**FLU FOR ALL**

Tom Dahlquist Lyle Frantz Phyllis Slayton

special acknowledgment to

Becky Dutkosky

**Class: **Algebra or higher

**Material: **Paper, cups, sodium hydroxide, phenolphthalein, student handouts and sample solution included.

**Goals: **Enhance student understanding of probability and sample taking. Students will also sharpen problem solving skills. This could be used to introduce probability or after it has been taught.

**Background: **To approach this problem, make the assumption that students are drinking and making their exchanges at the same time and call each time a Around@. Students will have a difficult time coming up with a solution for four or more exchanges. They should, however, be able to express a qualitative feel of the problem after working through the problem with three exchanges. The purpose is not actually to get a solution, but to show the beginnings of how to attack the problem of four or more exchanges. The students will benefit from thinking hard about this problem. The problem can be addressed using diagrams, but to solve formally requires some probability theory.

**Time Required:** Two or three days.

**Problem:** As a part of the homecoming week celebration students are having an ethnic food tasting party in the foreign language department. Many types of food and drink are arranged buffet style, and the students are having a swell time tasting food and swapping tastes from each other's cups. Each of the thirty students shares cups with three other students. Unknown at the time, one student was coming down with a virulent strain of flu, and although the student was contagious, he didn't show any symptoms of the flu....Yet!

Members of the class include the homecoming queen and the star halfback from the football team. What is the probability for each catching the flu from this party? What percentage of the class is likely to get infected with the flu at this party? For a class of thirty, what is the number of exchanges that will, on average, result in half the class being infected with the flu? One will assume that this strain of flu is so virulent that drinking out of an infected glass makes that person contagious. In other words, after drinking from a contaminated glass, all subsequent glasses that the person drinks from will be contaminated. We will also assume that a student does not drink from another person=s cup twice.

**Activity #1: **Students will either individually or in groups solve the above problem. This can be done in class or as a homework assignment. After students have had a chance to work on solutions, the possible solutions should be discussed in class. After discussing the solutions, do a simulation.

**Activity #2:** Set up thirty cups of a clear solution. The actual number can be adjusted to match the class size. Probably the original problem should match the simulation for a better comparison. In one of the cups put a clear chemical that can be traced. A weak Sodium Hydroxide solution would work well, but the chemistry teacher at your school should be able to make other suggestions. Number the cups so that students can trace the spread of the infection. Each student should take one cup and mix with four different people. To mix, the students should dump the contents of one cup into another and then pour half the contents back into the empty cup. While mixing, the students should record with whom they mixed in the correct order. After mixing, the cups are lined up on the table and the indicator, phenolphthalein, is added to show the spread of the contaminant. Students should note which of the cups have been infected.

**Extension of the Problem: **Statio Pendowski, an agent from the Center for Disease Control in Atlanta, has been sent to investigate the flu outbreak at the school. After interviewing the students, he compiled a list of the people who exchanged cups and the order in which they made contact. From this data and the list of persons infected he was able to deduce the source of the infection.

**Activity #3**: Using the information gathered during the simulation, work backwards and determine the cup that had the original infection.

**Sample Solution: **The key to finding a solution is to keep careful track what the infected persons are doing and view the uninfected as one highly weighted option. For three exchanges trace through the possible scenarios.

Henceforth the infected persons will be given a number indicating the order in which they were infected. In the event that more than one is infected in a given round the lower number is given to the person infected by the person with the lower number. For example if in round two, person #1 infects someone and person #2 infects someone, then the person infected by person #1 is #3 and the person infected by #2 is #4.

ROUND (0)

29(U) #1

ROUND (1)

28(U) #1,#2

ROUND (2)

26(U) #1,#2,#3,#4

Although which individuals have been infected can vary greatly, the number of infected persons after two rounds has to be four if the rules of interaction are followed. Round three has some variation to it because for the first time infected persons may interact with other infected persons, and this scenario does not spread the infection. The number of infections that may be spread during round three are zero, two, or four. Each case must be examined individually to consider the probability that it will occur.

**ROUND (3)**

**Scenario #1**

26(U) #1,#2,#3,#4

This can only happen if (**#1** pairs with **#4**) and (**#2** pairs with **#3**). This is a possible but unlikely event. This can only happen in this one way.

**Scenario #2**

24(U) #1,#2,#3,#4,#5,#6

This can happen in three ways. (**#1** pairs with **#4**) and (**#2** pairs with **U**) and (**#3** pairs with **U**) **or**

(**#2 **pairs with **#3**) and (**#1** pairs with **U**) and (**#4** pairs with **U**)** or**

(#3 pairs with **#4**) and (**#1** pairs with **U**) and (**#2** pairs with **U**)

This event (two additional exposures) can happen

(26)(25)+(26)(25)+(26)(25) = 1950 ways. This is a much more likely event than zero additional exposures.

