posted decimal odds: Home 2.45, Draw 3.60, Away 2.85 Raw implied probabilities:

• Home: 1/2.45 = 0.40816

• Draw: 1/3.60 = 0.27778

• Away: 1/2.85 = 0.35088

• Sum S = 0.40816 + 0.27778 + 0.35088 = 1.03682 Fair probabilities (divide each by S):

• Home: 0.40816 / 1.03682 = 0.3936

• Draw: 0.27778 / 1.03682 = 0.2680

• Away: 0.35088 / 1.03682 = 0.3384

Fair decimal odds:

• Home: 1 / 0.3936 = 2.54
• Draw: 1 / 0.2680 = 3.73
• Away: 1 / 0.3384 = 2.96

Tip for Ottawa/Canada bettors: display prices in decimal for transparent comparisons, convert to implied probability to evaluate value, and normalize multi-outcome markets to compare true prices across books.

Parlay Odds Calculator: how are combined probabilities and payouts computed?

A parlay multiplies the decimal odds of each independent leg to produce a total decimal price. If legs have decimal odds D1, D2, …, Dn and are independent, the parlay’s total decimal is Dtotal = D1 × D2 × … × Dn. The win probability is the product of each leg’s true hit rate: pwin = p1 × p2 × … × pn. Total return on a $1 stake is Dtotal; profit is Dtotal − 1. Any correlation, push, or void modifies these mechanics, so accurate calculators must reflect house rules and whether same-game legs are priced for correlation.

How do legs, correlations, and pushes affect a parlay?

Each additional leg increases payout multiplicatively but reduces win probability exponentially. Calculators assume independence by default; if legs are correlated or a leg pushes, the effective parlay changes.

• Legs (independence assumption): With independent legs, Dtotal = ∏Di and pwin = ∏pi. Example: legs at 1.95, 1.80, 2.20 yield Dtotal = 1.95 × 1.80 × 2.20 = 7.722; if true leg hit rates are 0.54, 0.55, 0.45, then pwin = 0.54 × 0.55 × 0.45 = 0.13365. Correlations:
• Natively forbidden correlations: Traditional parlays exclude correlated outcomes (e.g., team moneyline with its spread or highly dependent player props in the same game). Books either reject them or reprice via “same-game parlay” (SGP) engines.
• SGP pricing: When allowed, books embed correlation into the composite price. A standard parlay calculator that multiplies independent odds will overstate payout if applied to SGPs priced with correlation.
• Practical cue: If your book offers an SGP builder, use its quoted SGP price directly; do not multiply standalone market odds.

Pushes and voids:

• Push (voided leg): That leg’s multiplier becomes 1.00. A 3-leg parlay with one push settles as a 2-leg parlay using the remaining legs’ odds.
• Event cancellations/void markets: Voided selections reduce the parlay by one leg unless house rules state otherwise.
• Quarter/Asian lines: Quarter-handicaps (e.g., -0.25) split the stake into two sub-legs; a “half win” or “half push” is settled by splitting and combining sub-results. Some books exclude such legs from parlays or auto-split them; consult house rules before using a generic calculator.

Key takeaway: independence and grading rules drive accuracy. If any leg is correlated or can push/void, use a calculator that supports SGP pricing or split-stake logic, or translate the book’s quoted SGP price into decimal and treat it as a single composite leg.

What payout and ROI can I expect from my stake?

Use these formulas for any stake S (e.g., $10, $20, $25): Dtotal = ∏Di; payout on win = S × Dtotal; profit on win = S × (Dtotal − 1); win probability pwin = ∏pi (true leg hit rates, after removing vig if needed). Expected ROI = pwin × Dtotal − 1. Break-even requires pwin = 1 / Dtotal.

