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.


INTRODUCTION
Probability is a relatively familiar term that is widely used in this information age.For instance, the question 'what is the probability of obtaining heads when flipping a coin?' is asked ordinarily and most people answer " 1 2 ".This is because most people approach the question by thinking that a coin has a heads side and a tails side and the chance of the coin falling either heads or tails is the same.However, it is debatable whether the heads and tails sides of a coin are indeed obtained under the same condition or not.The designs on the heads and tails sides of a coin are different and it is uncertain whether the specific gravity in the coin is evenly distributed.One aspect that must be understood in the experiment of flipping a coin is that the assessment "the probability of each side is the same because they exist under the same condition" is not an absolute truth, but a selected assumption.In other words, the 1 2 probability of obtaining heads is not a transcendental and absolute truth, but it results from the assumption that each case has the same chance of occurring (Batanero, Henry, & Parzysz, 2005;Kim, 2008;Rubel, 2007;Woo, 1998).
Accordingly, the concept of transcendental and absolute probability does not exist.The classical definition of probability provided by Laplace in the 19 th century, the so-called mathematical probability, is based on the fundamental premise that each element within a sample space has the same chance of occurring (Alexander & Kelly, 1999;Gauvrit & Morsanyi, 2014;Gillies, 2000;Woo, 1998).According to the classical interpretation, probability is defined as the ratio of the number of elements of a certain event to the number of the elements of the sample space, and geometrical probability typically fits this interpretation as well (Lee, 1997).Considering this classical interpretation of probability, the importance of clarifying whether each element within the sample space is under the same condition has been widely recognized and is highlighted in school education (Batanero & Borovcnik, 2016;Garfield & Ahlgren, 1988).Many studies reported important results on students' probabilistic thinking (e.g., Agus, Peró-Cebollero, Penna, & Guàrdia-Olmos, 2015;Fischbein & Gazit, 1984;Fischbein & Schnarch, 1997;Green, 1983;Hawkins & Kapadia, 1984;Konold, 1989;Kwon & Lee, 2015;Kwon, Kim, & Lee, 2014;Rubel, 2007;Shaughnessy, 1977).Some researchers emphasized and focused on sample space that plays an important role in probabilistic tasks and situations (e.g., Chernoff, 2009;Chernoff & Zazkis, 2011;Jones, Langrall, & Mooney, 2007;Jones, Langrall, Thornton, & Mogill, 1997, 1999;Shaughnessy, 1998).
Despite these researchers' emphasizing the sample space, recently, Choi, Yun, and Hwang (2014) reported that pre-service mathematics teachers still have the difficulty to understand the sample space.They proposed that students should be provided with the opportunity to think about the sample space in probability and statistics education.We suggest that teacher educators and teachers would use Bertrand's chord problem as one of these opportunities.That's because we saw both groups of pre-service and in-service mathematics teachers who didn't understand the solutions of Bertrand's chord problem in our classrooms.Many college students and mathematics teachers didn't understand why there exist three solutions for this problem and why it is called a paradox.We thought that's because they didn't have the chance to meet its solutions exposing the term sample space although each solution represents chance and inherent equiprobability.Like this, the mathematical definition of probability can cause further confusion unless clearly defining the fundamental events that have equiprobability in probability space.Lee (1997) and Woo (1998) noted that this paradox is the limits to the mathematical interpretation of probability with its classical definition.In this article, we would represent the term Bertrand's chord problem instead of Bertrand's paradox.
