https://rcor.ro/tags/initial-conditions/ Recent content in Initial Conditions on rcor Hugo en Tue, 10 Nov 2026 00:00:00 +0000 https://rcor.ro/posts/2026-10-31-two-heads-in-a-row/ Tue, 10 Nov 2026 00:00:00 +0000 https://rcor.ro/posts/2026-10-31-two-heads-in-a-row/ <p>We flip a fair coin repeatedly and stop as soon as we observe <strong>two heads in a row</strong> <code>HH</code>. What is the expected number of flips?</p> <p>When flipping a fair coin until observing two consecutive heads (<code>HH</code>), a common initial intuition suggests an expected waiting time related to $ p = 1/4$ (i.e., 4 flips<sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup>). However, this naive calculation assumes independent blocks of size two, ignoring the <strong>sliding-window nature</strong> and the <strong>asymmetric failure resets</strong> inherent in sequence matching.</p>