A version of random interval arithmetic, balanced random interval arithmetic, is proposed. Random interval arithmetic, useful in interval based optimisation algorithms, utilizes a mix of standard and inner interval operations to obtain interval bounding of a function. The guarantee of enclosure which standard interval operations give is sacrificed but there is a very high probability of success and the computational cost is significantly reduced. Here, random interval arithmetic is generalized in the sense that different probabilities of standard interval operations are allowed in the mix of standard and inner interval operations instead of 0.5 in the original proposal. The influence of the probabilities of standard and inner interval operations on the estimates of bounds for the ranges of function values are investigated experimenting with a system of linear interval equations and with two typical test problems from global optimisation. The test results seem promising for exploitation with hybrid global optimisation algorithms based on the ideas of statistical inference and interval methods. As in stochastic global optimisation methods there will not be an absolute guarantee of finding the global minimum but there is a very strong likelihood. (C) 2004 Elsevier Ltd. All rights reserved.