Index

Discrete Mathematics Project
Counting Techniques/Probability Activity

Title

"What's In the Bag?" (Jim Arnow)

Goals

  1. Students will evaluate various methods for determining the contents of a bag using counting techniques and probability analysis.
  2. Students will work in small groups while gaining an understanding of how certain situations may be interpreted by probability models.
  3. By determining the significance of certain outcomes, students will become familiar with the certain types of statistical errors and gain an appreciation for what a statistician means by "reasonably certain."

Abstract

This activity requires students to determine how to interpret the results of a random sample. Students are given a small number of possible tests from which they must choose one to apply to a bag of colored blocks to determine its contents. Students must enumerate the possible results, their relative probabilities and how to interpret the results.

Problem Statement

At the front of the room is a bag. In that bag are eight blocks. Either they are divided into four block of one color and four of another or they are divided into six blocks of one color and two of another. Your job is to determine which of the two cases is true, however, you are only allowed to choose one of a possible set of experiments. You goal is to determine which experiment will best help you to make a reasonable decision about the contents of the bag.

Instructor Suggestions

  1. Give the class the "What's In the Bag?" handout.
  2. Divide the class into small groups as soon as there are no questions about the problem statement.
  3. Some students will wonder what the colors are. Tell them only that there are two colors, divided either 4 and 4 or 6 and 2. Giving any more information, such as what the color of the more common blocks is, can change the problem, usually making it simpler.
  4. Some students may find it useful to have a set of blocks to manipulate while they discuss the problem. Whether or not they ask for them, make them available from the start.
  5. A possible interpretation for Experiment 1 (Note that the convention used is that a draw of 6 of one color and 2 of another is represented as 6/2 while a distribution of 6 blocks of one color and two of another is represented as 6-2):

    Experiment 1: One Block at a time, 8 times

     
    Outcome:
    Probability if
    Distribution is 4-4
    Probability if
    Distribution is 6-2
    0/8 .0078 .1001
    1/7 .0625 .2673
    2/6 .2188 .3153
    3/5 .4375 .2307
    4/4 .2734 .0865

    As an example of how the entires were calculated, the entry for 3/5 with a distribution of 6-2 is found by finding the probability of drawing the more common block 3 times and the less common block five time plus the probability of drawing the more common block five times and the less common block 3 times:

    P(choosing 3/5 if dist. is 6-2) = C(8,3)(.75)3(.25)5 + C(8,5)(.75)5(.25)3 = .2307

    Now that these probabilities, we can look at the conditional probabilities across each row. For instance, the conditional probability that the distribution is 4-4 if we draw 3/5 is (.4375)/(.4375+.2307) = .655, so in that case, we the distribution is more likely 4-4. By examining the rows, we see that if we draw 0/8, 1/7 or 2/6, we should guess that the distribution being 6-2, however, if we draw 3/5 or 4/4 we should guess the distribution is 4-4.

  6. For any of the outcomes, we can calculate the possibility that our guess will be incorrect. This is a useful exercise, and brings up the idea of significance of a statistical test.
  7. This activity in it's present form is extremely difficult for students at a high school level. If it were to be implemented in an actual classroom, the situation should be significantly simplified. See the variations section for further suggestions.

Materials

"What's In the Bag?" handout, scientific calculators, paper bags filled with colored blocks, extra colored blocks, scratch paper, pencils.

Time

Discussion of activity (10 min.), small group exploration of problem (30 min. - 2 hr.), Small group presentation of results (30 min.), class discussion of results (20 min.)

Mathematics Concepts

Discrete Mathematics

discrete probability, independent events, conditional probability, binomial distribution

Related Mathematics

statistical significance, hypothesis

NCTM Standards Addressed

Problem Solving, Communication, Reasoning, Connections (both within mathematics and across disciplines), Statistics, Probability, Discrete Mathematics, Mathematical Structure

Colorado Model Content Standards Addressed

Number Sense (1), Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)

Curriculum Integration

This activity could only be implemented in a class with a fair degree of experience with discrete probability. This could serve as a transition from probability into statistical inference and the limitations of statistics to verify something absolutely.

Further Investigation


Variations/Comments

Send your variations and comments to this Web page's editor

References, Resources and and Related Web Links

Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete Mathematics Through Applications. New York: W. H. Freeman and Company.

Medenhall, W., Wackerly, D., & Scheaffer, R. (1990). Mathematical Statistics with Applications. Belmont, California: Duxbury Press.

Ross, S. (1985). Introduction to Probability Models. Orlando: Academic Press.

Tucker, A., (1984). Applied Combinatorics. New York: John Wiley & Sons.


Last updated January 15, 1997