# Loss Distribution Model for Metal Discovery Probabilities

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## Abstract

A loss distribution model is developed and applied to estimate the probabilities of finding enough new mineral deposits to meet demand, throughout the rest of this century, of the ten metals: Au, Ag, Cu, Mo, Pb, Zn, Ni, Co, Cr, and platinum-group elements (PGE). The model assumes that the necessary amount of metal exists in undiscovered mineral deposits and looks only at the probabilities of finding the required amount of metal in undiscovered resources. The probability density function that describes aggregate tonnage discovery over a specified period of time is the convolution of a mass density function that describes number of discoveries per unit time with a probability density function that describes deposit size distribution. Two alternatives for the deposit size distribution density function are used: pure lognormal or lognormal with a power-law upper tail, both taken from Patiño Douce (Nat Resour Res 25:365–387, 2016c). A Pascal (negative binomial) distribution is found to accurately reproduce number of yearly metal discoveries for the period 1950–2007, and is used to estimate future discovery probabilities. The convolution of the two functions is accomplished by means of a simple Monte Carlo method. Numerical experiments consisting of 10^{6} iterations are used to estimate the aggregate tonnage most likely to be discovered until the year 2100, as well as the largest deposit most likely to be discovered over the same time period, together with confidence intervals for the two quantities. The results for Au, Ag, Zn, and Cu strongly suggest that serious shortages of these metals are likely to occur before the year 2100. At the other end of the spectrum, the models suggest that supplies of Mo and Co are not likely to become critical over that time frame. Ni and Pb occupy intermediate positions, and the results for Cr and PGE are inconclusive, chiefly owing to the large variability found in their deposit size distributions. Using a pure lognormal distribution versus a lognormal distribution with a power-law upper tail for deposit sizes does not affect these conclusions. The results do not prove nor disprove that the required amount of metal exists in undiscovered resources, but provide concrete actionable intelligence about the intensity of the exploration efforts needed to find the metal, if the deposits exist.

## Keywords

Loss distribution Metal resources Mineral deposit discovery probability Statistics## Notes

### Acknowledgments

Thoughtful and thorough reviews by Frits Agterberg and an anonymous reviewer were most helpful in improving this manuscript and are gratefully acknowledged.

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