Strategies for solving introductory probability problems

1. Strategies for solving introductory probability problems Atsushi TERAO School of Social Informatics Aoyama Gakuin University

2. Motivation • Many students in Japan have to study hard for university entrance examinations. • Downside: many quit studying once they get in. • Many study-guide books (exam prep books) have been published.

3. Motivation • I found an old prep book for probability, “From permutation and combination to probability” (Fujimori, 1938), in Jimbo-cho, Tokyo. • One of a series of prep books published by KangaekataKenkyuSya • Out of print • The publisher become bankrupt long time ago.

4. Jimbo Town, Tokyo over 100 secondhand book stores

5. Motivation & Purpose • From the viewpoint of mathematics education, I’m curious to know • historical roles of this book • current value of this book • What does this book teaches? • To know this in accurate and detail, I plan to translate into production rules problem solving procedures or strategies taught in this book.

6. Motivation & Purpose • Form the viewpoint of cognitive science, through this translation, I want to do ground work for developing an intelligent tutoring systems for teaching introductory probability theory. • Making a list of production rules which students are expected to acquire in an introductory statistics course

7. Problem: Two person A and B draw a lottery ticket. Among the n (= number) tickets, x (= number) tickets are winning tickets. The person A draws first and person B second. Which person is in an advantageous condition? • From Fujimori, 1938 • The probability of the person A drawing a winning ticket is x/n. Find the probability the person B drawing a winning ticket. Is it smaller or larger than x/n? Or equal to x/n? • Suppose that n = 10 and x = 3

8. Problem solving Stages • Problem solving stages • Understanding: Constructing problem representation • Solution: Strategy choice and execution • the goal buffer in the model • =Goal> isa probability • =Goal> isa solution

9. Understanding Step 1 • Considering all possible cases, and find ones which match the problem description. • Win --- Win • Win --- Lost • Lost --- Win • Lost --- Lost W W L W L L

10. Initial state of the imaginalbuffer “The second person draws a winning ticket.”

11. Case Lost --- Win

12. Understanding Step 2 • Constructing a problem representation including • description of the critical cases • event categories • the number of elements in a category

16. The problem representation suggests this problem is a “sampling without replacement” problem. • The production rules in this model can be applied to any problems of this type. (I need to modify these rules to have a generality.)

17. Solution Step 1 • Calculate the probability of each case (e.g., Lost --- Win) • Find the probability of each event • Then find the product of them • Note that the type of events is “dependent.”

18. Probability of dependent trials First Trial Second Trial “win”2 = “win”1 – 1 Whole2 = Whole1 - 1

19. 3 - 1 10 - 1

20. Solution Step 2 • Sum up the probabilities of all critical cases

21. (p* find-first-case =goal> isa probability state start =imaginal> isa target-event target-1 =target-1 ;; win order-1 =slot1 ;; second target-2 =target-2 ;; blank order-2 =slot2 ;;none ==> =goal> state harvest-and-next =imaginal> +retrieval> isa case =slot1 =target-1 ;; second slot is "win" =slot2 =target-2 ;; none slot is blank ) Note: The P* function is useful. We can use variables for names of the slots.

22. Further Work • Keep going • Now, just one type of problem • When many types of problem are covered, I will test the ability of those production rules by giving them the probability problems currently used in university entrance exams • Evaluating current value of Fujimori’s prep book. • Developing an intelligent tutoring system

23. Thank you