Worked ROI example (3-leg, independent): Suppose odds are 1.95, 1.80, 2.20. Then Dtotal = 7.722. If your true leg hit rates are 0.54, 0.55, 0.45, then pwin = 0.13365. Expected ROI = 0.13365 × 7.722 − 1 = 0.0315 (3.15%). With a $20 stake, EV profit ≈ $0.63 and payout on a win is $154.44.

• Positive edge scales with legs: If each leg has edge (pi × Di > 1), ROI grows roughly as (∏(pi × Di)) − 1; more legs amplify your edge and variance.
• Negative edge scales against you: If pi × Di < 1 on average, adding legs reduces ROI and crushes hit rate.
• Vig awareness: Remove margin in multi-outcome markets (e.g., 1X2) to estimate true pi before computing pwin and ROI.

When should I use a round robin instead of a parlay?

A round robin (RR) breaks a set of selections into multiple smaller parlays of a chosen size (e.g., “2s from 4” creates 6 two-leg parlays). You trade maximum payout for higher hit frequency and smoother bankroll swings.

• Stability over spikes: Choose RR when you expect to win “most but not all” legs (e.g., 3 of 4, 2 of 3) and prefer partial returns instead of an all-or-nothing parlay outcome.
• Small but repeatable edges: With modest edges across several independent picks, RR 2’s or 3’s can convert consistency into steadier cash flow, at the cost of peak upside.
• Variance management: If your hit rate per leg is around break-even (e.g., ~53% at -110), a single big parlay has low pwin; RR cushions downswings by paying when subsets hit.
• When not to RR: If you have strong edges on every leg and can tolerate variance, a straight parlay often maximizes expected ROI because (pi × Di) compounding works in your favour.

Example (independent legs, -110 each, D = 1.9091, true pi = 0.55): Stake $24 as either (a) one 4-leg parlay or (b) a 2s-from-4 RR with six $4 doubles. (a) 4-leg parlay Dtotal ≈ 13.2833, pwin = 0.55^4 = 0.0915; EV profit ≈ 0.0915 × 13.2833 × $24 − $24 = +$5.18. (b) Each 2-leg double has D2 ≈ 3.6446, p = 0.3025; EV per $4 double ≈ 0.3025 × 3.6446 × $4 − $4 = +$0.41; across 6 doubles ≈ +$2.47. The parlay has higher EV and variance; the RR pays back more often with smaller swings. Match the tool to your edge profile and bankroll tolerance.

1X2 Odds Calculator: how do you price the draw and convert to moneyline?

In 1X2 (three-way) markets, you price home win (1), draw (X), and away win (2) over a fixed regulation period. In Canada, NHL three-way markets grade on 60 minutes only, while two-way moneylines include overtime and shootouts. In soccer, standard 1X2 settles on 90 minutes plus injury time—extra time and penalties are excluded unless explicitly stated. Understanding settlement windows is essential before converting 1X2 prices to two-way moneyline or double chance markets.

What changes with NHL overtime and soccer extra time rules?

Settlement rules define whether a draw is possible and which time frame counts. This directly affects the implied probabilities and any conversions between market types.

NHL (Canada):

• Three-way (1X2): Settled on regulation (60:00). Outcomes: Home, Draw, Away. Overtime/shootout do not count.
• Moneyline (two-way): Includes overtime and shootout. No draw option; one team must win.
• Period markets: Specify the period (e.g., 1st period 20:00). Overtime never applies to period bets.

Soccer:

• 1X2 (90 mins): 90 minutes plus injury time. Extra time and penalties excluded; draw is possible.
• To Qualify/To Advance: Includes extra time and penalties; no draw option (binary outcome).
• Extra time markets: Clearly labeled; apply only to extra-time period if played.

How do I remove the bookmaker margin to get fair 1X2 odds?

Books embed margin (vig) so the sum of raw implied probabilities exceeds 100%. To compare fairly or convert to other markets, first de-vig the 1X2 prices.

