This paper provides a bound on the number of numeric operations (fixed or floating point) that can safely be performed before accuracy is lost. This work has important implications for control systems with safety-critical software, as these systems are now running fast enough and long enough for their errors to impact on their functionality. Furthermore, worst-case analysis would blindly advise the replacement of existing systems that have been successfully running for years. We present here a set of formal theorems validated by the PVS proof assistant. These theorems will allow code analyzing tools to produce formal certificates of accurate behavior. For example, FAA regulations for aircraft require that the probability of an error be below 10-9 for a 10 hour flight.