Exchangeable copulas are used to model an extra-binomial variation in Bernoulli experiments with
a variable number of trials. Maximum likelihood inference procedures for the intra-cluster correlation are
constructed for several copula families. The selection of a particular model is carried out using the Akaike
information criterion (AIC). Profile likelihood confidence intervals for the intra-cluster correlation are constructed
and their performance are assessed in a simulation experiment. The sensitivity of the inference to
the specification of the copula family is also investigated through simulations. Numerical examples are presented.