The things we need, together.

For k selected services with access fractions p₁ … pₖ, the fraction with all selected services lies between:

max(0, p₁ + … + pₖ − k + 1) ≤ P(all) ≤ min(p₁ … pₖ)

These established Fréchet bounds are attainable: a nested arrangement of access reaches the upper endpoint; spreading the missing-service shares around a unit circle reaches the lower endpoint. A mixture preserves each marginal rate and fills the interval. No independence assumption is made.

Each tile is a display unit for one percentage point, not a sampled person. Service marks show a constructed access combination. When a tile straddles combinations, its background is split in proportion to their shares. The exact calculations use unrounded source values; displayed labels round to one decimal.

Only selected services enter the construction. A deselected service is not inferred to be present or absent. With one service, the interval collapses to its reported rate. The slider is a position between feasible endpoints, not a probability, forecast, intervention or confidence level.

All selected observations must exist in the same reference year for the same economy. We do not mix latest observations from different years or replace missing values with zero. The estimates refer to total population, treated as a common population for this conditional exercise. Provider surveys, models, reference frames and rounding can differ: a common year does not certify jointly harmonised observations.

The interval reflects missing overlap information only. Sampling error, model error, source revisions and comparability uncertainty are not included. Even a zero-width interval is not proof of exact real-world coverage. The country set is the 197-economy atlas cohort, not every territory in the world.