How do college students clarify five sample spaces for Bertrand's chord problem?

The concept of probability has a unique characteristic that causes confusion when a sample space is not clearly defined. This inherent nature of probability has been demonstrated in Bertrand's chord problem, which is well-known as the paradox of probability theory. This study demonstrated that a single probability in Bertrand's chord problem can be obtained by modifying it to clearly redescribe its sample space, and examined that how college students clarify the sample space. To this end, five modified questions were formed using Bertrand's chord problem to ensure that the sample space of each question was clearly expressed and were used to develop a survey questionnaire. The participants of the survey were 68 college students studying mathematics or mathematics education. The results of this study demonstrated that many college students have difficulty to seek out the sample spaces in some probability problems. Thus we suggested the importance of emphasizing to clarify the sample space in probability education.