This article presents the properties of complete randomization (e.g., coin toss) and of the random allocation rule (random permutation of n/2 of n elements). The latter is principally used in cases where the total sample size n is known exactly a priori. The likelihood of treatment imbalances is readily computed and is shown to be negligible for large trials (n greater than 200), regardless of whether a stratified randomization is used. It is shown that substantial treatment imbalances are extremely unlikely in large trials, and therefore there is likely to be no substantial effect on power. The large-sample permutational distribution of the family of linear rank tests is presented for complete randomization unconditionally and conditionally, and for the random allocation rule. Asymptotically the three are equivalent to the distribution of these tests under a sampling-based population model. Permutation tests are also presented for a stratified analysis within one or more subgroups of patients defined post hoc on the basis of a covariate. This provides a basis for analysis when some patients' responses are assumed to be missing-at-random. Using the Blackwell-Hodges model, it is shown that complete randomization eliminates the potential for selection bias, but that the random allocation rule yields a substantial potential for selection bias in an unmasked trial. Finally, the Efron model for accidental bias is used to assess the potential for bias in the estimate of treatment effect due to covariate imbalance. Asymptotically, this probability approaches zero for complete randomization and for the random allocation rule. However, for finite n, complete randomization minimizes the probability of accidental bias, whereas this probability is slightly higher with a random allocation rule. It is concluded that complete randomization has merit in large clinical trials.