Statistics: The Desert Island Epidemic

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 to NCTM standards of Mathematics as Problem Solving, Mathematical Connections, Number and Number Relationships, 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

Procedures:

1.      Introduce activity with a class discussion on infectious diseases, etc.  Pose the question of how to predict the spread of infectious diseases, and the need to do so. Talk about diseases they may have experienced, heard about, diseases in the news.

2.      Set up the scenario:

Six people live on an otherwise deserted island, in six different huts. An infectious disease strikes the island. The disease has a one-day infectious period and after that the person is immune (cannot get the disease again). Assume one of the people gets the disease (maybe from a piece of space debris, or a message in a bottle). He randomly visits one of the other islanders during his infectious period. If the visited islander has not had the disease, he gets it and is infectious the following day. The visited islander then visits another islander. The disease is transmitted until an infectious islander visits an immune islander, and the disease dies out. There is one islander visit per day. Assuming this pattern of behavior, how many islanders can be expected to get the disease?

3.      Ask students for their predictions.

4.      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 infection.

5.      Students now individually predict the number islanders who will be infected with the disease.

6.      Experiment:

·        Use a six-sided die. Each side represents an islander.

·        Roll the die to see which islandergets the disease.

·        Roll the die again to see which islanderis visited and gets the disease.

·        Continue rolling until an immune islander(one of the numbers that has already been rolled) is visited.

·        Note: If the same number is rolled one after the other, ignore the second roll, since these islandersdo not visit themselves.

·        Count how many different numbers were rolled (how many islandersgot the disease).

·        Repeat experiment five times.

·        Average the number of islanderswho got infected per outbreak.

·        Students make a final prediction of the class average.

7.      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?

8.      Discuss the expected outcome (3.5104) with your students. It is derived by multiplying the individual values by their expected probabilities, and adding them together:

1(0) + 2(1/5) + 3(8/25) + 4(36/125) + 5(96/625) + 6(24/625) = 3.5104

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?

9.      Questions for further discussion:

·        Why is it impossible for only one islander to be infected?

·        Should the average of the trials from your model be exactly 3.5104?

·        Does an expected value of 3.5104 mean that we expect 3 whole islanders and only part of a fourth islander to get sick each time?

·        What is the expected value or predicted value of this problem if there are 4 islanders? 14 islanders?

·        How would your predictions change if there was a two-day infectious period?

·        Is this exercise applicable in a real world situation?