  The detection of nonlocal correlations in a Bell experiment implies almost by
definition some intrinsic randomness in the measurement outcomes. For given
correlations, or for a given Bell violation, the amount of randomness predicted
by quantum physics, quantified by the guessing probability, can generally be
bounded numerically. However, currently only a few exact analytic solutions are
known for violations of the bipartite Clauser-Horne-Shimony-Holt Bell
inequality. Here, we study the randomness in a Bell experiment where three
parties test the tripartite Mermin-Bell inequality. We give tight upper bounds
on the guessing probabilities associated with one and two of the parties'
measurement outcomes as a function of the Mermin inequality violation. Finally,
we discuss the possibility of device-independent secret sharing based on the
Mermin inequality and argue that the idea seems unlikely to work.
