BTTS Probability Calculator from Goal Rates (λ)
Both Teams to Score, usually shortened to BTTS, asks whether the home and away teams each score at least one regulation-time goal. Under independent Poisson, BTTS probability follows directly from the two team goal rates. Dixon-Coles then modifies that probability through its treatment of 1-1.
- BTTS Yes wins when both teams score at least one goal.
- Under independence, P(BTTS) = P(home scores) × P(away scores).
- With Poisson rates, P(BTTS) = (1 − e−λh)(1 − e−λa).
- A negative Dixon-Coles ρ increases 1-1 and therefore increases BTTS probability.
- BTTS + Under 2.5 is exactly the 1-1 scoreline.
- BTTS + Over 2.5 excludes 1-1 and is unchanged by the standard Dixon-Coles correction at fixed λ.
- The match total alone cannot determine BTTS; how the total λ is divided between teams matters.
BTTS probability calculator from λ
Enter home and away expected goals. Set ρ to zero for independent Poisson or use a small empirically justified Dixon-Coles value.
What is Both Teams to Score?
BTTS is a binary market. It does not ask which team wins or how many total goals are scored. It asks whether each team scores at least once.
| Score | BTTS settlement | Over/Under 2.5 | Reason |
|---|---|---|---|
| 0-0 | BTTS No | Under | Neither team scores |
| 1-0 | BTTS No | Under | Only the home team scores |
| 1-1 | BTTS Yes | Under | Both score, but total goals equal 2 |
| 2-0 | BTTS No | Under | Two total goals do not imply BTTS |
| 2-1 | BTTS Yes | Over | Both score and total goals equal 3 |
| 3-0 | BTTS No | Over | Over 2.5 does not imply BTTS |
Market rules usually refer to regulation time unless the bookmaker states otherwise. Extra-time goals should not be added to a standard 90-minute BTTS market.
BTTS and Over 2.5 are related but not interchangeable. A 3-0 score wins Over 2.5 and loses BTTS Yes. A 1-1 score wins BTTS Yes and loses Over 2.5.
Poisson formula for BTTS probability
Let:
Under Poisson, the probability that a team scores zero is:
Therefore, the probability that it scores at least once is:
If the two team goal counts are independent, multiply the two scoring probabilities:
Expanding the product gives the inclusion–exclusion form:
BTTS No is the complement:
Four mutually exclusive scoring states
| State | Probability under independence |
|---|---|
| Neither team scores | e−(λh+λa) |
| Only home scores | (1 − e−λh)e−λa |
| Only away scores | e−λh(1 − e−λa) |
| Both teams score | (1 − e−λh)(1 − e−λa) |
The four states sum to 100%. This is a useful implementation check.
Team scoring probabilities are developed further in football team totals from odds.
Why the match total alone cannot determine BTTS
Under independent Poisson, total goals follow:
The Over/Under market constrains Λ, but BTTS also depends on how Λ is split between the teams.
| λ home–away | Total λ | BTTS Yes | Over 2.5 | Interpretation |
|---|---|---|---|---|
| 1.40–1.40 | 2.80 | 56.76% | 53.05% | Balanced scoring rates maximize the chance both score |
| 2.00–0.80 | 2.80 | 47.61% | 53.05% | Same total, but the weaker attack lowers BTTS |
| 2.50–0.30 | 2.80 | 23.79% | 53.05% | High total can coexist with low BTTS in a one-sided match |
For a fixed total Λ, BTTS probability is largest when the two team rates are equal. This follows from symmetry and the fact that one very small λ makes one team unlikely to score.
Over 2.5 mostly measures total scoring environment; BTTS measures shared scoring. The same total-goal expectation can support very different BTTS probabilities.
Worked example from market-implied λ
Using the standard ImpliedScore demonstration market:
| Home | Draw | Away | Over 2.5 | Under 2.5 |
|---|---|---|---|---|
| 2.10 | 3.50 | 3.40 | 1.85 | 2.00 |
A Dixon-Coles fit with ρ = −0.03 gives approximate rates:
For the comparison below, both calculations hold these λ values fixed. The Poisson result is an independence baseline using the Dixon-Coles-fitted rates, not a separately re-fitted Poisson model. Re-fitting under independence may produce slightly different λ values and BTTS probabilities.
