Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 3K / a / Solution

For a $q$-ary code with codeword lengths $l_1,\dots ,l_m$, Kraft inequality isChoose an infinite random $q$-ary string with independent uniform symbols. The event that it begins with codeword $i$ has probability $q^{-l_i}$. For a prefix-free code, these events are disjoint, so their probabilities sum to at most one. This proves necessity.