Correlation changes the joint event
Examples of related legs include a team to win and the same team to win 1-0, a player to score and that player's team to score, or over match goals and both teams to score. The relationship can be positive, negative or logical rather than merely linear.
OpenStax states that multiplication of marginal probabilities requires independence. For dependent events A and B, use:
P(A and B) = P(A) * P(B given A)
Illustrative conditional calculation
Suppose a model estimates:
- P(Home win) = 0.55
- P(Score is 1-0 given a home win) = 0.25
Then:
P(Home win and 1-0) = 0.55 * 0.25 = 0.1375, or 13.75%
A naive multiplication using an unconditional P(1-0) would answer a different question. The values are illustrative and not a match forecast.
Product treatment
bet365's current related-contingency page gives an operator example in which ordinarily related selections cannot be accepted as a standard accumulator, while a specifically priced Bet Builder can be an exception. DraftKings' current rules also define treatment of correlated selections and separately governed same-game products.
| Product state | What to retain |
|---|---|
| Combination rejected | Rejection message and attempted legs |
| Combination accepted at a quoted price | Product name, combined price and timestamp |
| Standalone prices multiplied manually | Evidence that independence is justified |
| Same-game product repriced after a void | Original and revised receipt |
Dependence audit
- Draw arrows showing which leg changes another's probability.
- Separate logical containment from statistical association.
- Use a joint score, event-path or conditional model where appropriate.
- Compare the model joint probability with the quoted product price.
- Include void, push and repricing states in expected value.
- Backtest the complete combination, not only its individual legs.
OpenStax's expected-value method applies after the joint outcome probabilities and net payoffs have been defined.
Next step
Use Same Game Multi for the next part of this topic.
Separate four kinds of dependence
Use a label more precise than correlated:
| Relationship | Example | Suitable representation |
|---|---|---|
| Logical containment | Team wins 1-0 and team wins | Direct joint event; the narrower event implies the broader one |
| Shared event path | Player scores and team scores | Conditional or event-level model |
| Common external cause | Rain affects goals and corners | Joint model with the shared input |
| Shared model error | Two legs use the same biased rating | Error-dependence analysis |
NIST defines the correlation coefficient as a measure of linear relationship, so it cannot by itself establish every nonlinear or logical dependence. Validate the complete joint probability against held-out combinations where sufficient comparable data exists; for chronological data, scikit-learn's TimeSeriesSplit documentation explains why future observations must not enter earlier training sets.
OpenStax's treatment of intersections and mutually exclusive events supports useful model invariants: a joint probability cannot exceed either marginal probability, a contained event has the same joint probability as the contained event itself, and mutually exclusive legs have joint probability zero. These checks catch mapping defects before product prices are compared.
Continue learning
- Next guide: Goalscorer Accumulators
- Related guide: Half-Time Result Accumulators
- Definition: Correlation in Betting
Assumptions and limitations
Correlation does not automatically create value. Operator prices may already account for dependence, models can misstate conditional probabilities, and product rules can change.
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