Step 1: probability each team scores
Step 2: independent BTTS probability
Step 3: fair odds
Scoring-state decomposition
| State | Probability | Fair odds |
|---|---|---|
| Neither team scores | 6.42% | 15.56 |
| Only home scores | 24.12% | 4.15 |
| Only away scores | 14.61% | 6.84 |
| Both teams score | 54.85% | 1.82 |
The “neither” row above uses the independent 0-0 probability for the four-state decomposition. Dixon-Coles raises 0-0 and changes the joint states while preserving each team’s separate scoring probability.
Enter 1X2 and Over/Under odds to derive λ, compare Poisson and Dixon-Coles and view BTTS alongside the full score matrix.
Open the football score calculatorHow Dixon-Coles changes BTTS probability
The standard Dixon-Coles correction changes four score cells:
| Score | τ multiplier | BTTS category |
|---|---|---|
| 0-0 | 1 − λhλaρ | BTTS No |
| 1-0 | 1 + λaρ | BTTS No |
| 0-1 | 1 + λhρ | BTTS No |
| 1-1 | 1 − ρ | BTTS Yes |
The independent probability of 1-1 is:
Dixon-Coles changes BTTS by the same amount that it changes 1-1:
With λhome = 1.559, λaway = 1.186 and ρ = −0.03:
| Market | Independent Poisson | Dixon-Coles | Change |
|---|---|---|---|
| BTTS Yes | 54.85% | 55.20% | +0.36pp |
| BTTS No | 45.15% | 44.80% | −0.36pp |
| BTTS Yes fair odds | 1.82 | 1.81 | Lower odds after probability increase |
Why team-to-score probabilities do not change directly
Dixon-Coles preserves each team’s Poisson marginal distribution at fixed λ. Therefore:
- P(Home scores) remains 1 − e−λh;
- P(Away scores) remains 1 − e−λa;
- their joint probability changes because independence has been relaxed.
Under dependence, P(A and B) is not necessarily P(A)P(B). Both teams retain the same separate scoring probabilities, but the probability that both score changes.
See the Dixon-Coles model guide for the complete τ correction.
BTTS with Over and Under 2.5
BTTS Yes + Under 2.5
If both teams score and the total is below 2.5, the only possible score is 1-1:
Under independent Poisson:
Under Dixon-Coles:
BTTS Yes + Over 2.5
This market includes every BTTS score except 1-1:
Under independent Poisson:
Why Dixon-Coles leaves this combination unchanged at fixed λ
Dixon-Coles increases BTTS through 1-1. The same adjusted 1-1 probability is then subtracted when the market requires Over 2.5. The two changes cancel:
| Combination | Independent Poisson | Dixon-Coles ρ = −0.03 |
|---|---|---|
| BTTS Yes + Under 2.5 | 11.88% | 12.24% |
| BTTS Yes + Over 2.5 | 42.97% | 42.97% |
Do not multiply BTTS probability by Over 2.5 probability. The events are dependent. The correct joint probability must be calculated from the score matrix or the exact formula above.
How to remove margin from direct BTTS odds
The prices below are illustrative and are not a synchronized BTTS quote from the same bookmaker and timestamp as the earlier 1X2 and Over/Under example. Suppose a bookmaker quotes:
| Selection | Odds | Raw implied probability |
|---|---|---|
| BTTS Yes | 1.78 | 56.18% |
| BTTS No | 1.95 | 51.28% |
| Total | 107.46% |
Under proportional margin removal, the current ImpliedScore production baseline:
| Selection | Fair probability | Fair odds |
|---|---|---|
| BTTS Yes | 52.28% | 1.91 |
| BTTS No | 47.72% | 2.10 |
The model example above gives 55.20% under Dixon-Coles, 2.92 percentage points above the de-vigged direct-market probability.
This is a measurable disagreement, not automatic value. Possible causes include different odds timestamps, market-specific margin, model misspecification and rounding.
For complete-market margin removal and alternative methods, use the No-Vig Calculator.
Can BTTS odds identify the two team goal rates?
One BTTS probability gives one equation:
There are two unknown rates. Therefore, BTTS alone does not identify a unique pair. Infinitely many λ combinations can generate the same BTTS probability.
Add a total-goals probability
A match-total market constrains:
Given Λ and BTTS, the split can often be solved numerically. For a fixed total rate, the BTTS probability is maximised when the rates are equal:
A solution exists only when the supplied BTTS probability does not exceed this balanced-rate maximum. Probabilities below the maximum generally produce two mirror solutions, (λhome, λaway) and (λaway, λhome), while the maximum itself produces the single balanced solution λhome = λaway = Λ/2. The BTTS formula is symmetric: swapping λhome and λaway gives the same result.
