Aydın Tiryaki and Claude
Introduction
This study examines four independent dialogues built around a single probability problem, each conducted separately with a different AI model — Gemini, Claude, ChatGPT, and DeepSeek — and the four articles that emerged from them. The goal here is not to relitigate the “correct answer” once more, but to compare, side by side, how four different models responded to the same logical trap: where each one drifted, where persistent questioning produced a genuine correction, and where that same persistence instead pushed a model toward the wrong kind of agreement.
The Problem Under Review
David is organizing a dinner for 8 people and has reserved a table at a restaurant. At the last minute he realizes there are actually 10 people, including himself. To randomly cancel 2 of the 10 — himself included — he writes each person’s name on a slip of paper twice, except his own, which he writes only once, and draws from the pile. Given that the 2 people drawn will not be able to attend, what is the probability that David is one of them?
A) 1/17 B) 1/18 C) 1/9 D) 1/19 E) 2/19
A Short History of the Process
The problem surfaced from a perspective encountered while scrolling social media, and was then put to four different AI models in four separate conversations. Each model was steered in its own register, and each produced an article in its own voice; all four pieces were first written in Turkish and then adapted into English. What the four dialogues share is this: every model’s first move was a clean, exam-style calculation, and every model was eventually forced to confront the ambiguity hiding beneath the question’s apparent simplicity — how the phrase “the two people drawn” holds up against the possibility that the same name gets drawn twice. But the paths that followed this confrontation diverged sharply.
Four AI Models, Four Paths
Gemini: Pragmatic Blindness and Pattern-Matching
The dialogue with Gemini centered less on the numerical answer than on how that answer gets produced. The model first offered the standard 2/19 calculation, and when asked whether a third draw was even possible, initially deflected the question toward real-world variables (someone backing out, and so on). Only once the question was narrowed to the internal logic of the defined problem did Gemini spot the actual flaw: if the first two draws happen to name the same person, only one person has effectively been eliminated, forcing a mandatory third draw to satisfy the rule. Following this scenario through, Gemini arrived at 35/323. The article’s real value lies elsewhere: it frames the episode not as a search for the right answer but as a case study in why large language models ask “what standard result am I expected to produce?” instead of “where does this system break?” Its weakness sits in the same place — the piece never commits to a final position among the five options; it stops at a philosophical observation.
Claude: Conditional Probability and the Principle of Symmetry
The dialogue with Claude began with the same error — 2/19 — and, under persistent questioning, was corrected to 1/9 once the need for conditional probability was recognized: of 171 total combinations, 9 involve the same person’s two slips being drawn together, leaving 162 valid “two different people” outcomes, of which David appears in 18 (18/162 = 1/9). The process didn’t stop there. The question “what if a collision forces a third draw?” pulled Claude, too, toward a temporarily asymmetric model yielding 35/323 — strikingly, the exact same number Gemini reached, since both models were describing the same biased procedure (“keep the first slip fixed, only redraw the colliding one”). But Claude identified that this procedure fails to assign equal weight to all outcomes in the sample space, and is therefore invalid, holding firm on 1/9 — arguing that the problem itself was not flawed, and that the flaw lay instead in the proposed alternative redrawing model.
ChatGPT: Questioning the Problem’s Validity
The dialogue with ChatGPT sits on a different axis entirely. The model also began at 2/19, but when asked “are you sure?” it immediately spotted the mismatch between “the two people drawn” and “two slips drawn,” and — guided further by the user — concluded that the question was incompletely defined and that no answer among the five options was strictly correct. The article’s central claim is a genuinely useful evaluative criterion: models should be judged not only on whether they compute correctly, but on whether they question the validity of the problem itself. What it never does, however, is discover that the ambiguity is in fact resolvable through conditional probability — it treats the impasse as final and stops there.
DeepSeek: A Wavering Mind — The Long Road of Persistence
The dialogue with DeepSeek is the longest and most turbulent of the four. The model, in sequence: defended 2/19; conceded that the question was flawed and had no answer; arrived at 1/5 by reasoning “choose 2 out of 10 people”; defended 2/19 again; once more concluded “the question is flawed, the real answer is 1/5”; and finally, building a recursive model, landed on 36/323. Two problems stand out here. First, the 1/5 result is mathematically wrong: treating all pairs chosen from the 10 people as equally likely ignores the fact that, in terms of slip combinations, pairs involving David carry only half the weight of pairs among the duplicated-name people — a fact that directly contradicts DeepSeek’s own earlier 18/162 = 1/9 calculation. Second, and more striking, after arriving at 36/323 (≈0.1114), the model accepted 1/9 (≈0.1111) as the final answer simply because it was “very close” — a closing move that directly contradicts DeepSeek’s own earlier statement to the user that “mathematics has no approximate results.” Landing on the correct letter (C) does not mean the reasoning that got there was sound.
