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

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.

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 Kangae kata Kenkyu Sya – Out of print – The publisher become bankrupt long time ago.

Jimbo Town, Tokyo over 100 secondhand book stores

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.

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

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

Problem solving Stages Problem solving stages 1.Understanding: Constructing problem representation 2.Solution: Strategy choice and execution the goal buffer in the model 1.=Goal> isa probability 2.=Goal> isa solution

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

“The second person draws a winning ticket.” Initial state of the imaginal buffer

Case Lost --- Win

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

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.)

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.”

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

Solution Step 2 Sum up the probabilities of all critical cases

(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.

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

Thank you