  In this paper, we develop a family of bivariate beta distributions that
encapsulate both positive and negative correlations, and which can be of
general interest for Bayesian inference. We then invoke a use of these
bivariate distributions in two contexts. The first is diagnostic testing in
medicine, threat detection, and signal processing. The second is system
survivability assessment, relevant to engineering reliability, and to survival
analysis in biomedicine. In diagnostic testing one encounters two parameters
that characterize the efficacy of the testing mechanism, {\it test
sensitivity}, and {\it test specificity}. These tend to be adversarial when
their values are interpreted as utilities. In system survivability, the
parameters of interest are the component reliabilities, whose values when
interpreted as utilities tend to exhibit co-operative (amiable) behavior.
Besides probability modeling and Bayesian inference, this paper has a
foundational import. Specifically, it advocates a conceptual change in how one
may think about reliability and survival analysis. The philosophical writings
of de Finetti, Kolmogorov, Popper, and Savage, when brought to bear on these
topics constitute the essence of this change. Its consequence is that we have
at hand a defensible framework for invoking Bayesian inferential methods in
diagnostics, reliability, and survival analysis. Another consequence is a
deeper appreciation of the judgment of independent lifetimes. Specifically, we
make the important point that independent lifetimes entail at a minimum, a
two-stage hierarchical construction.
