Parlay dependence explorer
The separate probabilities of two events do not determine the probability that both occur. Multiplication gives the parlay probability only under independence; dependence can move the joint probability anywhere inside exact limits imposed by the two individual probabilities.
The individual legs remain fixed throughout this experiment. Only their overlap changes. The resulting fair parlay odd can therefore change radically while both leg probabilities remain exactly the same.
The complete feasible range
The limits are the Fréchet bounds: max(0, P(A) + P(B) − 1) and min(P(A), P(B)). Values beyond them would require a negative probability in the joint table and are impossible.
Where the probability goes
| Leg B wins | Leg B loses | Total | |
|---|---|---|---|
| Leg A wins | 30.0%Both win | 30.0%A only | 60.0% |
| Leg A loses | 20.0%B only | 20.0%Neither | 40.0% |
| Total | 50.0% | 50.0% | 100% |
The row and column totals never change unless an individual leg changes. The four interior cells move because dependence redistributes probability without altering either marginal probability.
This is one point in the feasible range, not a fact implied by the separate legs.
No difference from independence.
Interpretation: this explorer does not estimate the real dependence between events. It shows the range compatible with the stated leg probabilities. A bookmaker may already account for dependence in a quoted same-event parlay; dependence creates value only when the quotation misprices it.