A popular way to sample from the posterior distribution of a Bayesian probit regression model is given by a well-known data augmentation scheme. Despite its convenience and easiness of implementation, in practice it is often not clear how many iterations are needed for the corresponding Markov chain to converge: this might lead to a too short or too long burning phase, with a resulting loss of accuracy or computational resources. In this document we explore how to use couplings to upper bound the Total Variation distance from stationarity at each iteration: the usefulness of the methodology is illustrated on simulated data.