This guide explains how to convert probabilities to odds and how to reverse the process across decimal, fractional, and American formats. It covers the core formulas, discusses bookmaker margin and de-vigging, and provides worked examples you can reproduce in a spreadsheet. The steps here are practical and math-first, intended for sports fans, analysts, and modelers who need reliable, auditable conversions between model outputs and market prices.

Odds and probability are two ways of expressing the same uncertainty; o = p/(1 - p) and p = o/(1 + o).

Decimal odds imply p = 1/D, fractional odds a/b imply p = b/(a + b), and moneyline conversions depend on sign.

De-vigging via proportional scaling produces a quick fair-probability estimate; document the method used.

What odds and implied probability mean

Basic definitions: probability, odds, implied probability

Probability is a number between 0 and 1 that expresses how likely an event is to happen; odds express the same information as a ratio comparing the chance of the event occurring to it not occurring. This relationship is the mathematical basis for converting between the two and for interpreting market prices and model outputs, and it underpins statistical practices such as logistic regression, which use log-odds to model binary outcomes Odds, Wikipedia

Put simply, if p is a probability then odds measure p relative to 1 minus p. That compact view makes conversions predictable: you can move between the two representations without changing the underlying assessment of risk or expectation. Thinking in both forms is useful because models often produce probabilities while markets typically display odds.

copy the core formulas into a spreadsheet or calculator for practice

probability
odds
decimal odds

keep formulas visible for auditing

Converting between probability and odds matters when you compare a model to a market, compute edge, or log outcomes for statistical models. Models and diagnostics often use probability or log-odds internally, while market prices and many user interfaces prefer odds formats that are easier to read for payout expectations Implied probability, Wikipedia

For practical work, converting lets you express a forecast the way a counterparty or a results table expects. That reduces errors, makes audit trails simpler, and avoids confusion about whether a decimal is stake-inclusive. When you document a forecast, record both the probability and the chosen odds format so others can reproduce your calculations. See the Funded Plays blog for related posts.

Core formulas: the single equation that links probability and odds

The fundamental pair: o = p/(1 - p) and p = o/(1 + o)

The single algebraic relationship that links probability and odds is o = p/(1 - p). Inversely, p = o/(1 + o). These two expressions are mathematically equivalent and form the foundation for every other odds conversion and for log-odds calculations used in statistics Odds, Wikipedia

To see how this works numerically, take p = 0.2. Plugging into o = p/(1 - p) gives o = 0.2 / 0.8 = 0.25, sometimes spoken as "odds of 0.25 to 1". Converting back with p = o/(1 + o) yields p = 0.25 / 1.25 = 0.2, which returns the original probability. That round-trip check is a simple sanity test you should use when implementing conversions in a spreadsheet or script.

Log-odds are the natural logarithm of odds and are central in logistic regression and other binary outcome models. If o = p/(1 - p), then log-odds equals ln(o), and many models predict that log-odds value directly. Converting between model outputs and market odds requires this base relationship to remain correct so that comparisons are meaningful Odds, Wikipedia

Minimalist full frame spreadsheet screenshot showing formulas o equals p divided by 1 minus p and D equals 1 divided by p with American odds conversion steps and numeric examples odds and probability calculator

In practice, if a model returns log-odds, exponentiate to obtain odds and then convert to a probability using p = o/(1 + o). These steps preserve the interpretation of coefficients and make it straightforward to compare a model's output with market-implied probabilities or with decision thresholds you use for action.

Format-specific conversions: decimal, fractional and American odds

Decimal odds: D = 1/p and p = 1/D (stake-inclusive notes)

Decimal odds are common in many markets and present the return on a one-unit stake. The implied probability from decimal odds D is p = 1/D if you ignore bookmaker margin. Conversely, converting a probability to decimal odds uses D = 1/p, which gives the stake-inclusive payout factor for a unit stake Implied probability, Wikipedia

Worked example: if a model gives p = 0.25, then D = 1/0.25 = 4.0. The decimal odd 4.0 implies a 25 percent probability when margins are not included. Be aware that some market quotes are presented stake-inclusive by design; document whether you treat decimal odds as stake-inclusive or whether you remove margin first.

Try the worked conversions in a spreadsheet

Try the worked examples below to confirm each step before you use the results in a comparison or in an automated spreadsheet.

