The Cache Coherent (CC) and the Distributed Shared Memory (DSM) models are standard shared memory models, and the Remote Memory Reference (RMR) complexity is considered to accurately predict the actual performance of mutual exclusion algorithms in shared memory systems. In this paper we prove a tight lower bound for the RMR complexity of deadlock-free randomized mutual exclusion algorithms in both the CC and the DSM model with atomic registers and compare\&swap objects and an adaptive adversary. Our lower bound establishes that an adaptive adversary can schedule $n$ processes in such a way that each enters the critical section once, and the total number of RMRs is $\Omega(n \log n/\log\log n)$ in expectation. This matches an upper bound of Hendler and Woelfel (2011).