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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ia/paper-2/12f/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 12F / d / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

Divide the keystrokes into disjoint blocks of five, and let

A_{n}

be the event that block

n

is exactly HELLO. The events are independent events, and

P (A_{n}) = 2 6^{- 5} .

(1)

Consequently

\sum_{n = 1}^{\infty} P (A_{n}) = \infty.

(2)

Part (c) shows that infinitely many of these block events occur almost surely. Each occurrence is an occurrence of HELLO in the full typed sequence, so

P (HELLO appears infinitely often) = 1 .

(3)

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