Display uncertainty on spatialy distributed proportions (visualisation)

This question is related to Distribution of estimator of multiple (spatially related) proportions. We consider here the /visualisation/ issue.

Consider a spatial random process $$Z(s)$$, where $$s$$ denotes the spatial location.

Our objective is to delineate the zone $$\mathcal{Z}$$ where the probability that $$Z$$ exceed a given threshold $$\zeta$$ is above a specified probability $$r$$:

$$\mathcal{Z} = \{s | \mathrm{Pr} (Z(s) > \zeta) > r\} .$$

Displaying the estimate of the zone is straighforward using two colours (“exceed” and “not exceed”), or a drawing a fontier.

• or confidence bounds (for each considered location $$s$$.
How would you display the exceedance zone such that a non mathematician can use the map to take decision regarding where to sparingly perform an expensive action wherever the threshold is exceeded. Put another way, assume $$r$$ is low (1-5%) and we want to ward of the error of non acting despite threshold exceedance.
Following my answer to the other question, you can use the local value of $$\hat{P}(Z>\xi)$$ and display it with a colormap, plus a contour of the zone where it exceeds $$r$$.