With this backdrop, this study used Bertrand's chord problem to demonstrate that the limits to the mathematical definition of probability in some probability problems can be overcome by clarifying the sample space, and to emphasize that it is important to teach students to clearly define the sample space in probability problem.To this end, each solution of Bertrand's chord problem was represented using the term sample space and this problem was modified in five ways to ensure the existence of only one sample space.These modified problems were given to college students studying mathematics and mathematics education to demonstrate that how well they could clarify the sample spaces and obtain a single probability.
Bertrand's chord problem.We draw at random a chord onto a circle.What is the probability that it is longer than the side of the inscribed equilateral triangle?(Aerts & Sassoli de Bianchi, 2014, p. 1) Bertrand (1989) proposed this problem which leads to the different results P(A) = 1 2 , P(B) = 1 3 , and P(C) = 1 4 .These probabilities are according to each of the three assignments for 'equally possible situations'; (A) the linear line between centers of chord and circle, (B) angles of intersections of the chord on the circumference, and (C) the center of the chord over the interior area of the circle (Jaynes, 1973).Even though Bertrand presented three answers for it, Jaynes (1973) concluded that Bertrand's chord problem is well posed and has the unique probability P(A) = 1 2 .This conclusion was followed by his viewpoint toward probability theory that "the only valid basis for assigning probabilities is frequency in some random experiment (p.2)".Using an analogy with a cylindrical cake cutting, Rowbottom (2013) also insisted that Bertrand's original problem is vague and all Bertrand's three solutions are not the effective potential ways.
variations lead to different solutions can be solved.Porto, Crosignani, Ciattoni, and Liu (2011) basically accepted that Bertrand's results are all correct as well as many other possible ones because a chord drawing randomly cannot be uniquely defined.They provided a realistic physical experiment associating with it as alternative to Bertrand's chord problem.Furthermore, Klyve (2013)  Meanwhile, Aerts and Sassoli de Bianchi (2014) showed that Bertrand's chord problem includes an easy problem and a hard problem.They reported that the easy problem is solvable by clarifying Bertrand's chord problem in precise terms.They presented one example for a specific physical realization of Bertrand's chord problem which has the same to three different answers proposed by Bertrand, and then insisted that Bertrand's three solutions should be more easily explained via three different conditional probabilities.Thus, Aerts and Sassoli de Bianchi concluded that Bertrand's chord problem should not anymore be considered as a paradox.Also, they remade the hard problem using modified Bertrand's chord problem in that two points instead of a straight line are randomized.This hard problem became solvable by calculating a uniform average, which they called a universal average, over all possible ways of selecting an interaction.
Unlike above researchers who focused on Bertrand's chord problem's solutions and answers, Gyenis and Rédei (2014) suggested a new interpretation of it and investigated the relation between Bertrand's chord problem and the classical interpretation of probability.They argued that this paradox is harmonized with how the science uses mathematical probability theory to model phenomena, without making any damage the principle of indifference and the classical interpretation of probability.

