本研究最主要的目的是想瞭解中學生一些基本的機率概念，是否與文獻中所提的判斷偏誤有關。本研究所謂的判斷偏誤是指國外學者Tversky & Kahneman, Fischbein, Konold等所提的「代表性」、「可利用性」、「結果取向」、「複合事件等機率迷思概念」，以及在「同時投擲與N次投擲」的相同機率架構中是否有混淆的情況。本研究則延伸他們的研究，一來探討我國的中學生是否也會有受這些判斷偏誤影響的情況，並進而探討機率判斷偏誤是否會與中學生之「獨立」與「樣本空間」這兩個基本機率概念有關。以量的分析與半結構式訪談，來探討分析判斷偏誤與基本機率概念的關係。
本研究的研究對象，是以台北地區尚未學過「實驗機率」單元的國二學生160人，以及尚未學過「古典機率」單元的高中生513人。透過兩份自編的評量工具蒐集資料，並進行量的分析，探討判斷偏誤與機率概念的關係。此外尚以半結構訪談的方式，從673位研究對象中選出具代表性的5人，進一步確認量的研究結果。以下是本研究主要的發現：
1. 雖然國高中受試者之機率先備知識有很大差異，且施測樣本數有很大不同，但本研究中多元線性迴歸的結果，卻有許多國高中迴歸結果非常相近的情形，然而這是研究者從未預先想到的。尤其是，最能預測國高中受試者在複合事件等機率能力的預測變項皆為樣本空間，且所有自變項約可解釋複合事件等機率之變異數的三成五左右。最能預測國高中受試者在代表性判斷偏誤的預測變項為獨立，且所有自變項約可解釋代表性之變異數的五成左右；最能預測國高中受試者在一次投擲的預測變項皆為獨立，所有自變項共可解釋國高中受試者一次投擲能力之變異數的二至三成；國高中受試者在可利用性與結果取向中所有自變項可解釋判斷偏誤能力之變異數均未到兩成，且每個預測變項的效應均很小。
2. 除了可利用性次量表之外，受試者有許多選項正確，但所持理由與正式機率理解相距甚遠的情形。
3. 量的施測與訪談中發現，部分受試者有基模轉換與概念衝突情形發生，但受試者往往沒有察覺此現象。
4. 部分國中受試者與少數高中受試者有在本研究中許多工具一致選擇「無法預測」選項，以及一致選擇「兩者/三者相同」選項之情形。雖然在某些題目中此為正確選項，但受試者的想法與正式機率理解有很大差距。所持理由多符合結果取向之表徵。
5. 部分國高中受試者提及過去投擲經驗，但在本研究中，提及過去經驗不一定對選正確選項有所幫助，可能是受了可利用性判斷偏誤之影響。且部分國高中受試者持有「投擲有由人為控制的」想法，特別是在「獨立」與「一次投擲」次量表中，此現象尤為顯著。
6. 部份國中受試者持有「都是相同骰子/硬幣，所以機率相同」、「都是相同彩券，所以機率相同」的想法，並以「是否公平」為本研究中所有工具的答題判準；少數國高中受試者持有「只要是有關於投擲骰子的題目，機率都是1/6；只要是有關於投擲硬幣的題目，機率都是1/2」的想法；少數高中受試者持有「機率這種東西，就是要越多次越可信」的想法，並以此為本研究中所有工具的答題判準。以上這些想法有「僅考慮單一因素，沒有考慮除了相似性以外的因素」的特性，屬於代表性判斷偏誤表現。
7. 本研究與國外學者的結果相同的是，受試者有很多複合事件等機率的表現。但研究者進一步探討複合事件等機率的成因，發現可以更深入的分為：可能是受到「結果取向」的影響，或受到「代表性判斷偏誤」的影響。

There are three main purposes of this study. The first one is to find out the level of understanding of the secondary students in Taiwan on fundamental probability concepts. The second purpose is to verify if the five judgmental heuristics and probabilistic misconception, identified by Tversky & Kahneman(1972,1973), Fischbein(1991,1997) and Konold(1989), could also be observed among our secondary students. The third purpose is an attempt to find out if there exists any relation between “probability background” and “judgmental heuristics and probabilistic misconception”. A total of three instruments were developed to achieve these purposes.
