The difference ΔF between free energies has applications in biology, chemistry, and pharmacology. The value of ΔF can be estimated from experiments or simulations, via fluctuation theorems developed in statistical mechanics. Calculating the error in a ΔF estimate is difficult. Worse, atypical trials dominate estimates. How many trials one should perform was estimated roughly by Jarzynski [Phys. Rev. E 73, 046105 (2006)PLEEE81539-375510.1103/PhysRevE.73.046105]. We enhance the approximation with the following information-theoretic strategies. We quantify "dominance" with a tolerance parameter chosen by the experimenter or simulator. We bound the number of trials one should expect to perform, using the order-∞ Rényi entropy. The bound can be estimated if one implements the "good practice" of bidirectionality, known to improve estimates of ΔF. Estimating ΔF from this number of trials leads to an error that we bound approximately. Numerical experiments on a weakly interacting dilute classical gas support our analytical calculations.