Solutions to Bertrand's Chord Problem using the Term Sample Space
In general, many researchers and practitioners cited three typical solution methods that lead to the results Aerts & Sassoli de Bianchi, 2014;Borovcnik, & Bentz, 1991, Drory, 2015;Lee, 1997;Marinoff, 1994;Porto et al., 2011;Woo, 1998).Kim (2008) introduced other two other solutions which lead to the results of impossibility and indefinite, respectively.However, they described the results and probabilities without using the term sample space in three different solutions for it.In this section, we tried below five solutions to help mathematics teachers and students understand the solutions of Bertrand's chord problem in terms of sample space.Also, these were supposed to be the basis of five modified Bertrand's chord questions given as testing tool in the next section.
Solution 1. Suppose that a line segment passes through midpoint D of the opposite side of the vertex A of an equilateral triangle inscribed in a circle, as shown in Figure 1 and that the point where the line segment meets with the circle is B. Assume that the set of all chords that are perpendicular to line segment AB is the sample space.The length of such perpendicular chords in this probability space is determined by the location of point P, which is defined by the intersection of line segment AB with a perpendicular chord.
Suppose that C is a point where line segment AB meets with a chord that has the same length as the triangle side within the sample space and that C is not located on the opposite side of vertex A. When a point Plies on line segment CD, the length of the chord is greater than the side of the equilateral triangle inscribed in the circle.In other words, the probability in this case is the same as the probability that point P is on line segment CD.Therefore, the length of line segment CD is 1 2 of the length of diameter AB.Therefore, we obtain a probability of  Suppose that the other two vertices of the triangle inscribed in the circle are B and C. When the endpoint P of a chord is on arc BC, the length of the chord is greater than the side of the triangle inscribed in the circle.In other words, the probability in this problem is the same as the probability that point P is on arc BC.Therefore, the length of arc BC is 1 3 of the length of the circumference.Therefore, the probability is 1 3 .Solution 3. Suppose that the midpoint of a randomly drawn chord is M, as shown in Figure 3, and that the radius of a circle inscribed in an equilateral triangle is ′.A circle circumscribed about the equilateral triangle is denoted by  and the circle with radius ′ by ′.The probability of the question is the same as the probability that the midpoint M is within the circle ′ in a probability space of which a sample space is defined as a set of midpoints M of randomly drawn chords.Therefore, the area of circle ′ is Suppose that an extension of chord AP having A as one endpoint meets with tangent line  at the point P′.Each chord AP has a one-to-one correspondence with each point P′ on tangent line .In other words, the statement "draw chord AP from vertex A" is equivalent with "select point P′ on tangent line ".Therefore, suppose that the set of all points defined by the intersections of the extensions of all chords drawn from vertex A of the triangle with tangent line  is the sample space of a probability space.The chords contained in this probability space are determined by the locations of points P′s.
Suppose an equilateral triangle ACD inscribed in a circle and that the contact points of the extensions of the two chords AC and AD with tangent line  are C′ and D′, respectively.When the contact point of the extension of a chord with tangent line  lies on line segment C′D′, the length of the chord is greater than the side of the equilateral triangle.In this case, however, the length of the line of the sample space is infinite; thus, if its probability is defined by the length, this leads to a contradiction.Therefore, it is impossible to solve the problem under this condition.Consider an equilateral triangle ABC inscribed in a circle and that the contact points of the extensions of the two chords AC and ADwith the closed curve are C′ and D′, respectively.When the contact point of the extension of a chord with the closed curve is on curve segment C′D′, the length of the chord is greater than the side of △ ABC.Suppose that the length of curve segment C′D′ is , the length of the closed curve is , and the length of a partial curve is .Therefore, the probability is   .However, considering that the closed curve circumscribed from point Acan be randomly drawn, the lengths  and  can vary.Therefore, the answer is indefinite.
As shown by the five solutions above, the cause of confusion in Bertrand's chord problem results from the ambiguous expression 'a chord randomly drawn', which does not clarify 'in which probability space its probability distribution is provided' or 'which fundamental events occur under the same condition'.In other words, this shows the importance of clarifying the premise of a sample space under the same condition in Bertrand's chord problem.Against this backdrop, we discussed below how well college students clarify each sample space in five modified Bertrand's chord questions asking a single answer.

