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

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

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

• How to statisticians decide whether a result is significant?
• Why do results from surveys and polls list a "margin of error?"
• What modifications can be made to the possible experiments to increase our chances of being correct (without checking all of the blocks at once?

• As stated above, this problem is very difficult in it's present form. Some possible ways to make it more accessible could include:
1. Reduce the situation to a smaller number of blocks. If possible, build up from one to many.
2. Make the experiments simpler -- the experiment drawing two blocks three times is quite difficult to analyze. It probably should be avoided except by groups who have already completed the analysis of the other two experiments.