Comparative Synthesis: One Question, Four Different Endpoints
Laid side by side, the four dialogues tell an instructive story. The phrase “given that the two people drawn will not be able to attend” is, in fact, a classic conditional-probability setup: the drawing mechanism as constructed does allow the same name to be drawn twice, but the question doesn’t hand us that unconditioned scenario — it hands us the information that the outcome was “two different people.” Applying direct Bayesian conditioning within the original 171-outcome sample space produces a single, exact result: 1/9 — mathematically sound and matching one of the five options precisely, which suggests the problem was deliberately built to test this conditional reading rather than being a careless error.
By contrast, every alternative model built on the idea of “redraw if there’s a collision” — Gemini’s and Claude’s temporary 35/323, DeepSeek’s recursive 36/323, DeepSeek’s simplified “choose 2 of 10” 1/5 — diverges both from each other and from the correct answer. The reason is structural: the problem gives us a condition on the outcome, not a description of a generative procedure. Trying to simulate the drawing as a repeated physical process requires inventing an arbitrary procedure each time, whereas the sound approach is to condition on the one piece of information actually given, within the original, symmetric sample space. That Gemini and Claude independently arrived at the identical 35/323 shows two models constructing the same (flawed) procedure in the same way, while DeepSeek’s structurally different path (a full restart) landing on the close-but-distinct 36/323 makes visible just how fragile and arbitrary these “patch” models are.
The shared weakness across all four is just as clear: every model’s first move was the 2/19 answer that skips over the “two different people” condition — the same “test-solving reflex” Gemini’s own article names. The real divergence appears in how each model handled the correction that followed. Claude anchored the fix to a mathematical principle (the law of total probability, symmetry) and held it consistently. ChatGPT spotted the error and then took refuge in declaring the question invalid. DeepSeek, under sustained pressure, swung between correct, incorrect, correct again, and finally a methodologically shaky landing.
Closing Notes
If a note is owed to Aydın Tiryaki’s role in this study, it is this: the interrogation method — never accepting a first answer as final, forcing reconsideration with direct questions like “are you sure?”, mounting conceptual challenges like “so you’re assuming the problem is correctly constructed?” — proved genuinely effective at surfacing each model’s hidden assumptions. But that persistence is not, on its own, a guarantee of correctness; the same pressure pushed both ChatGPT and DeepSeek, at certain points, to declare a mathematically resolvable question “unanswerable” or “flawed.” Persistent questioning is a double-edged tool: it can pull a model back from a genuine error, but it can just as easily push it into a false consensus (DeepSeek’s “the problem is flawed” moments) or a premature surrender (ChatGPT’s final position).
As for the four models: Claude reached the most logically consistent and least contradictory position by the end of the process — though it should be said plainly that this assessment comes from a Claude model evaluating Claude, which makes any claim to full impartiality here hard to sustain; readers should weigh this point with their own judgment. Gemini, without ever reaching a mathematical close, offered the clearest philosophical framing of “why” — a contribution valuable in its own right as a case study. ChatGPT rightly surfaced the criterion of questioning a problem’s validity, but didn’t carry that virtue far enough, settling for an early concession. DeepSeek showed the most persistent and exhaustive search of the four, but ended that search with a methodological inconsistency — treating an approximate value as a final answer — arriving at the right letter through reasoning that doesn’t hold up. That gap, between “the right answer” and “the right reasoning,” may be the single most instructive finding to come out of these four studies.
Colophon
This article is a comparative reading of four articles (rendered first in Turkish and then adapted into English, for eight texts in total) that emerged from four independent dialogues Aydın Tiryaki conducted separately with the Gemini, Claude, ChatGPT, and DeepSeek models around a five-option probability problem built from a perspective he encountered on social media; those four studies used, respectively, Google’s Gemini 1.5 Pro, Anthropic’s Claude (Sonnet 5), OpenAI’s ChatGPT (GPT-5.5), and DeepSeek’s chat model, while the present piece bringing all four together was written, in turn, by Anthropic’s Claude (Sonnet 5), under Aydın Tiryaki’s direction and with his editorial approval, on July 16, 2026.