Testing Tool
We modified Bertrand's chord problem to clearly show five fundamental events that have the same possibility to occur; in other words, to clearly find out the sample space which was intended in each question.Using the testing tools shown in Table 1, this study investigated how well the college students could correctly determine the sample space and the probability that satisfies the conditions of individual questions.We provided appropriate figure in each modified problem to help students find out all chords which were intended as the elements of the sample space.
The sample spaces designed in five modified questions in Table are as follows.In question No. 1, the sample space is defined as the set of all chords perpendicular to a fixed diameter of a circle.In question No. 2, the sample space is defined as the set of all chords drawn from one fixed vertex of a triangle.In question No. 3, the sample space is defined as a set collecting all midpoints Ms of randomly drawn chords.In question No. 4, the sample space

Participants
An offline survey using the above testing tools was planned to be conducted for the 3 rd -and 4 th -year students who major mathematics or mathematics education at the J national university.We recruited 68 volunteered students to participate in this survey using two department bulletin boards during one week.Table 2 shows the information of the survey participants, including their major subject, grade, and gender.Among the participants, the numbers of the 3 rd -and 4 th -year college students were 45 and 23, respectively.Three of the 68 participants responded that they had studied Bertrand's chord problem before and the remaining 65 students that they had not studied it yet.

Data Collection and Analysis
The survey was conducted in their department lecture rooms in October 2014.The survey date depended on grades and major.The participants were supplied with the survey questionnaire and were instructed to solve the questions in the given order and not to change their solution process and answers; this method was intended to clarify how well they approach individual questions.The total response time for the five questions was unlimited to ensure that the students could provide their answers without restrictions.
To analyze how well the students responded on the five modified Bertrand's chord questions, we input their all answers in Microsoft Office Excel and calculated the frequency and the response percentage on each answer.We also qualitatively analyzed their incorrect answers to interpret how they approached to each question and to infer whether they had errors in selecting a sample space or in the concept of probability itself.We presented the students' wrong solutions as real examples.

Responses to Modified Question No. 1
Question No. 1 was modified to define a set containing all chords perpendicular to a fixed diameter of a circle as the sample space.The distribution of the students' responses was as shown in Table 3.Approximately 72% of the surveyed students answered that the probability was 1 2 .In addition, two of the three students who responded that they had studied Bertrand's chord problem before answered " 1 2 " and the rest did not respond to Question No. 1.This indicated that most of the surveyed students could solve the probability question without much confusion when the sample space was defined as the set of all chords perpendicular to a fixed diameter of the circle.Because all incorrect answers, except one incorrect answer of 0, were presented without any information on the solution process, it was difficult to determine the causes of errors.We just could infer that they would have any difficulty to solve some geometrical probability problems or to understand some mathematical concepts involved in this question.The student who answered "0" considered a right-angled triangle inscribed in the circle instead of an equilateral triangle, owing to a lack of understanding of the question.

Modified Question No. 2
Question No. 2 was modified to define the sample space as a set of all chords drawn from one vertex of a triangle.The participants' response distribution was as shown in Table 4. Approximately 72% of the surveyed students answered that the probability was 1 3 .In addition, all three students who had studied Bertrand's chord problem before answered " 1 3 ".Among the 49 students who answered correctly to Question No. 1, 40 students answered correctly again.In other words, approximately 59% of the total surveyed students answered correctly both to Questions Nos. 1 and 2. This indicated that the surveyed students could solve the probability question without much confusion when its sample space was defined as the set of all chords drawn from one vertex of the triangle.To determine the causes of the incorrect answers to Question No. 2 of those who answered correctly to Question No. 1, their processes of solving the two questions were reviewed.Table 5 shows the answers of one student who answered correctly to Question No. 1 but incorrectly to Question No. 2. This shows that some students made an error in calculating the length of the partial curve that meets the conditions, not in selecting a sample space or one related to the concept of probability.

Modified Question No. 3
Question No. 3 was modified to define a set containing the midpoints of all random chords as the sample space; the response distribution is shown in Table 6.The number of students who correctly answered " 1 4 " to Question No. 3 was a mere 19% of the total sample.In addition, only one of the three students who had studied Bertrand's chord problem before answered " 1 4 " and the other two answered " 1 2 " and " 1 3 ".In particular, only seven of the 13 participants who answered correctly to Question No. 3 also answered correctly both to Questions No. 1 and No. 2. This indicates that more than 80% of the surveyed students calculated the probability in Question No. 3 incorrectly.
Unlike the two previous questions, Question No. 3 showed a relatively high incorrect answer rate.To determine the cause of this observation, some notable incorrect cases were reviewed, as shown in Table 7.
The cases (A) and (B) were the most common as participants' answers of this question.It was possible to assume that most of the students who incorrectly answered had a difficulty in finding the sample space of this question.These students revealed the error of recognizing only a part of the sample space intended in this question because they did not consider all random chords and limited it to the set of some chords.In other words, these students could understand the concept of probability but had difficulty to clarify the sample space satisfying the condition.From this result, we could reconfirm the reason why many pre-service and in-service mathematics teachers had difficulty to focus on the position of midpoint M which uniquely determines the chord among original three solutions of Bertrand's chord problem.It can be inferred that they had difficulty to seek out the set to all midpoints because of not considering all random chords.
In the cases (C) and (D), only one student answered incorrectly, as shown in the Table 7; the student succeeded in defining the sample space but made errors in calculating probability and in determining the figure of the event.define a sample space satisfying the axiom of probability in this question.This result seems to be attributable to the fact that cases with infinite length and the area of a sample space had not been addressed in the process of teaching the concept of probability.
The case shown in Figure 6 indicates the reason why most of the students answered " 1 3 " to this question.This case shows that most of participants failed to recognize the set of the contact points P′ between lines AP and tangent lines  as the sample space of this problem.In other words, they made an error in recognizing the set of the endpoints of all the chords drawn from the vertex A of the triangle as the sample space, like in modified Question No. 2. However, one student determined the sample space and event correctly, as shown in Figure 7.The student understood that the sample space in this question was the line segment C′D′ on the tangent line but failed to calculate the probability because the length of the line starting from vertex A was ∞ and the length of the line segment was finite.

