Suppose elements a1,…,ak of a boolean algebra A are assigned probabilities π1,…πk. Boole asked to determine the possible probabilities of some other element a∈A, given the assignment ai↦πi,(i=1,…,k). De Finetti's solution of Boole's problem yields a closed interval Ia⊆[0,1] such that the set of possible probabilities of a coincides with Ia. Ia is nonempty iff the assignment is consistent in de Finetti's sense. Now suppose the probability of ak undergoes a small perturbation πk↦πk+dπk. We study the resulting modification of Ia. For instance, we prove that the one-sided derivatives ∂length(Ia)/∂πk± always have a rational, (generally non-integer) value. In the particular case when each probability πi is rational and A is presented as a Lindenbaum algebra, all these derivatives are Turing computable. Their existence domain is decidable by the Tarski-Seidenberg algorithm for polynomial equations and inequalities in real closed fields. Our results build on de Finetti's consistency notion and his solution of Boole's problem, and extend Hailperin's polyhedral methods for combining bounds on probabilities.