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USING PROBABILITY FOR PROFIT
MASTER TEACHER Lucio Calzada
GRADES 6-8
MATH/SCIENCE
OVERVIEW
This lesson requires students to apply the mathematics of
probability to a real world situation involving insurance. Through class
interactions, video segments, hands on activities, cooperative learning, and
role playing, students will learn the relevant uses of probability. The
students will conduct an experiment to prove the probability of an event using
coin tosses and they will determine insurance premiums based on factors
affecting the insuring of property.
ITV SERIES
Mathmedia: Probability
LEARNING OBJECTIVES
The student will be able to:
*determine the probability of an independent event
*use statistics to determine risk
*decide what factors will affect premiums for the property they
are going to insure
*establish premiums for insuring property
MATERIALS
For groups of 3 or 4
a copy of Heads or Tails record sheet
coins with differentiated heads & tails
copy of probability activity sheet
PRE-VIEWING ACTIVITIES
The students, already in their learning groups, are shown a
coin. One student is identified with the question, "Pick heads or
tails". Once the student makes the selection, ask why that particular
selection was made. "Why did you pick _____? Do you have
knowledge about this coin that I should know?" (The student will
probably respond that they just chose because of preference.) Then ask the
class what the chances are that the coin will land with the student's selection
facing up. ( Most students should respond with 50% or 50-50.) Ask the
students to explain circumstances when 50% is considered "good or
bad". (Examples could be: 50% chance of rain; .500 batting
average; 50% chance of losing in a game; 50% chance of losing a bet for $5, $10,
all the money you have.) What if the Surgeon General warned that you had a
fifty percent chance of contracting cancer if you smoke. Would you smoke?
Ask the class how they know that the probability of a coin
landing on a certain selection is 50%. After listening to all responses,
lead them to realize that one way to discover probability with a coin is to
conduct an experiment. Hand them a recording sheet (or ask them to record
their results on any sheet of paper). Have each group (use cooperative
learning if possible) flip a coin 100 times and have them record the results.
Have each group share the results with the class. The results shared
should be in terms of raw data and then in a percentage. Have each group
explain how they arrived at their calculations for percentage. The teacher
should also record this information on the chalkboard or overhead projector for
each group, then for the entire class. The teacher would then
capitalize on this situation to demonstrate how the larger the sample
size, the more accurate the results will be (the results should prove to be 50 -
50, heads to tails). Now ask the students what kinds of factors could
affect results involving probability with a coin. A possible answer could
be "foul play." The teacher should now ask the students for
ideas on the practical (real world) use of probability and statistics.
FOCUS FOR VIEWING
To give students a specific responsibility while viewing say,
"Insurance companies use probability and statistics to determine risk when
insuring people and property. When watching this video segment, I want you
to watch for factors that companies think about when setting premiums. I
want each of you to think about questions you might ask your clients if you
owned an insurance company."
VIEWING ACTIVITIES
Start Video, Mathmedia: Module II, Probability,
Risky Business, a segment at the beginning of the program . Pause
video after the actor says, "Why is a young person like me charged more
than my Aunt Thelma?" Ask the students to list, with their group why
they think the young man would be charged more for his insurance than his Aunt
Thelma. Have the students share their ideas. Say to the students,
"I want you to listen and list the factors that the insurance person will
gather to justify the young person's premium. Each group will share what
it has heard. Let's see who catches all the factors that the insurance
agent addresses."
Resume tape.Pause video after the actor says,
"and before that, a fender bender." Ask the students to share
what factors they think should affect the young man's premiums and why.
Say to the students, "I want you to listen for a word that the insurance
business uses to substitute for the word, probability."
Resume tape. Pause video after the actor
says, "Can you think of any other factors?" Ask the students
what word the insurance companies identify with probability.(answer: risk)
"With the help of your groups, list other factors that an insurance company
might consider when setting premiums for a client." Have the students
share their lists.
Resume tape. Pause video after the actor says,
"Which of these might increase or decrease your insurance cost?
Why?" Ask the students to discuss and write which of the listed
factors would increase or decrease the insurance premium and why they would
increase or decrease the premium. Have the students share their results
with the class. Ask the students to listen closely for the probability of
people getting into accidents and how that changes based on the person's age.
Resume tape. Stop video after the actor
says, "...and what the cost might be to the company?"
POST-VIEWING ACTIVITY
Ask the students, "What did the insurance agent say the
probability was of a person under the age of twenty-five getting into an
accident?" (answer: one out of four) Ask the students,
with the help of their groups, to translate one out of four into a fraction, a
decimal, and then a percent. Ask one of the groups to share their
responses and how they arrived at those calculations.