Modified Question No.5
In terms of modified Question No. 5, the probability is indefinite because the closed curve, the sample space in this case, is circumscribed at point A and it can be randomly drawn.The response distribution of the surveyed students was as shown in Table 9.Approximately 66% of the participants answered " The reason why most of the surveyed students answered " 1 3 " to this question can be assumed from the case shown in Figure 8.This case shows that most of the participants failed to recognize the set of the contact points P′s between lines AP and the closed curve as the sample space in this case, like in Question No. 4. In other words, they made the same error as in Question No. 4.However, some students determined the sample space and event correctly and generalized the probability, as shown in Figure 9.
In Figure 9, the student correctly understood that the sample space was a random closed curve and that the event was the union of partial curves of this closed curve.This particular student also correctly determined the sample space in Question No. 4 and had not studied Bertrand's chord problem before.These results show that most of the students had difficulties in seeking out the sample space which was intended in the question, which is essential for defining the probability in some cases.

CONCLUSION AND IMPLICATIONS
The concept of probability has a unique characteristic that causes confusion when no clear definition is provided for the sample space, which is a universal set of elements with the same chance to occur.We can identify this inherent nature of probability via Bertrand's chord problem, which is known as a probability paradox.This problem has three solutions because it doesn't clarify sample space which is intended.So, we modified Bertrand's chord problem to have the single probability, and we examined how college students studying mathematics and mathematics education seek out the sample spaces represented in five modified Bertrand's chord questions.Additionally the study highlighted the importance of clarifying the sample space in defining the probability space.From the results of the survey, the following implications for probability education were obtained.
First, in some modified Bertrand's chord questions (No. 1 & No. 2) in which the sample space is clearly defined, approximately 70% of the surveyed college students correctly determined the sample space and event and calculated its probability.These results imply that many students can get a single probability by clarifying the sample spaces in some questions without any confusion.We suggest that it is necessary to clearly expose the sample space in probability questions unless we does not use Bertrand's chord problem as an open-ended question that can generate multiple answers with various approaches.
Second, in questions with clearly defined sample spaces (No. 3,No. 4,& No. 5), approximately 80% of the surveyed students failed to determine the sample spaces and events correctly.This means that many students still have difficulty in determining all possible outcomes that could occur in some probabilistic tasks even though they understand the concept of probability.It is in agreement with previous studies which suggested that many students have difficulty in seeking out the sample space (e.g., Borovcnik & Bentz, 1991;Borovcnik, & Kapadia, 2009;Konold, 1989;Speiser & Walter, 1998).Probability causes many controversies every moment when students accept it (Freudenthal, 1973) and students' probabilistic thinking improves by education (Fischbein, 1975;Fischbein & Schnarch, 1997;Kwon, Kim, & Lee, 2014;Rubel, 2007).With these reports, the result of this study indicates that the importance of the sample space should be highlighted in probability education and that the process of clearly recognizing sample spaces should be introduced.Under the current secondary school curriculum, probability questions are limited only to sample spaces with finite numbers of elements; however, it is necessary to highlight the importance of clearly recognizing sample spaces when teaching the concept of probability.At this stage, it is recommended to use Simpson's paradox (Borovcnik et al., 1991) when considering sample spaces with finite numbers of elements.
Finally, we are looking forward that this study helps mathematics teachers and students to understand Bertrand's chord problem and its solutions in terms of sample space.In particular, we suggest that the modified question No. 4 and No. 5 may be used to help a deeper understanding of the probability and its sample space because most of students had difficulties to clarify the sample spaces and to calculate probability in these questions.Also we agreed with the conclusion of Aerts and Sassoli de Bianchi ( 2004) that Bertrand's chord problem is not anymore a paradox.
defended Bertrand's intention against Rowbottom (2013)'s interpretation of Bertrand's chord problem.Klyve noted that Rowbottom misinterpreted what Bertrand said about the random chord although his conclusion is correct according to his own description.Drory (2015) discussed that it has inherent ambiguity and depends on explicitly defining the selection procedure for the random chords.

