The Galerkin truncated inviscid Burgers equation has recently been shown by the authors to be a simple model with many degrees of freedom, with many statistical properties similar to those occurring in dynamical systems relevant to the atmosphere. These properties include long time-correlated, large-scale modes of low frequency variability and short time-correlated "weather modes" at smaller scales. The correlation scaling in the model extends over several decades and may be explained by a simple theory. Here a thorough analysis of the nature of predictability in the idealized system is developed by using a theoretical framework developed by R.K. This analysis is based on a relative entropy functional that has been shown elsewhere by one of the authors to measure the utility of statistical predictions precisely. The analysis is facilitated by the fact that most relevant probability distributions are approximately Gaussian if the initial conditions are assumed to be so. Rather surprisingly this holds for both the equilibrium (climatological) and nonequilibrium (prediction) distributions. We find that in most cases the absolute difference in the first moments of these two distributions (the "signal" component) is the main determinant of predictive utility variations. Contrary to conventional belief in the ensemble prediction area, the dispersion of prediction ensembles is generally of secondary importance in accounting for variations in utility associated with different initial conditions. This conclusion has potentially important implications for practical weather prediction, where traditionally most attention has focused on dispersion and its variability.