• Convert posted odds to raw probabilities: praw = 1 / D for each of Home (H), Draw (D), Away (A).
• Find the overround: S = pH,raw + pD,raw + pA,raw.
• Normalize to fair probabilities: pi,fair = pi,raw / S.
• Convert to fair odds: Di,fair = 1 / pi,fair.

Worked example (posted 1X2 decimal: Home 2.50, Draw 3.40, Away 2.80):

• Raw implied probabilities: pH,raw = 1/2.50 = 0.4000; pD,raw = 1/3.40 = 0.2941; pA,raw = 1/2.80 = 0.3571.

• Overround: S = 0.4000 + 0.2941 + 0.3571 = 1.0513.

• Fair probabilities: pH = 0.4000/1.0513 = 0.3805; pD = 0.2941/1.0513 = 0.2798; pA = 0.3571/1.0513 = 0.3397.

• Fair decimal odds: H = 1/0.3805 = 2.63; D = 1/0.2798 = 3.57; A = 1/0.3397 = 2.95.

• Alternative de-vig methods: Proportional (above) is standard. Advanced approaches (e.g., Shin or power transforms) reallocate margin asymmetrically if suspect markets, but proportional is typically sufficient for calculators.

Can I compare 1X2 implied probability with double chance markets?

Yes. After de-vigging 1X2 to fair probabilities, you can derive double chance (DC) probabilities directly and compare fairly priced odds.

From fair 1X2 probabilities:

• p(1X) = p(H) + p(D)

• p(X2) = p(D) + p(A)

• p(12) = p(H) + p(A) = 1 − p(D)

• Convert to fair odds: D(DC) = 1 / p(DC).

• Compare: If a sportsbook’s DC price exceeds your fair D(DC), you’ve found positive expected value (before considering limits and correlation).

Continuing the example (fair p(H)=0.3805, p(D)=0.2798, p(A)=0.3397):

• p(1X) = 0.3805 + 0.2798 = 0.6603 ⇒ fair odds ≈ 1.51

• p(X2) = 0.2798 + 0.3397 = 0.6195 ⇒ fair odds ≈ 1.61

• p(12) = 1 − 0.2798 = 0.7202 ⇒ fair odds ≈ 1.39

• Approximate NHL moneyline from 1X2: If overtime/shootout are assumed 50/50 when tied, fair ML probabilities are pH,ML ≈ p(H) + 0.5×p(D) and pA,ML ≈ p(A) + 0.5×p(D). With the example: pH,ML ≈ 0.3805 + 0.1399 = 0.5204 (fair decimal ≈ 1.92; American ≈ -109); pA,ML ≈ 0.3397 + 0.1399 = 0.4796 (fair decimal ≈ 2.09; American ≈ +109). If you have team-specific OT/SO edges, replace 50/50 with those estimates.

Practical checklist for Ottawa/Canada bettors: confirm the settlement window first (regulation vs OT/SO or 90 mins vs To Qualify), de-vig the 1X2, derive DC or ML probabilities as needed, and only then compare prices across books for true value.

Poker Odds Calculator vs Sports Betting Calculator: what’s different?

A sports betting calculator works with fixed prices and event probabilities that are external to you. A poker odds calculator must infer probabilities from card combinatorics, hand ranges, stack sizes, and future betting. That means poker math is state-dependent (board texture, number of players, positions), while sports math is price-dependent (decimal/American odds and implied probabilities). Understanding outs, equity, pot odds, and the limits of calculators is crucial to avoid systematic mistakes at the table.

How are outs, equity, and pot odds calculated in Texas Hold’em?

Outs are unseen cards that improve you to (what you believe is) the winning hand. Equity is your chance to win at showdown given the current state and ranges. Pot odds compare the price of a call to the potential reward. Use exact combinatorics for precision and the “Rule of 2 and 4” for quick approximations.

