A method of deriving the relation between radar-observed reflectivities Ze and gauge-measured rain intensity, R is presented. It is based on matching the probabilities of the two variables. The observed reflectivity is often very different from the true reflectivity near the surface due to the averaging of the real reflectivity field aloft by the beam, path attenuation, and variations in the drop-size distribution (DSD) between the pulse volume and the surface. The probability-matching method (PMM) inherently accounts for all of these differences on average. The formulation of the Ze − R functions is constrained such that 1) the radar-retrieved probability density function (PDF) of R is identical to the gauge-measured PDF, and 2) the traction of the time that it is raining is identical for both the radar and for simultaneous, collocated gauge measurements. This ensures that the rain measured by the radar is equal to that observed at the gauges. The resultant Ze − R functions are not constrained to be power laws. The method was applied to data obtained by a 1.65° beamwidth C-band radar and 22 gauges located near Darwin, Australia. The data were stratified by range and also by rainfall type. The resultant Ze − R functions manifest the nature of the PDF of R and the manner in which the beam effects the PDF of Ze through beam averaging and the average effects of C-band attenuation. In this regard, the Ze − R functions reflect the nature of the precipitation. This provides the hope for greatly improved rainfall measurements, especially for climatic purposes and for sufficiently large space-time domains. In those climates where the storms closely resemble one another, the relations may be used for individual storms over their lifetime or for a few storms at any one moment. The time-space domain in this study was sufficient for the instability driven convective storm in Darwin but too small for the synoptic-scale systems. The functions show a particularly strong range dependence for rain types characterized by large reflectivity gradients; that is, those in which the beam-average Ze differs most from the actual reflectivity Z. Because of the dominance of the beam effects relative to those due to variations in DSD, the use of radar polarimetry will improve the accuracy of rainfall retrievals, based on specification of the DSD alone, only with narrow beams or at short ranges and for small-scale, short-term purposes when the probability based relations may not be representative. Polarimetry is also valuable for smaller space-time domains than those for which the probability matched Ze − R relations may be valid.