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A completely randomized design assigns treatments directly to available experimental units using complete randomization, without blocking. Fixed group sizes can be enforced. The residual then includes uncontrolled between-unit variation.

Complete randomization chooses uniformly among all treatment assignments having prespecified treatment-arm sizes.

Under complete randomization, every assignment vector with exactly $n_z$ patients receiving treatment $z$ is equally likely. Thus, for $z=(z_1,\dots ,z_n)$ and $N_r(z)={\sum }_i{1}_{\{z_i=r\}}$,The reciprocal probability $n!/(n_0!n_1!n_2!)$ is the multinomial coefficient counting assignments with those arm sizes.

Under independent assignment,The arm-count vector has a multinomial distribution. Conditional on $N_r=n_r$ for $r=0,1,2$, every compatible vector has the same factor ${\prod }_r{\pi }_r^{n_r}$, sofor compatible $z$, and zero otherwise. Conditioning independent assignment on its arm sizes therefore recovers complete randomization.

A randomized controlled trial assigns experimental units to competing treatments by a specified random mechanism and compares their outcomes. Under complete randomization of $N$ units, exactly $n_1$ receive assignment one and $n_0=N-n_1$ assignment zero. With fixed potential outcomes, each unit has assignment probability $n_1/N$. ThereforeThis proves design-unbiasedness of the mean difference for the finite-population average treatment effect, assuming consistency in causal inference and no interference in causal inference.

Randomization controls the assignment distribution; it does not force every realized baseline covariate difference to be zero. Equal assignment probabilities yield unconditional exchangeability in the design, whereas covariate-dependent randomization can require conditional exchangeability and an analysis respecting that design.^[ref]

When assigned and received treatments differ, an intention-to-treat analysis estimates an assignment-policy effect. A treatment-receipt effect needs additional assumptions. Allocation concealment, blinded outcome assessment and handling of missing follow-up address other sources of distortion beyond random assignment itself.