Formulation of probability theory problem with subtle condition

TL;DR

This paper presents four challenging probability problems emphasizing the importance of understanding conditions, discusses their solutions with numerical and programming links, and evaluates chatbot responses to these problems.

Contribution

It introduces four probability problems highlighting subtle conditions, provides detailed solutions with numerical and programming insights, and tests chatbot performance on these problems.

Findings

Chatbots showed varying understanding of subtle probability conditions.

Problems effectively reveal the importance of precise problem comprehension.

Solutions demonstrate the connection between probability conditions and programming logic.

Abstract

Problems in probability theory prove to be one of the most challenging for students. Here, we formulate and discuss four related problems in probability theory that proved difficult for first to fourth-year undergraduate students whose first language was not English. These examples emphasize how crucial it is to understand the conditions and requirements of the problems precisely before starting to solve them. We discuss the solutions to those problems in detail, complement them with numerical estimations, and link the conditions in the problems to the logical statements in Python programming language. We also tested two widely used chatbots (GPT-4o and Claude 3.5 Sonnet) by checking their responses to these problems.