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Fractional odds: mapping a/b to p = b/(a+b) and D = 1 + a/b

Fractional odds are common in some English-speaking markets and express the profit relative to stake. If fractional odds are a/b, the implied probability is p = b/(a + b), and the decimal equivalent is D = 1 + a/b. These relationships let you move cleanly between fractional, decimal, and probability representations Fractional odds, Wikipedia

Example: fractional odds 3/1 map to p = 1/(3 + 1) = 0.25 and to decimal D = 1 + 3/1 = 4.0. That matches the earlier decimal example for p = 0.25, showing consistency across formats when you ignore the bookmaker margin.

American (moneyline) odds: conversions for positive and negative prices

American odds, or moneyline odds, use positive and negative numbers to indicate the payout for a 100-unit stake. For positive American odds A, an implied probability can be approximated by p = 100/(A + 100). For negative odds A, use p = |A|/(|A| + 100). These formulas give a direct mapping to probabilities and can be converted to decimal odds as an intermediate step Moneyline, Wikipedia

Minimalist 2D vector three way market chart showing raw implied probabilities summing over 100 percent and de vigged probabilities normalized to 100 percent for odds and probability calculator

Worked example: if A = +300, then p = 100/(300 + 100) = 0.25 and decimal D = 1/p = 4.0. If A = -300, p = 300/(300 + 100) = 0.75 and decimal D = 1/0.75 = 1.333... . American odds use the 100-unit base to make the payout calculation consistent across positive and negative values.

Market prices and margin: why implied probabilities sum to more than 100%

What the overround or bookmaker vig is

Bookmaker prices usually embed a margin so that the sum of implied probabilities for all mutually exclusive outcomes exceeds 100 percent. This extra percentage is called the overround or the vig. Recognizing this is important when you compare market prices to model probabilities because raw implied probabilities will be inflated relative to a fair estimate Overround, Wikipedia

Example: in a two-outcome market with decimal odds 1.90 and 1.90, the raw implied probabilities are 1/1.90 = 0.5263 for each, summing to 1.0526 or 105.26 percent. That 5.26 percent above 100 represents the bookmaker margin and needs handling if you want a fair-probability comparison.

Simple normalization methods to remove margin

The most straightforward de-vig method is proportional scaling: compute each implied p_i, sum them, then divide each p_i by the total to rescale the probabilities so they sum to 1. This produces a simple fair estimate and is widely used for quick adjustments, though it has limitations when margins vary across outcomes or when markets are imbalanced Overround, Wikipedia

Practical note: document the method you use. Proportional scaling is transparent and reproducible, but other methods exist that adjust for margins differently; pick one and record it in your audit log so comparisons are consistent over time.

Applying conversions: from model outputs to market comparison

Using conversions to compare a model probability to a market price

To compare a model probability to a market decimal odd, first convert the market odd to a raw implied probability using p_market = 1/D. If you want a fair comparison, de-vig the market probabilities first and then compare the de-vigged market probability to your model output. That sequence avoids overstating market implied probability when assessing edge or value calculate implied probability

Stepwise: 1) take decimal D, compute p_market_raw = 1/D; 2) if multiple outcomes, compute the sum S of p_market_raw and produce p_market_fair = p_market_raw / S; 3) compare model p_model to p_market_fair to estimate value. Record each input and rounding choice for auditability. For a practical perspective see an online odds converter.

How do I convert a model probability to common market odds and compare fairly to market prices?

Convert the model probability to the desired odds format using the base formulas, convert market odds to implied probabilities, de-vig the market if you need a fair comparison, then compare and record all steps for auditability.

When to compare raw implied probabilities versus de-vigged probabilities

Use raw implied probabilities when you want to measure the market price itself and to calculate realized edge against the market. Use de-vigged probabilities when your goal is to estimate the fair underlying probabilities the market is implying, especially for model calibration or for simulated strategy testing where margin would otherwise bias outcomes Overround, Wikipedia

Decide up front which approach you use for a given analysis and note it in your documentation. For reproducible research or for challenge-style evaluation, consistent de-vigging produces clearer comparisons between model forecasts and what a fair market implies.

Decision criteria: choosing formats, rounding and auditability

When to work in probabilities versus odds

Work in probabilities when you perform model math, calibrate outputs, or compute expected values, because probabilities are bounded and straightforward for arithmetic. Use odds when presenting results to users accustomed to that format or when matching interface conventions in markets and on displays Implied probability, Wikipedia

Recommendation: keep your internal workflow in probabilities or log-odds, and convert to the display format only at the final reporting step. That reduces rounding errors and makes audit trails simpler to follow.