• Counting outs: Common draws: gutshot (4 outs), two overcards (≈6 clean outs if all are good), open-ended straight draw (8), flush draw (9), combo draws (12–15 when outs don’t overlap). Remove “dirty” outs that could still leave you second-best (e.g., pairing your low card on a higher pair board, or non-nut flush outs in multi-way pots).
• Exact equity (single card to come): On the turn with o outs, P(hit on river) = o / 46. On the flop with two cards to come, P(hit by river) = 1 − C(47 − o, 2) / C(47, 2), where C(n, k) is a combination.
• Rule of 2 and 4 (approx.): From the turn, equity ≈ 2 × outs (%). From the flop to river, equity ≈ 4 × outs (%). Example: 9 outs ≈ 18% (turn) and ≈ 36% (flop to river); exact values are ~19.57% and ~34.96% respectively.
• Pot odds and break-even equity (single-card call): If current pot before facing a bet is P, opponent bets B, and you must call C = B, then break-even equity for a call is E* = C / (P + B + C). If your draw’s equity ≥ E*, the call is +EV (ignoring future action and rake).
• Worked example (turn, flush draw): Pot P = $90, villain bets B = $60, you call C = $60. Break-even E* = 60 / (90 + 60 + 60) = 28.57%. With 9 outs, exact equity = 9/46 = 19.57% → fold without implied odds; you must expect future winnings to compensate (see next section).

Notes: Percentages assume all outs are clean and that no card removal beyond counted outs changes equity (a close approximation in heads-up pots). In multi-way pots or with dominated draws (e.g., non-nut flushes), reduce effective outs before applying the math.

When do implied odds and reverse implied odds matter?

Implied odds capture extra money you expect to win on future streets when you hit. Reverse implied odds capture extra money you expect to lose because the made hand can be second-best or pays off poorly. They turn borderline folds into calls—or profitable calls into folds—depending on stack depth and hand quality.

• Implied odds (needed extras): For a turn call with equity p, required additional money you must expect to win when you hit (beyond the current pot and bets) is I ≥ C × (1/p − 1) − (P + B). Using the earlier example (P = $90, B = $60, C = $60, p = 9/46 = 0.1957): I ≥ 60 × (1/0.1957 − 1) − 150 ≈ $96.6. If stacks and opponent tendencies won’t yield ≈$97 more when you hit, folding is prudent.

• Stack-to-Pot Ratio (SPR): Deeper stacks increase implied odds for nutty draws (e.g., nut flush/straight draws). Shallow stacks diminish implied odds and often justify jamming earlier with high-equity draws. Reverse implied odds risks:

• Dominated draws: Calling with a 9-high flush draw versus a range containing many higher flushes risks paying off when you hit; discount outs that make dominated hands.

• Paired boards: Outs that pair the board may give opponents boats; reduce outs accordingly.

• Top pair weak kicker (TPWK): Hitting a weak top pair can be costly against tighter ranges; account for poor realization of equity.

• Multi-way pots: Implied odds can improve (more callers to pay you) but reverse implied odds also rise (more nutted hands in ranges). Tighten draw-calling standards without position and nut potential.

What can’t a poker odds tool account for (ranges, position, live reads)?

Calculators excel at card math but cannot fully capture strategic context. Treat outputs as baselines, then adjust using opponent modeling, position, and future action forecasts.

• Ranges vs single hands: Equity versus a precise hand differs from equity versus a realistic range. You must define villain ranges by position, preflop action, and tendencies; tools won’t infer them automatically.
• Position and fold equity: Out-of-position callers realize equity worse; in-position aggressors realize better and generate fold equity with raises. A raw equity edge may be unprofitable if you realize it poorly.
• Dirty outs and domination: Tools often count all outs as clean unless you specify filters. Adjust for higher flushes, better kickers, board pairing, and redraws.
• Bet sizing trees: Future bet sizes and frequencies shape implied and reverse implied odds. Static calculators don’t model full decision trees or exploitative lines.
• Multi-way complexity and blockers: Multi-way equities swing sharply; blocker effects and removal matter. Many simple tools assume heads-up scenarios.
• Rake and payout structures: In cash games, rake reduces small-pot EV; in tournaments, ICM and payout ladders alter risk/reward; basic equity tools ignore these.

