https://cs.lozic.me/areas/probability/Recent content in Probability on CS TheoremsHugoen-usFri, 12 Nov 2027 12:00:00 +0100https://cs.lozic.me/posts/t073-bloom-filters/Fri, 12 Nov 2027 12:00:00 +0100https://cs.lozic.me/posts/t073-bloom-filters/<h2 id="symptom">Symptom</h2> <p>Every request to your service checks whether a user ID exists before doing anything else. The check hits the database. Ninety-eight percent of the time the answer is no, because most of the traffic is bots probing for accounts that were never created.</p>https://cs.lozic.me/posts/t035-bpp-pseudorandomness-and-derandomization/Fri, 23 Jul 2027 12:00:00 +0100https://cs.lozic.me/posts/t035-bpp-pseudorandomness-and-derandomization/<h2 id="symptom">Symptom</h2> <p>You need to test whether two arithmetic circuits compute the same polynomial. The deterministic approach is to expand both into normal form, and the expansion is exponentially large.</p> <p>The randomized approach takes four lines. Pick a random point, evaluate both circuits there, compare. If they differ as polynomials, the Schwartz-Zippel lemma says a random point catches it with probability at least $1 - d/|S|$. Repeat a few times and the error is negligible.</p>https://cs.lozic.me/posts/t104-universal-hashing/Fri, 23 Apr 2027 12:00:00 +0100https://cs.lozic.me/posts/t104-universal-hashing/<h2 id="symptom">Symptom</h2> <p>Your service went down under a hash collision attack.</p> <p>Somebody noticed your web framework put POST parameters into a hash table, found thousands of distinct keys colliding under its hash function, and posted a form with 20,000 of them. Every insert walked a chain. Quadratic behaviour, one CPU pinned per request, service dead. This actually happened, across PHP, Python, Ruby, Java and .NET in 2011, and again against Rust&rsquo;s default <code>HashMap</code> before it switched to SipHash.</p>https://cs.lozic.me/posts/t023-kolmogorov-complexity/Fri, 09 Apr 2027 12:00:00 +0100https://cs.lozic.me/posts/t023-kolmogorov-complexity/<h2 id="symptom">Symptom</h2> <p>You have two files, each exactly one megabyte.</p> <p>The first is a megabyte of the byte <code>0x00</code>. The second is a megabyte from <code>/dev/urandom</code>. Gzip the first and you get a few hundred bytes. Gzip the second and you get slightly more than a megabyte, because the header costs something and there is nothing to exploit.</p>https://cs.lozic.me/posts/t009-balls-into-bins/Fri, 12 Feb 2027 12:00:00 +0100https://cs.lozic.me/posts/t009-balls-into-bins/<h2 id="symptom">Symptom</h2> <p>You have 100 backends and a load balancer hashing request IDs to pick one. Expected load per backend is exactly 1%, and you have checked the hash is good.</p>https://cs.lozic.me/posts/t008-concentration-bounds/Fri, 18 Dec 2026 12:00:00 +0100https://cs.lozic.me/posts/t008-concentration-bounds/<h2 id="symptom">Symptom</h2> <p>Your service handles four billion requests a month and you want the error rate. Computing it exactly means a job over four billion log lines. Someone suggests sampling ten thousand of them.</p>https://cs.lozic.me/posts/t007-linearity-of-expectation/Fri, 27 Nov 2026 12:00:00 +0100https://cs.lozic.me/posts/t007-linearity-of-expectation/<h2 id="symptom">Symptom</h2> <p>You are analyzing a hash table. You want the expected number of buckets that end up empty. The bucket occupancies are all tangled together — one key landing in bucket 3 makes every other bucket slightly less likely to be chosen — and the dependencies look like they will make the sum intractable.</p>https://cs.lozic.me/posts/t074-the-birthday-bound/Fri, 06 Nov 2026 12:00:00 +0100https://cs.lozic.me/posts/t074-the-birthday-bound/<h2 id="symptom">Symptom</h2> <p>You need a short ID for uploads. Eight hex characters feels generous, so you take the first 32 bits of a hash and move on.</p> <p>At about 80,000 uploads, two files collide, and one of them silently overwrites the other, and the bug report says the customer&rsquo;s invoice contains someone else&rsquo;s line items.</p>