Say to the students, "All of you have just been named to
the ________Middle School Health Insurance Board. Our task is to determine
health insurance premiums for our students in this classroom." Walk
the students through a brainstorming session on what they should do next.
Possible responses may be:
1. Determine the goals of the company.
2. Determine what health items will be covered.
3. Determine cost of health treatment.
4. Conduct surveys to determine health throughout the
classroom.
5. Estimate annual costs for the company.
6. Predict annual pay-outs to clients based on calculated
probability.
Conduct a class discussion about surveys and generate a
survey to assist in determining health risks for students in the classroom.
Then use compiled data or generate fictitious data for generating expenses
the company may have. Example: 20 of the 30 students in this
classroom have had to visit a medical facility this year and paid an average
cost of $48. What is the probability that a student will need to visit a
medical facility next year? (answer: 2 out of three students will visit a
medical facility.) Ask the students to express their responses in words,
in a decimal, in a fraction, and a percent. Discuss with the class other
costs that an insurance company will have to consider beyond the cost of medical
attention for their clients. Possible answers might be: employee
salaries, office supplies, taxes, and rent. For this exercise, the
students may want to hire a part time employee for perhaps $500 a month to
manage operations. Office supplies and rent may cost another $500 a month.
Given all this information, have the students, with the help of their groups,
generate total annual costs (including a small profit of $1,200 annually for the
teacher!) for the company and then have them set the premiums for their 30
clients and announce their rates to the class. Have them explain their
results. A possible solution may be:
20 visits x $48 = 960 annually
$1000 monthly for employee & bills
$1000 x 12 months = $12,000 annually
$960 + $12,000 + $1,200(profit) = $14,160.
If divided evenly among all clients, the annual cost to each
client would be:
$14,160 / 30 = $472 which is $39.33 a month.
Ask the students, "Does it pay to purchase this
insurance plan? Why or why not? Do we have to charge the same rate
to everyone? If we didn't charge the same rate to everyone, on what data
would we base our premiums?" To the last question, some student might
say that we could base premiums on health history. Have the students
explain what that means. Say to the students, "Another factor that we
might look at is gender. Maybe the boys went to the doctor more often than
girls or vice versa." Give the students a scenario in which the class
of 30 is composed of 15 boys and 15 girls. Also mention that of the 20
visits to the doctor, 13 visits were by a girl. Ask the students,
"How might we change our premiums to reflect that data?" Walk
the class through this solution:
13 visits by girls / 20 total visits = .65 or 65%
65% of the visits were by girls.
Ask the class, "What percentage of the visits were by
boys, and how did you get that solution?"
Possible solutions:
100% - 65% = 35%
35% of the visits were by boys.
OR
7 / 20 = .35 or 35%
Ask the class, "How would you use this information to
determine premiums?" Listen to their solutions and then walk them
through this solution:
Since 65% of the visits were by girls, we could charge the
girls 65% of the costs, therefore, $14,160 x 65% = $9,204.
Take the $9,204 and divide it by 15. You get, $613.60/yr.
which is $51.13 a month for the girls.
Ask the class to determine the cost for the boys with the help
of their groups.
Possible Solution:
$14,160 x 35% = $4,956
$4,956 / 15 = $330.40
$330.40 / 12 = $27.53 a month.
Say to the class, "We have now determined possible
premiums for our clients. I want each group to discuss and write which
premium you would vote to implement and why." Have each group share
their opinions.
ACTION PLAN
Assign each learning group the task of creating its own
insurance company insuring the entire school student body. They are
responsible for insuring student personal property such as Walkmans, Discmans,
eyeglasses, shoes, etc. This assignment should be completed over several
days so that surveys could be generated and compiled. Risk of theft and
risky behaviors should be considered when determining premiums. Ask the
students to create and submit a preliminary plan so that you can monitor their
progress. At the agreed upon deadline, each group should present its
premiums and on what they based their premiums. Encourage students to
retrieve necessary data from the school administration, existing insurance
companies, and their parents.
EXTENSIONS
LANGUAGE ARTS
Students may write an informative composition about the
increasing cost of elderly health care and their opinion on possible solutions.
SCIENCE
Scientists sometimes have to determine the population of certain
animals in a very large region through population samples of a smaller
region. They often use proportion and probability to estimate and project
population growth. Students can research and/or participate in the methods
used to determine an estimated population of certain animals in a given area.
MATH
In their math journals, have the students write about their
experience with one application of probability and have them list other
applications for probability.
MIDDLE SCHOOL HEALTH INSURANCE BOARD Students
Covering Students Because We Care
1. List the tasks that you think an insurance board of
directors should do to start its company.
2. Generate a survey that will assist in determining
health risks that may affect your company.
3. Calculate total annual costs and set premiums for
your insurance company using the information provided to you by the teacher.
Updated: April 01, 2008
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