Bottom line: Use the calculator for precise base rates (outs and equity), then layer in pot odds, implied/reverse implied odds, position, and range logic—just as a sports odds tool must be paired with market selection and vig removal to produce actionable decisions.

How did odds formats evolve in Canada, and why does decimal dominate?

Fractional odds arrived in Canada via British bookmaking traditions and horse racing, American (moneyline) odds spread through U.S. media and cross-border books, and decimal odds were popularized by European online sportsbooks and provincial lottery operators. Since the federal legalization of single-event sports betting in 2021 and the launch of Ontario’s regulated iGaming market in 2022, most Canadian-facing sportsbooks default to decimal. Decimal shows total return per $1 stake, multiplies cleanly for parlays, and maps directly to implied probability, which makes it the most transparent working format for bettors in Ottawa and across Canada.

The other side of the coin: should you rely on an odds calculator?

Use calculators as precision tools for price conversion, probability, and expected value—not as substitutes for handicapping, market context, or risk management. They are only as good as the inputs and settlement assumptions you supply.

• Strengths: Fast, error-free conversions (decimal/American/fractional/probability), de‑vig of multi-outcome markets, parlay math, and expected value/ROI under clear assumptions.
• Limits: Calculators do not know house rules (push handling, overtime/extra time), correlation between legs (same-game parlays), or current market liquidity and limits. They cannot judge model quality or data bias.
• Risk of overreliance: Small rounding differences and stale prices can flip a marginal “edge.” Without de‑vig, multi-outcome comparisons are distorted. For NHL and soccer, mixing regulation-only 1X2 with OT-included moneylines leads to wrong inferences.
• Best practice: Confirm settlement windows, remove vig before fair comparisons, and sanity-check any edge against multiple books to avoid one-off misprices or low-limit traps.

Implied probabilities are exact (p = 1/decimal). American values are rounded to the nearest whole number; fractional expresses net return to 1 unit staked (decimal − 1).

Why do some edge calculators mislead without removing vig?

In multi-outcome markets (e.g., 1X2), posted implied probabilities sum to more than 100% due to margin (overround). If you add raw probabilities directly or derive double chance from them, you will understate fair odds and manufacture phantom “value.” Always de‑vig first, then aggregate or compare.

• Mechanics: For posted decimals H=2.60, D=3.25, A=2.70, raw probabilities are 0.3846, 0.3077, 0.3704; they sum to S=1.0627. Proportional de‑vig gives fair p(H)=0.36197, p(D)=0.28945, p(A)=0.34858.
• Double chance trap: Naively p(1X)raw=0.3846+0.3077=0.6923 ⇒ “fair” DC ≈ 1/0.6923 = 1.444. Correctly de‑vigged p(1X)=0.36197+0.28945=0.65142 ⇒ fair DC ≈ 1/0.65142 = 1.535. If a book offers 1X at 1.52, the raw method calls it value (1.52 > 1.44), but the correct fair says it’s slightly −EV (1.52 < 1.535).
• Parlay and SGP distortions: Multiplying leg odds assumes independence and no embedded correlation pricing. Same‑game parlay engines reprice correlation; multiplying standalone market odds overstates EV.
• Rounding drift: Converting American to decimal and back (and book-specific rounding to 2 decimals) can shift edges by a few basis points. Use consistent precision when evaluating tight lines.

How do calculators support—not replace—bankroll and risk strategy?

Price and probability tools inform staking but do not decide it. Use them to compute break-even rates, ROI, and suggested stakes under a chosen risk framework, then temper with limits, variance, and personal risk tolerance.

• Break-even and ROI: For decimal D and your win probability p, break-even is p* = 1/D. ROI per unit stake is ROI = p × D − 1.