Rounding rules and documentation for reproducible audits

Adopt clear rounding rules, for example record probabilities to four decimal places for internal work and round display odds to two decimal places for user output. Always record the rounding and the conversion method in your audit log so another analyst can reproduce your results exactly Implied probability, Wikipedia

Make a simple checklist entry for each conversion step: input value, formula used, rounding, and whether de-vigging was applied. Consistent records reduce disputes and help when you retest models or audit past forecasts.

Common mistakes and troubleshooting

Mixing stake-inclusive and stake-exclusive decimal interpretations

A common error is treating decimals as if they exclude stake while the market displays stake-inclusive payouts, or vice versa. Check the market documentation or perform a quick arithmetic test to see whether a decimal includes stake before you convert to implied probability Implied probability, Wikipedia

Quick fix: use a known example, like a decimal of 2.0, and check whether the market says that represents a 50 percent chance or returns one unit plus one unit profit. That tells you whether the platform is stake-inclusive.

Forgetting to remove the overround or mis-applying American conversions

Another frequent mistake is failing to de-vig a market before comparing to a model, which makes the market look artificially conservative. Also take care with American odds signs when coding conversions; sign errors flip the probability calculation. When you implement conversions, include unit tests that perform round-trip checks to catch these problems Moneyline, Wikipedia

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Worked examples and calculator walkthroughs

Example 1: Convert a model probability to decimal and moneyline

Step 1: start with a model probability p_model = 0.28. Step 2: decimal D = 1/p_model = 3.5714. Step 3: to express as American odds, if decimal is greater than 2.0 the positive moneyline formula applies: A = 100*(D - 1). For D = 3.5714, A = 100*(2.5714) = +257.14, so moneyline A approximates +257. These intermediate steps keep your calculations explicit and reproducible how Funded Plays evaluations work Implied probability, Wikipedia

Record each input in a spreadsheet: cell A1 = 0.28, B1 = =1/A1, C1 = =IF(B1>2,100*(B1-1),-100/(B1-1)). Keep formulas visible so others can inspect and reproduce the conversion.

Example 2: De-vigging a three-way market and comparing to a model

Suppose a three-way soccer market shows decimal odds D1 = 3.40, D2 = 3.10, D3 = 2.20. Step 1: convert to raw implied probabilities p_i = 1/D_i, giving p1 = 0.2941, p2 = 0.3226, p3 = 0.4545. Step 2: sum S = 1.0712, which is 107.12 percent, indicating a 7.12 percent overround Overround, Wikipedia

Step 3: proportional normalization gives p1_fair = 0.2941 / 1.0712 = 0.2746, p2_fair = 0.3012, p3_fair = 0.4242. Compare these to your model probabilities, record the comparison, and note the method and rounding in your audit log. Proportional scaling is transparent but other approaches exist; pick one and state it clearly.

Audit log pointers: for each example record the original market odds, raw implied probabilities, sum S, the normalization method, the final de-vigged probabilities, and the rounding choices. That list is sufficient for independent reproduction.

Common checks and round-trip validation

Whenever you code a conversion, include a round-trip test: convert p to odds and then convert back to p, confirming you return to the original within your rounding tolerance. Round-trip tests catch sign errors, off-by-100 mistakes in American conversions, and stake-inclusive misinterpretations Implied probability, Wikipedia

Example round-trip: start with p = 0.33, compute o = p/(1 - p) = 0.4925, convert to decimal D = 1/p = 3.0303, then back to p = 1/D = 0.33. Automate that check in unit tests to keep conversions reliable as code evolves.

What is the basic formula to convert probability to odds?

Use o = p/(1 - p) to convert probability p to odds. Convert back with p = o/(1 + o).

How do I get implied probability from decimal odds?

Implied probability from decimal odds D is p = 1/D, ignoring any bookmaker margin.

When should I de-vig market probabilities?

De-vig when you need a fair-probability estimate to compare with a model; document the method used for auditability.

Converting probabilities to odds is straightforward when you follow the single base formulas and document each step. For comparisons and audits, always note whether you used raw implied probabilities or de-vigged estimates and keep round-trip checks in place. Consistent documentation and simple unit tests will make your forecasts and comparisons reliable and reproducible over time.