A total of 673 high school subjects in Taipei were involved in this study. Among them are 163 eighth graders who have not studied the topic of “experimental probability” and 513 tenth graders who have not studied the topic of “classical probability”. The first part of this study focused on how these two groups of students perform on item related to the two basic probability concepts, “independence” and “sample space”. The second part of this study focused on how these two groups of students perform on items related to the five judgmental heuristics and probabilistic misconception, namely, “Representativeness”, “Availability”, “Outcome approach”, “Equiprobability Bias in Compound Event” , “Tossing N dice simultaneously vs Tossing one die N times consecutively”. The third part of this study focused on identifying possible relationship between students’ understanding of probability concepts and the five judgmental biases.
In this study, quantitative analysis was the primary means for data analysis. Subsequent qualitative analysis was also pursued in order to supplement the findings.
Regarding the three purposes, the following results were obtained:
1. Although the prior knowledge and the sample sizes of senior and junior high subjects are quite different, there are many similar results in the multiple regression analysis for these two groups. Among which, the most predictable variable on the performance of “Equiprobability Bias in Compound Event” by our senior or junior high subjects is “sample space”, in a model that explained 35% of the variance of the dependent variable. the most predictable variable on the performance of “Representativeness” by our senior or junior high subjects is “independence”, in a model that explained 50% of the variance of the dependent variable. The most predictable variable on the performance of “Tossing all at once” by our senior or junior high subjects is “independence”, in a model that explained 20% to 30% of the variance of the dependent variable.
2. Except in the “Availability” subscale, it was found that though many subjects chose the correct options, their reasons given revealed major misunderstanding.
3. In both the quantitative and qualitative analysis, we found that some of the subjects shown the schema-transferred and concept-conflicted performance, which they were not aware of.
4. Some junior high subjects and a few senior high subjects consistently chose the options “can not be predicted” or “ all the same” in many items of this study. Their understanding of probability is far from satisfactory, even though on those occasions when the 2 options were the right answers. Their performances could be attributable to the “Outcome approach” heuristic.
5. Some high school subjects mentioned their past experience of dice-tossing, but that did not help them in choosing the correct items. This phenomenon may be attributable to the “Availibility” heuristic. Besides, some of them hold the idea of “dice-tossing can be controlled by human being”. This phenomenon was very evident in the “Independence” and “Tossing N dice simultaneously vs Tossing one die N times consecutively ”, sub-scales.
6. Some junior high subjects held the idea of “ the same dice(coin), the same probability”, “the same lottery ticket , the same probability” and use the rule, “fair or not” as their probability judgmental critiria. A few subjects held the idea “as long as the problems are related to dice-tossing, the probability of the events are all equal to 1/6” and “as to probability, the more frequencies, the more convincement” and used these as their probability judgmental critiria. These subjects had the same characteristic. They only considered the factor of similarity, and not any other factors. This may be attributable to the “Representativeness” judgmental bias.
7. Some subjects showed heavy signs of “Equiprobability Bias in Compound Event” performance, which verify what was discussed in the literature. After a series of further inquisitions, it was found such performances could be further explainable by the “Outcome approach” and the “Representativeness” judgmental bias.

一、中文部分
陳順宇、鄭碧娥（民82）：國中生機率能力評量分析。中華民國第八屆科學教育學術研討會論文彙編, 39-67
謝展文（民89）：直覺法則對於數學及科學學習的影響—以國小四、五、六年級學生為對象。台北市：國立台灣師範大學科學教育研究所碩士論文。
張景中（民89）：數學與哲學。台北市：九章。
李佳奇(民90): 高中生對條件機率解題策略與錯誤類型之探討。台北市：國立台灣師範大學數學研究所碩士論文。
國民中學數學科教科書第六冊(民89):國立編譯館主編。台北市:九十家書局。
高級中學數學科教科書第四冊(民89):國立台灣師範大學科學教育中心主編。台北市:九十家書局。
高級中學數學科教科書第四冊(民89):余文卿主編。台北市:龍騰書局。
高級中學數學科教科書第四冊(民89):林福來主編。台北市:南一書局。
Philip, J. D., & Reuben, H. (1996)，常庚哲、周炳蘭譯（民85）：笛卡爾之夢--從數學看世界。台北市：九章。
Moore, D. S. (1998)，鄭惟厚譯（民89）：統計，讓數字說話。台北市：天下文化。
Cole. (1998)，丘宏義譯（民89）：數學與頭腦相遇的地方。台北市：天下文化
Reborah Bennett. (2000)，王業鈞譯（民90）：你賭對了嗎？。台北市：新新聞
二、英文部分
Castro, C. S. (1998). Teaching probability for conceptual change. Educational Studies in Mathematics, 35, 233-254.
