 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-66/4/ii/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

Every word in the typical set has probability at least

2^{- n (H (Z) + δ)}

. Consequently

1 \geq \sum_{z^{n} \in T_{δ}^{n}} P_{Z}^{n} (z^{n}) \geq ∣ T_{δ}^{n} ∣ 2^{- n (H (Z) + δ)} .

Rearranging proves the upper estimate in the typical-set cardinality bounds:

∣ T_{δ}^{n} (P_{Z}) ∣ \leq 2^{n (H (Z) + δ)} .

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