**Scenario #3**

22(U) #1,#2,#3,#4,#5,#6,#7,#8

This can happen if each of the infected persons infects an uninfected person. This can happen in (26)(25)(24)(23) = 358,800 ways. This is by far the most likely outcome occurring almost 99.5% of the time.

The probability of any one person **catching** the flu is approximately .955(7/29)+.005(5/29)=.241 or 24.1% chance of catching the flu. The original perpetrator cannot catch the flu at the party because he already has it. This leaves a population of 29 possible infectees.

Increasing the number of exchanges to four very quickly makes the problem much more difficult. The likelihood of infected to infected interactions rises greatly and with varying probabilities of interaction. The problem would have to be addressed either using a computer model that counted the interactions or using a simulation, which is what this exercise does. The huge number of possible outcomes makes it very difficult to do by hand in a timely or accurate manner.

It will take 5 exchanges before at least half the class will be infected. Five interactions will overshoot one half of the class considerably. Four interactions comes closest to infecting exactly one half of the class on average.

**Evaluation:** Students will turn in their solutions which will be judged as outlined below.

15% (1) Statement of the problem to show that the problem is understood

20% (2) A list of assumptions made to make the problem simple enough to solve

20% (3) Conversion of problem into mathematical terms

20% (4) Solve mathematical relationships

15% (5) Clarity and neatness of explanation

10% (6) The actual numbers

**Possible Follow-up Discussions: **How does this model relate to the spread of real diseases such as AIDS and other STD's, flu, chicken pox? How do the factors of virulence and immunity change the problem? How does a disease spread over a larger area such as a city, state, or country?

**Extensions:** Add some wrinkles to the problem by making a percentage of the class immune to the disease. Make the disease less virulent by having only a percentage of those exposed get the disease. Use a larger group and limit the number of exchanges that some groups make. Perhaps a portion of the class will begin exchanging after the rest of the class has made a couple of exchanges.

**STUDENT HANDOUT**

**Description of Problem:** As a part of the homecoming week celebration, students are having an ethnic food tasting party in the foreign language department. Many types of food and drink are arranged buffet style and the students are having a swell time tasting food and swapping tastes from each other's cups. Each of the thirty students shares cups with three other students. Unknown at the time one student was coming down with the flu, and although the student was contagious, he didn't show any symptoms of the flu....Yet!

Members of the class include the homecoming queen and the star halfback from the football team. **(1)** What is the probability for each getting the flu from this party? **(2)** What percentage of the class is likely to get infected with the flu at this party? **(3)** If the number of exchanges is changed to four, how does the difficulty in solving the problem change? **(4)** For a class of thirty, what is the number of exchanges that will, on average, result in half the class being infected with the flu? One will assume that this strain of flu is so virulent that drinking out of an infected glass makes that person contagious. In other words, after drinking from a contaminated glass, all subsequent glasses that the person drinks from will be contaminated. Also assume that a student does not drink from another=s cup twice.

Evaluation: Students will turn in their solutions which will be judged as outlined below.

15% (1) Statement of the problem to show that the problem is understood

20% (2) A list of assumptions made to make the problem simple enough to solve

20% (3) Conversion of problem into mathematical terms

20% (4) Solve mathematical relationships

15% (5) Clarity and neatness of explanation

10% (6) The actual numbers

**STUDENT HANDOUT**

**Activity:** In one of the cups is a clear chemical that can be traced. The cups are numbered so that you can trace the spread of the infection. Each student should take one cup and mix with four different people. To mix, dump the contents of one cup into another and then pour half the contents back into the empty cup. While mixing, record with whom you mixed and the order. After mixing, line up the cups on the front table so the indicator can be added and the extent of the contamination recorded. Students should note which of the cups have been infected.

Use the chart below to record your data.

**Your Cup #_______________**

Person cup#

1.______________ _____

2.______________ _____

3.______________ _____

4.______________ _____

**STUDENT HANDOUT**

**Extension of the Problem: **Statio Pendowski, an agent from the Center for Disease Control in Atlanta, has been sent to investigate the flu outbreak at the school. After interviewing the students, he compiled a list of the people who exchanged cups and the order in which they made contact. (These students, being foreign language students, had a knack for memorizing detail.) From this data and the list of persons infected he was able to deduce the source of the infection.

**Activity #3:** Using the information gathered during the simulation, work backwards and determine the cup that had the original infection.

**Evaluation:** Students will turn in their solutions which will be judged as outlined below.

15% (1) Statement of the problem to show that the problem is

understood

20% (2) A list of assumptions made to make the problem simple enough to solve

20% (3) Conversion of problem into mathematical terms

20% (4) Solve mathematical relationships

15% (5) Clarity and neatness of explanation

10% (6) The actual numbers

*Funded in part by the National Science Foundation and Indiana University 1995*