Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 224 / 2 / c / Solution

Conditionally on $X_1^n=x_1^n$, the uniform index $J$ selects the symbol $a$ with probability ${\hat{P}}_{x_1^n}(a)$. The law of total probability therefore givesThus $\overline{P}$ is the conditional mean type of a string in $B$.