A directional market such as 1X2 or Asian handicap is still required to determine which team receives the larger rate.
BTTS is useful as an additional fitting target. It can expose a score model that fits 1X2 and totals but allocates too much or too little joint scoring probability.
How to compare a BTTS model with the market
Compare fair probabilities, not raw reciprocal odds
Remove the Yes/No market margin first. Comparing a margin-free model probability with raw bookmaker implied probability overstates the market estimate.
Use prices from the same timestamp
A BTTS market collected after a lineup announcement should not be compared with λ fitted from earlier 1X2 and totals.
Inspect all source-market residuals
If the score model uses 1X2 and Over/Under as fitting targets, compare its output with the direct BTTS market as an out-of-target residual:
This residual is especially useful because BTTS is sensitive to the split between λ values and to goal dependence.
Expected return at offered odds
For model probability p and decimal odds O:
The result is conditional on the model probability being correct. A positive value does not validate the model.
How to validate BTTS probabilities
- Freeze every forecast before kickoff. Store λ values, model, ρ, BTTS probability, market prices and timestamp.
- Use a binary outcome. Set y = 1 when both teams score in regulation time and y = 0 otherwise.
- Plot calibration. Forecasts near 60% should produce BTTS in about 60% of matches.
- Calculate Brier score. For each match, use (p − y)².
- Calculate log loss. Penalize confident incorrect Yes and No probabilities.
- Compare paired models. Score independent Poisson, Dixon-Coles and direct de-vigged market probabilities on the same fixtures.
- Check probability ranges and leagues. Aggregate accuracy can hide systematic bias among favourites, balanced games or high-total matches.
- Test combinations separately. BTTS + Over 2.5 is a different binary outcome and needs its own validation.
Profit or ROI can supplement evaluation at offered prices, but it is not a proper probability score and is highly variable in small samples.
See football probability calibration for reliability diagrams, Brier score and log loss.
Limitations of a Poisson BTTS model
λ values can be wrong
Historical ratings, projected xG and market-implied rates can all miss current lineup, tactical or contextual information.
Independence may fail
Score state, red cards and shared match tempo can link the two scoring processes.
Dixon-Coles is only a local correction
It changes BTTS through 1-1 but does not directly alter higher joint-scoring cells such as 2-2, 3-2 or 3-3. More flexible redistribution patterns and alternative marginals are discussed by Michels, Ötting & Karlis (2025).
Poisson fixes marginal variance equal to the mean
A team that alternates between low- and high-tempo regimes can have a different scoring distribution from one fixed Poisson rate.
Direct BTTS margin removal is method-dependent
Proportional normalization is the current ImpliedScore production baseline, but it is not the only way to distribute overround; alternative de-vig methods should be treated as explicit sensitivity choices rather than silently mixed across samples.
Market settlement must match the model
Regulation time, extra time, abandoned matches and official goal attribution can affect settlement.
In-play BTTS needs updated rates
After a goal, red card or substantial elapsed time, pre-match λ values are no longer the current remaining-goals model.
Broader structural risks are covered in where Poisson football models fail.
Practical conclusion
- BTTS depends on two team scoring rates, not only the total expected goals.
- Under independent Poisson, multiply each team’s probability of scoring at least once.
- Dixon-Coles changes BTTS through the adjusted 1-1 probability.
- BTTS + Under 2.5 is exactly 1-1.
- BTTS + Over 2.5 must be calculated jointly and is not the product of two market probabilities.
- Direct BTTS odds should be de-vigged before comparison with model output.
- Validate frozen probabilities with calibration, Brier score and log loss.
References
- Maher, M. J. (1982). Modelling Association Football Scores . Statistica Neerlandica, 36(3), 109–118.
- Dixon, M. J. & Coles, S. G. (1997). Modelling Association Football Scores and Inefficiencies in the Football Betting Market . Journal of the Royal Statistical Society: Series C, 46(2), 265–280.
- Michels, R., Ötting, M. & Karlis, D. (2025). Extending the Dixon and Coles Model: An Application to Women’s Football Data . Journal of the Royal Statistical Society: Series C, 74(1), 167–186.
- ImpliedScore. Methodology: From Odds to a Score Probability Matrix.
- ImpliedScore. Football Team Totals from Betting Odds .
- ImpliedScore. Poisson vs Dixon-Coles vs Bivariate Poisson .