Markov chain Monte Carlo methods do not compute partition functions Z, yet they allow ratios of quantities like Z to be estimated. For example, consider a random-walk Metropolis algorithm in a state space where the energy is zero in a connected accessible region, and infinitely large everywhere else; and imagine that the accessible space can be chopped into two regions connected by one or more corridor states. The fraction of times spent in each region at equilibrium is proportional to the volume of the region. How does the Monte Carlo method manage to do this without measuring the volumes?