Cohen, J. (1957). Subjective probability. Scientific American, 197, 128-138.
Fischbein, E., & Gazit, A. (1984). Does the teaching of probability improve probabilistic intuition? Educational Studies in Mathematics, 15, 1-24.
Fischbein, E., Nello, M. S.,& Marino, M. S. (1991). Factors affecting probabilitic judgments in children and adolescents. Educational Studies in Mathematics, 22, 523-549.
Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic, intuitively based misconceptions. Journal for research in mathematics education, 28, 98-105.
Garfield, J., & Ahlgren, A. (1988). Difficulties in learning basic concepts in probability and statistics: implications for research. Journal for Research in Mathematics Education, 19(1), 44-63.
Green, D. (1997). Recognizing randomness. Teaching statistics, 19( 2), 36-39.
Hawkins, A. & Kapadia, R. (1984). Children’s conceptions of probability: A psychological and pedagogical review. Educational Studies in Mathematics, 15, 349-377.
Jones, G., Langrall, C. W., Thornton, C. A, & Mogill, A. T. (1997). A framework for assessing and nurturing young children’s thinking In probability. Educational Studies in Mathematics, 32, 101-123.
Jones, G., Langrall, C. W., & Thornton, C. A. (1999). Students’ Probabilistic thinking in instruction. Journal for research in mathematics education, 30( 5), 487-519.
Kahneman, D., & Tversky, A.. (1972). Subjective probability: A judgment of representativeness. Cognitive Psychology, 3, 430-453.
Kahneman, D., & Tversky, A. (1973b). Availability: A heuristic for judging frequency and probability. Cognitive Psychology, 5, 207-232.
Konold, C. (1989). Informal conceptions of probability. Cognition and instruction, 6, 59-98.
Konold, C. (1991). Understanding students’ beliefs about probability. Radical constructivism in mathematics education, 139-156.
Konold, C., Pollatsek, A., Well, A., Lohmeier, J., & Lipson, A. (1993). Inconsistencies in students’ reasoning about probability. Journal for research in mathematics education, 24( 5), 392-412.
Konold, C. (1994). Teaching probability through modeling real problems.
Mathematics Teacher, 87, 232-235.
Konold, C. (1995). Issues in assessing conceptual understanding in probability and
statistics. Journal of statistics education, 3(1). http://www.amstat.org/publications/jse/v3n1/konold.html
Lecoutre, M. (1992). Cognitive models and problem spaces in “purely random”situations. Educational Studies in Mathematics, 23, 557-568.
Pollatsek, A., Konold, C. (1987). Understanding conditional probabilities. Organizational behavior and human decision processes, 40, 225-269.
Shaughnessy, J. M. (1977). Misconceptions of probability :an experiment with a small-group,activity-based,model building approach to introductory probability at the college level. Educational Studies in Mathematics, 8, 295-316.
Shaughnessy, J. M. & Dick, T. (1991). Monty’s delimma:should you stick or switch? Mathematics Teacher, 84, 252-256.
Shaughnessy, J. M. (1993). Research in probability and statistics. In D. Grouws(Ed.), Handbook of Research on Mathematics Teaching and Learning, (pp. 465-494). New York:Macmillan.
Steinbring, Heinz. (1991). The concept of chance in everyday teaching: aspects of a social Epistemology of mathematical knowledge. Educational Studies in Mathematics, 22, 503-522.
Tarr, J. E., & Jones, G. A. (1997). A framework for assessing middle school students’ thinking in conditional probability and independence. Mathematics Education Research Journal, 9(1), 39-59.
Tversky, A., & Kahnemam, D. (1974). Judgment under uncertainty: Heuristics and Biases. Science, 185, 1124-1131.
Verschaffel, L., Corte, E., & Lasure, S. (1999). Children’s Conceptions about the Role of Real-World Knowledge in Mathematics Modeling: Analysis and Improvement. In W. Schnotzm S. Vosniadou, & M. Carretero (Eds.), New Perspectives on Conceptual Change(pp. 175-189). New York:Pergamon.
Watson, J. M. (1995). Conditional probability: It’s place in the mathematics curriculum. Mathematics teacher, 84, 12-17.