Strategies for solving introductory probability problems

Atsushi TERAOSchool of Social InformaticsAoyama 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 stages1. Understanding: Constructing problem

representation2. Solution: Strategy choice and execution

• the goal buffer in the model1. =Goal> isa probability2. =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.”

2

2""

1

1""

Whole

win

Whole

win

“win”2 = “win”1 – 1Whole2 = Whole1 - 1

First Trial Second Trial

Probability of dependent trials

3 - 1

10 - 1

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