We study the effect of varying strength delta of bond randomness on the phase transition of the three-dimensional Potts model for large q. The cooperative behavior of the system is determined by large correlated domains in which the spins point in the same direction. These domains have a finite extent in the disordered phase. In the ordered phase there is a percolating cluster of correlated spins. For a sufficiently large disorder delta>deltat this percolating cluster coexists with a percolating cluster of noncorrelated spins. Such a coexistence is only possible in more than two dimensions. We argue and check numerically that deltat is the tricritical disorder, which separates the first- and second-order transition regimes. The tricritical exponents are estimated as betat/vt=0.10(2) and vt=0.67(4). We claim these exponents are q independent for sufficiently large q. In the second-order transition regime the critical exponents betat/vt=0.60(2) and vt=0.73(1) are independent of the strength of disorder.