• Kelly criterion (for optimal growth): With decimal D and b = D − 1, full Kelly fraction is f* = (b × p − (1 − p)) / b. Many bettors use fractional Kelly (e.g., 25–50%) to reduce variance. Examples:

• D = 2.10, p = 0.49 ⇒ ROI = 0.49 × 2.10 − 1 = +2.9%; b = 1.10 ⇒ f* = (1.10 × 0.49 − 0.51) / 1.10 ≈ 2.64% of bankroll.

• D = 2.45, p = 0.42 ⇒ ROI = 0.42 × 2.45 − 1 = +2.9%; b = 1.45 ⇒ f* ≈ (1.45 × 0.42 − 0.58) / 1.45 ≈ 2.00% of bankroll.

• Risk controls: Cap stake size (units), diversify across uncorrelated markets, and beware of low-limit or heavily juiced props where theoretical edges are hard to realize in practice.

• Process: Convert to implied probability, remove vig, estimate your edge, then choose stake via a consistent plan (fixed %, Kelly fraction, or flat units). Reconcile results against actual closing lines to validate your model.

Frequently Asked Questions

How do I calculate the bookmaker margin (overround) and fair probability on a two‑way line like -110/-110?

Add the raw implied probabilities from each side, then normalize each by that sum; the excess over 100% is the margin. At -110/-110 the decimals are 1.9091, so each raw p is 1/1.9091 = 52.381%, the sum S = 104.762%, margin = 4.762%, and fair probabilities are 50.000% each (fair decimal 2.00).

What is closing line value (CLV) and how do I measure it in decimal odds?

CLV is the edge you captured versus the market at close; measure it as the improvement in break‑even probability or expected ROI relative to the closing price. Example: you bet 1.91 (implied p* = 52.381%) and close is 1.83 (p* = 54.645%); your CLV in probability terms is (54.645 − 52.381) / 54.645 = 4.15%, and if the close reflects true win chance p = 1/1.83, your ROI edge is 1.91/1.83 − 1 = 4.37% per unit staked.

How can I find and size a no‑risk arbitrage across sportsbooks?

For any market, convert all offers to decimal and check if the sum of reciprocals is below 1; if so, stakes can be sized to lock profit. Example two‑way arb: Book A 2.10 vs Book B 2.10 gives 1/2.10 + 1/2.10 = 0.9524; with a $100 total stake, target payout is $100 / 0.9524 ≈ $105, so stake $50 on each side, guaranteed profit ≈ $5 (5.0%).

How do odds boosts change expected value and break‑even probability?

A percentage boost multiplies the net return; if D is decimal and boost k% applies to winnings, boosted odds are D′ = 1 + (D − 1) × (1 + k). The new break‑even is p*′ = 1/D′. Example: 20% boost on 2.50 yields D′ = 1 + 1.50 × 1.20 = 2.80; EV gain per $1 at true p = 0.42 is 0.42 × (2.80 − 2.50) = +0.126 (12.6%).

How do I value free bets or bet credits where the stake is not returned?

The EV of a stake‑not‑returned bonus is EV = bonus × p × (D − 1); the break‑even p is 1/(D − 1). A $25 bonus at D = 2.20 with p = 0.50 is worth $25 × 0.50 × 1.20 = $15. If you consistently place bonuses at D ≈ 3.00 with p ≈ 0.40, the EV per $100 in bonuses is $100 × 0.40 × 2.00 = $80 before any restrictions.

What is my expected loss to vig on common North American prices like -110 or -115?

If the true chance is 50%, expected ROI at -110 (D = 1.9091) is 0.50 × 1.9091 − 1 = −4.545%, which is a $4.55 loss per $100 bet. At -115 (D = 1.8696) the ROI is 0.50 × 1.8696 − 1 = −6.522% (−$6.52 per $100), while +100 (D = 2.00) is exactly break‑even at 50%.

