On the acceptance ratio in Monte Carlo computer simulations

The functional form of acceptance probabilities in Monte Carlo algorithms bears a resemblance to the distance functions which are specifically defined to be bracketed by the unit interval. This observation led us to seek the average distance between any two points on the unit interval and this by analogy resulted in a suggestion of an upper and a lower bound of 1/2 and 1/3, respectively, for the acceptance ratio or the average acceptance probability in Monte Carlo computer simulations.