Statistics: Cereal Box Problem

Key Concepts: Probability, Prediction/Expected Value

Goal: Students will begin developing their own probabilistic intuition and will better understand the relation between predicted values and expected values.

Standards: This project ties directly to NCTM standards of Mathematics as Problem Solving, Mathematical Connections, Number and Number Relationships, Computation and Estimation, Statistics, and Probability.

Objectives:

·        To show that they can make a numerical prediction based on their own assumptions about probability.

·        To show that they can use the Monte Carlo method (random number generator) to test their own numerical predication.

·        To show that they understand the difference between a prediction and the expected value in a probability situation.

·        To show that they understand how to compare experimental results with mathematical expected values.

Materials:

• Prediction and experimental results tables
• Six-sided dice
• Cereal boxes (optional – for demonstration)

Procedures:

1.      Introduce activity with a class discussion on cereal box prize giveaways, etc.  Pose the question of how to predict the amount of boxes one would have to buy in order to get all the prizes that are included.

2.      Ask students for their own predictions on how many boxes they would need to buy in order to get all the prizes, when there are six possible.

3.      Discuss the use of a six-sided die as a random number generator, and a good predictor for simple probability problems. Ask students how this could relate to prediction of cereal prizes. (One roll of the dice represents one trip to the supermarket)

4.      Students now individually predict the number of cereal boxes they will have to buy in order to get all six prizes. Students make prediction on their results pages.

5.      Experiment: Students roll their dice repeatedly, placing a tally mark in the appropriate column of row #2, for prize one, two, three, four, five or six. Important: They must stop after they have placed a tally mark in the last empty box. Continuing will produce invalid results. Students fill out their entire results table (five trials), and then make new predictions of the number of boxes they would have to buy, recording this in the final box.

6.      Compile results of all students’ trials, either on a chart or through class discussion. Is the class average close to their predictions? Whose prediction was the closest?

7.      Discuss the expected outcome (14.7) with your students. It is derived by calculating the sum of the reciprocals:

6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 14.7.

Is this equal to the experimental outcome? Is it equal to the predicted outcomes? Which value is more valid? How could this formula be used to predict other experimental results?

8.      Questions for further discussion:

·        If you bought 15 boxes of cereal, but still did not get all six prizes, do you have a right to complain to the manufacturer? If so, how would you justify your complaint?

·        Suppose you suspected that one prize was in shorter supply than the others, is there any way you could prove it?

·        What other kinds of problems could you model in this way?

·        What tools, in addition to dice, would be good for generating random outcomes? Suppose you had to generate random numbers between 1 and 12, how would you do it?