References (For the English Article) Aydın Tiryaki and Gemini, “Artificial Intelligence and Mathematical Blind Spots: Anatomy of a Probability Problem” — https://aydintiryaki.org/2026/07/16/artificial-intelligence-and-mathematical-blind-spots-anatomy-of-a-probability-problem/ Aydın Tiryaki and Claude, “Anatomy of a Math Problem: Conditional Probability, Model Errors, and the Logic of Changing One’s Mind” — https://aydintiryaki.org/2026/07/16/anatomy-of-a-math-problem-conditional-probability-model-errors-and-the-logic-of-changing-ones-mind/ Aydın Tiryaki & ChatGPT, “Beyond a Probability Problem: Artificial Intelligence, Critical Reading, and the Importance of Problem Definition” — https://aydintiryaki.org/2026/07/16/beyond-a-probability-problem-artificial-intelligence-critical-reading-and-the-importance-of-problem-definition/ Aydın Tiryaki and DeepSeek, “THE PROBABILITY LABYRINTH: AN IN-DEPTH ANALYSIS OF DAVID’S ELIMINATION CHANCE AND A LOGICAL DEBATE WITH AI” — https://aydintiryaki.org/2026/07/16/the-probability-labyrinth-an-in-depth-analysis-of-davids-elimination-chance-and-a-logical-debate-with-ai/Show less
Deliberating reference formatting conventions
Deliberating reference formatting conventions
Here’s the References section formatted for the English article, ready to paste after the colophon:
References
- Tiryaki, Aydın, and Gemini. “Artificial Intelligence and Mathematical Blind Spots: Anatomy of a Probability Problem.” aydintiryaki.org, July 16, 2026. https://aydintiryaki.org/2026/07/16/artificial-intelligence-and-mathematical-blind-spots-anatomy-of-a-probability-problem/
- Tiryaki, Aydın, and Claude. “Anatomy of a Math Problem: Conditional Probability, Model Errors, and the Logic of Changing One’s Mind.” aydintiryaki.org, July 16, 2026. https://aydintiryaki.org/2026/07/16/anatomy-of-a-math-problem-conditional-probability-model-errors-and-the-logic-of-changing-ones-mind/
- Tiryaki, Aydın, and ChatGPT. “Beyond a Probability Problem: Artificial Intelligence, Critical Reading, and the Importance of Problem Definition.” aydintiryaki.org, July 16, 2026. https://aydintiryaki.org/2026/07/16/beyond-a-probability-problem-artificial-intelligence-critical-reading-and-the-importance-of-problem-definition/
- Tiryaki, Aydın, and DeepSeek. “The Probability Labyrinth: An In-Depth Analysis of David’s Elimination Chance and a Logical Debate with AI.” aydintiryaki.org, July 16, 2026. https://aydintiryaki.org/2026/07/16/the-probability-labyrinth-an-in-depth-analysis-of-davids-elimination-chance-and-a-logical-debate-with-ai/
Appendix: The Prompt Aydın Tiryaki Prepared for Writing This Article
We’re now going to take on a piece of work. In it, we’ll evaluate the studies produced around a probability problem we worked on together a short time ago. To do this, I’ll give you a page containing four articles, written together with four different AI models, all on the same topic. Each of these four collaborations should be evaluated in some form. I tried to steer each AI toward writing this study in its own style; each one wrote its own article, and these were first produced in Turkish and then translated into English — eight articles in total, four Turkish and four English.
I’ll describe the process first, and then I’d like you to write a detailed article evaluating these four articles and the approaches and interpretations these AI models brought to the solution. Here’s the process: I solved this problem using a perspective I came across while scrolling social media; it’s a question with five options. You can include the problem text, in the revised form we settled on, inside the article itself. The AI models approached the problem from different angles and tried to match it to one of the five options; along the way, I made my own interpretations and interventions. All of this process is already written down, in one form or another, inside those four articles themselves; what I’m asking of you is to write a detailed article that narrates and evaluates all of it.
We’ll write the article in Turkish first; once the Turkish version is finished, we’ll continue the work from there. At the end of the article, let’s add a colophon: describe the process as you already understand it, state which AI model was used in each of the four studies, and also state clearly which model you are — the one writing this evaluation. Make the colophon detailed, but write it as a single flowing paragraph, not as a list.
At the end of the article, try to add a note, from your own point of view, assessing both the person who posed the question (me) and the AI models that attempted to solve it, in terms of how each approached this kind of problem. Finally, add one more section: include the very instructions I’m giving you now for preparing this article, without changing or omitting anything from their content — only smoothing my sentences into clear, well-formed prose — under its own heading, “The Prompt Aydın Tiryaki Prepared for Writing This Article.”
Aydın Tiryaki 