Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 221 / 1 / i / b / Solution

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.