This paper is a continuation of a recent ASME Conference paper entitled "Design of a Python-Based Plug-in for Bench-marking Probabilistic Fracture Mechanics Computer Codes with Failure Event Data" (PVP2009-77974). In that paper, which was co-authored by Fong, deWit, Marcal, Filliben, Heckert, and Gosselin, we designed a probability-uncertainty plug-in to automate the estimation of leakage probability with uncertainty bounds due to variability in a large number of factors. The estimation algorithm was based on a two-level full or fractional factorial design of experiments such that the total number of simulations will be small as compared to a Monte-Carlo method. This feature is attractive if the simulations were based on finite element analysis with a large number of nodes and elements. In this paper, we go one step further to derive a risk-uncertainty formula by computing separately the probability-uncertainty and the consequence-uncertainty of a given failure event, and then using the classical theory of error propagation to compute the risk-uncertainty within the domain of validity of that theory. The estimation of the consequence-uncertainty is accomplished by using a public-domain software package entitled "Cost-Effectiveness Tool for Capital Asset Protection, version 4.0, 2008" (http://www.bfrl.nist.gov/oae/
or NIST Report NISTIR-7524), and is more fully described in a companion paper entitled "An Economics-based Intelligence (EI) Tool for Pressure Vessels & Piping (PVP) Failure Consequence Estimation," (PVP2010-25226, Session MF-23.4 of this conference). A numerical example of an application of the risk-uncertainty formula using a 16-year historical database of probability and consequence of main steam and hot reheat piping systems is presented. Implication of this risk-uncertainty estimation tool to the design of a risk-informed in-service inspection program is discussed.