Figure 1 .
Figure 1.A chord vertical to diameter AB

4 .
of circle .Therefore, this probability is 1 Solution 4. Suppose that a line segment passes through the midpoint of the opposite side of the vertex A of an equilateral triangle inscribed in a circle and that this line segment meets the circle at point B, as shown in Figure 4. Draw a tangent line  that passes through point B.

Figure 2 .
Figure 2. A chord fixed at point A

Figure 4 .
Figure 4.A point where the extension of a chord fixed at a point on a circle meets with a tangent line

Questions 1 .
Suppose that a line segment passes through the midpoint D of the opposite side of the vertex A of an equilateral triangle inscribedin a circle and that the line segment meets the circle at point B, as shown in the figure on the right.What is the probability that the length of a chord whose extension is drawn perpendicularly to diameter AB is longer than the side of the triangle?2. Suppose a chord that defined by a random point P on the circumference of a circle and by the vertex A of an equilateral triangle inscribed in the circle, as shown in the figure.What is the probability that the length of a chord is greater than the side of the triangle inscribed in the circle?3. Suppose an equilateral triangle inscribed in circle  and another circle ′ inscribed in the triangle.What is the probability that the midpoint M of a random chord of circle , as shown in the figure, is located inside circle ′? 4. Consider an equilateral triangle ACD inscribed in a circle, as shown in the figure, and the tangent line  that is parallel to side CD.Suppose that the contact points of sides AC and AD with tangent line  are C′ and D′, respectively.If a chord AP starts from vertex A and passes through a random point P on the circumference of the circle, what is the probability that the contact point P′ of line AP with tangent line  is located on line segment C′D′? 5. Consider an equilateral triangle ACD inscribed in a circle, as shown in the figure, and a random closed curve  circumscribed in a circle that contacts the circle at vertex A. Suppose that the contact points of lines AC and AD with curve  are C′ and D′, respectively.If line AP starts from vertex A and passes through a random point P on the circumference of the circle, what is the probability that the contact point P′ of line AP with the closed curve is located on curve segment C′D′?
most of them failed to correctly determine the sample space and event, like in modified Question No. 4. Unlike in Question No. 4, some students correctly described the probability as a generalized ratio of the length of the figure of the sample space to the figure of the event.

Table 1 .
Testing questions

Table 3 .
Distribution of responses to modified Question No. 1

Table 4 .
Distribution of responses to modified Question No. 2

Table 5 .
The case of a student who answered correctly to Question No. 1 and incorrectly to Question No. 2

Table 6 .
Distribution of responses to modified Question No. 3

Table 8 .
Distribution of responses to modified Question No. 4 Figure 6.Case of incorrect answer Question No. 4 Figure 7. Case of correctly finding the sample space Question No. 4

Table 9 .
Distribution of responses to modified Question No. 5