How do I remove vig on asymmetric two‑way moneylines like -135/+125?

Convert each side to raw implied p, sum them, then divide each by the sum for fair p. For -135 (D = 1.7407) and +125 (D = 2.25), raw p are 57.47% and 44.44%, sum S = 101.91%; fair p are 57.47/101.91 = 56.41% and 43.59%, giving fair decimals 1.773 and 2.294.

How much can rounding change implied probability and expected value?

Rounding shifts probability by basis points that compound over volume; for example, 1.8333 rounded to 1.83 increases implied p from 54.545% to 54.645% (a 10 bp change). On an underdog, 2.38 vs 2.40 changes implied p from 42.0168% to 41.6667%; if true p = 42.0%, ROI at 2.38 is ≈ −0.04% while at 2.40 it is ≈ +0.80%, an $0.84 swing per $100 stake.

How do I hedge a futures ticket or late‑stage parlay to lock a guaranteed profit?

If your ticket pays P on a win (return including stake) and the opposing side is priced at decimal D, a symmetric hedge that equalizes outcomes is Hedge stake H = P / D. Net profit on either result is P − initial stake − H. Example: $50 at +500 (D = 6.00) pays P = $300; if the opponent is 1.80, set H = 300/1.80 = $166.67 to lock $300 − $50 − $166.67 = $83.33 either way.

How do I convert 1X2 prices to Draw No Bet (DNB) once I’ve removed the vig?

Use the fair 1X2 probabilities and condition on “no draw”: p(H|¬D) = p(H) / (1 − p(D)) and p(A|¬D) = p(A) / (1 − p(D)), with fair DNB odds D = 1 / p(·|¬D) = (1 − p(D)) / p(·). If p(H) = 0.44, p(D) = 0.25, p(A) = 0.31, then fair DNB‑Home = 0.75/0.44 = 1.7045 and fair DNB‑Away = 0.75/0.31 = 2.4194.

How can I tell if a same‑game parlay (SGP) price is over or under‑paying versus independent legs?

Compare the SGP’s implied probability to the product of de‑vigged single‑leg probabilities; a higher implied p than independent means correlation has reduced payout. If two legs each have fair p = 50%, independence gives p = 25% and fair D = 4.00; if the SGP quote is D = 3.30, its implied p is 1/3.30 = 30.3% and the payout is 17.5% lower than the independent 4.00 benchmark.

How many bets do I need to verify a small edge with statistical confidence?

For a binomial win rate near 50%, the 95% confidence margin m requires n ≈ 1.96² × 0.25 / m² samples. To pin your true hit rate within ±3 percentage points you need about 1,067 bets; for ±2 points you need about 2,401 bets, which bounds whether a 2–3% ROI is distinguishable from noise.

In poker, how do I set the minimum equity for an all‑in call with two cards to come?

Use the same break‑even formula E* = C / (P + C) but compare it to your by‑river equity; call if equity ≥ E*. If the pot is $100, villain shoves $100, and you must call $100, E* = 100/300 = 33.33%. A 12‑out draw on the flop has exact by‑river equity 44.96%, which clears the threshold before considering rake or future action.

How big is the equity hit from counting ‘dirty’ outs in poker (e.g., non‑nut flushes)?

Reducing 9 flush outs to 7 clean outs on the flop drops by‑river equity from 34.96% to 27.85%, a 7.11 percentage point loss. The exact value uses combinations: with 7 outs, equity = 1 − C(47 − 7, 2) / C(47, 2) = 1 − 780/1081 = 27.85%.

How do Asian handicap quarter‑lines like +0.25 settle, and how can I model them in a calculator?

A +0.25 bet splits equally into +0 and +0.5; on a draw it returns half the stake and wins half at the posted price, so the settlement multiplier is 0.5 × 1.00 + 0.5 × D. For D = 1.90 and a draw, the return per $1 staked is 1.45, which you can treat as the effective parlay multiplier for that leg.