Given a nucleic acid sequence, a recent algorithm allows the calculation of the partition function over secondary structure space including a class of physically relevant pseudoknots. Here, we present a method for computing base-pairing probabilities starting from the output of this partition function algorithm. The approach relies on the calculation of recursion probabilities that are computed by backtracking through the partition function algorithm, applying a particular transformation at each step. This transformation is applicable to any partition function algorithm that follows the same basic dynamic programming paradigm. Base-pairing probabilities are useful for analyzing the equilibrium ensemble properties of natural and engineered nucleic acids, as demonstrated for a human telomerase RNA and a synthetic DNA nanostructure.