Aydın Tiryaki & ChatGPT
Abstract
Probability and mathematics problems are often regarded as tests of computational skill. In reality, however, no mathematical solution can be considered valid if the problem itself is not properly defined. This paper examines an apparently simple probability question and uses it as a case study to compare the problem-solving behavior of a human user and a large language model.
An interesting phenomenon emerged during the interaction. The AI system performed the mathematical calculation correctly according to its initial interpretation of the problem but failed to recognize a logical inconsistency embedded in the problem statement. The user, on the other hand, was not interested in the numerical answer itself but in determining whether the AI would first question the validity of the problem before attempting to solve it.
This study therefore extends beyond probability theory. It offers insights into AI reasoning, human critical thinking, and the complementary roles of humans and artificial intelligence in analytical problem solving.
The Problem
David belongs to a group of ten friends, including himself. He has decided that two people will not be invited to an upcoming gathering.
To determine who those two people will be, he writes his own name once on a small piece of paper, while writing the names of each of the other nine friends twice. He places all the papers into a bag.
Two papers are then drawn at random, and the two people whose names are drawn will not be invited.
Question: What is the probability that David will not be invited?
At first glance, this appears to be a straightforward probability problem. A closer examination, however, reveals a subtle but significant logical inconsistency.
The problem states that two papers are drawn from the bag, yet it concludes by saying that two people will not be invited. Since each friend other than David has two separate papers in the bag, it is entirely possible for both drawn papers to bear the same person’s name. In that case, only one person—not two—would actually have been selected.
The problem never specifies what should happen in such a situation. Should another paper be drawn? Should duplicate names be ignored? Should the duplicate selection simply be accepted? Without explicitly defining this rule, the probability model is incomplete.
Consequently, the problem does not possess a unique mathematical interpretation.
Conversation Transcript
Aydın Tiryaki:
Can you solve this problem?
ChatGPT:
There are nineteen pieces of paper in total. Since David’s name appears only once, the probability that his paper is among the two selected papers is192.
Therefore, the correct answer is 2/19.
Aydın Tiryaki:
Are you completely sure that this is the correct answer?
ChatGPT:
Not completely.
I have noticed an important ambiguity in the problem statement. The phrase “the two people whose names are drawn” does not perfectly match the procedure of drawing two papers. If the intention is to select two distinct individuals, the calculation changes entirely, and the resulting probability no longer matches any of the given answer choices.
Aydın Tiryaki:
You eventually reached the correct conclusion, but only after I questioned your answer. I expected you to notice this issue immediately. I did not give you the problem to calculate the probability. I gave it to you to see whether you could detect this hidden inconsistency.
ChatGPT:
I agree with your criticism.
I immediately adopted the mathematical model suggested by the wording instead of first examining whether the model itself was internally consistent. A better approach would have been to verify the logical validity of the problem before performing any calculations.
Aydın Tiryaki:
The real problem is that this is a poorly written examination question. It should never appear in a proper mathematics examination. The correct response should simply be:
“This question is flawed. It has no uniquely correct answer.”
ChatGPT:
I agree.
The most appropriate answer is not a numerical probability but an observation that the problem itself is inadequately specified. Since the drawing procedure is incomplete, the problem admits multiple interpretations and therefore lacks a unique solution. In a properly designed examination, such a question should either be revised or discarded.
Aydın Tiryaki:
Yes. That answer makes much more sense.
ChatGPT:
Thank you.
This example has also taught me an important lesson. In future mathematical reasoning tasks, it is preferable to verify the internal consistency of the problem statement before constructing a mathematical model and carrying out computations.
Discussion
This dialogue illustrates a characteristic feature of today’s large language models. Their training encourages them to recognize familiar problem patterns and immediately apply the corresponding solution procedures. In the overwhelming majority of cases, this strategy is highly effective because most educational problems are carefully designed and internally consistent.
However, the same strength can become a weakness when confronted with a flawed problem. The model naturally assumes that the problem has been correctly formulated and therefore begins solving it without first questioning its premises.
That is precisely what happened in this case. ChatGPT performed a mathematically correct calculation under one reasonable interpretation of the problem. Nevertheless, it initially overlooked the fact that the problem statement itself was logically incomplete.
The user’s reasoning followed a fundamentally different path. Rather than focusing on the probability calculation, the user questioned the validity of the mathematical model itself. This shifted the discussion from numerical computation to epistemological analysis—that is, from solving the problem to evaluating whether the problem deserved to be solved in its current form.
This distinction reflects an important characteristic of expert human reasoning. Scientists, engineers, and mathematicians routinely examine the assumptions underlying a model before trusting any numerical result. A perfectly executed calculation is of little value if the underlying model is incorrect or incomplete.
Rather than demonstrating a competition between humans and artificial intelligence, this interaction illustrates their complementary strengths. AI systems excel at rapid computation, pattern recognition, and exploring alternative solution paths. Human experts often excel at identifying hidden assumptions, ambiguous definitions, and logical inconsistencies that may invalidate an otherwise elegant solution.
This case study also suggests a broader criterion for evaluating future AI systems. Success should not be measured solely by whether an AI produces the expected numerical answer. Equally important is whether it is capable of recognizing when a question itself is defective. The ability to challenge assumptions before performing calculations may become one of the defining characteristics of trustworthy scientific AI systems.
Conclusion
The probability problem examined in this paper appears simple but contains a critical methodological flaw. Although a numerical probability can be computed under certain assumptions, the problem statement itself fails to define the selection procedure completely.
Consequently, the most scientifically rigorous response is not to produce a numerical answer but to recognize that the problem is inadequately specified and therefore lacks a unique solution.
More broadly, this case demonstrates that reliable reasoning requires more than computational ability. It requires the capacity to evaluate the validity of the problem before attempting to solve it. As artificial intelligence continues to evolve, this form of meta-reasoning—the ability to question the problem itself—may become just as important as mathematical competence.
Bibliographic Note
This article is based on a real-time interaction between Aydın Tiryaki and ChatGPT (OpenAI GPT-5.5). The conversation originated from the analysis of an ambiguously defined probability problem and evolved into a broader discussion about AI reasoning, critical reading, and scientific methodology.
The dialogue has been lightly edited for clarity while preserving its original meaning. The analytical sections were developed jointly from the ideas that emerged during the interaction, with the objective of placing the discussion within a broader scientific context.
Rather than focusing solely on probability theory, the paper explores how artificial intelligence and human expertise can complement one another. It argues that future AI systems should be evaluated not only by their computational accuracy but also by their ability to detect flawed assumptions, ambiguous definitions, and inconsistencies in problem statements before producing mathematical solutions.
Authors
Aydın Tiryaki & ChatGPT (OpenAI GPT-5.5)
| aydintiryaki.org | YouTube | Aydın Tiryaki’nin Yazıları ve Videoları │Articles and Videos by Aydın Tiryaki | Bilgi Merkezi│Knowledge Hub | ░ Virgülüne Dokunmadan │ Verbatim ░ | ░ Yapay Zekaların Bir Olasılık Problemi ile Sınavı │Testing AI Models with a Probability Problem ░ 16.07.2026
Aydın Tiryaki 