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Geometry
Lesson Activities
Proving That All Circles Are Similar
The Lesson Activities will help you meet these educational goals:
• Content Knowledge—You will prove that all circles are similar.
• Mathematical Practices—You will use appropriate tools strategically.
• Inquiry—You will perform an investigation in which you will make observations, draw
conclusions, and communicate your results in written form.
• 21st Century Skills—You will carry out technology-assisted modeling.
Directions
Please save this document before you begin working on the assignment. Type your
answers directly in the document.
_________________________________________________________________________
Teacher-Graded Activities
Write a response for each of the following activities. Check the Evaluation section at the end
of this document to make sure you have met the expected criteria for the assignment. When
you have finished, submit your work to your teacher.
1. Proving That All Circles Are Similar
In this activity, you will explore how similarity transformations establish similarity for all
circles. Open the GeoGebra geometry tool, and complete each step below. If you need
help, follow these instructions for using GeoGebra.
a. Start by creating two different circles:
• Create a point, and label it A. To make the remainder of the activity easier, choose
integers for the x and y coordinates of point A.
• Create a circle with its center at point A and with a radius of your choice. To make
the remainder of the activity easier, choose an integer value for the radius.
• Create another point, and label it B. To make the remainder of the activity easier,
choose integers for the x and y coordinates of the point.
• Create a circle with its center at point B and with a radius of your choice that is
different from the radius chosen for circle A. To make the remainder of the activity
easier, choose an integer value for the radius.
Take a screenshot of the two circles you created, and paste it in the space provided.
Type your response here:
1
© 2013 EDMENTUM, INC.
b. Recall what you know about similarity. If circle B is similar to circle A, what must exist?
Type your response here:
c. What is the difference of the x-coordinate of point A and the x-coordinate of point B?
Type your response here:
d. In terms of rigid transformations (reflections, rotations, and translations), what does
this difference represent?
Type your response here:
e. What is the difference of the y-coordinate of point A and the y-coordinate of point B?
Type your response here:
f. In terms of rigid transformations (reflections, rotations, and translations), what does
this difference represent?
Type your response here:
g. What is the ratio of the radius of circle A to the radius of circle B?
Type your response here:
h. In terms of nonrigid transformations, what does this ratio represent?
Type your response here:
i.
Write a sequence of dilations and translations that maps circle B onto circle A.
Type your response here:
j.
What is the relationship between circle A and circle B?
Type your response here:
2
k. You have established that circles A and B are similar. What conclusion can you make
about any two given circles?
Type your response here:
l.
For assurance that the similarity proof can be generalized to all circles, try more
examples of creating two circles and determining the similarity transformations that
map one circle onto the other circle. Take a screenshot of two such circles, labeled C
and D, that are different from the ones created in part a, and paste it in the space
provided.
Type your response here:
m. Write a sequence of dilations and translations that maps circle D onto circle C and that
shows the two triangles you created are similar.
Type your response here:
3
Evaluation
Your teacher will use this rubric to evaluate the completeness of your work as well as the
clarity of thinking you exhibit.
Total Points: 100
Task 1: Proving That All Circles are Similar
Task points: 100
a.
Draw circle A and circle B.
5
b.
Identify the relationship for similarity.
5
c.
Find the difference of the x-coordinates.
5
d.
Identify what the difference represents.
10
e.
Find the difference of the y-coordinates.
5
f.
Identify what the difference represents.
10
g.
Determine the ratio between the radii.
5
h.
Identify what the ratio represents.
10
i.
State a sequence of dilations and translations.
10
j.
State a relationship between circle A and circle B.
10
k.
Draw a conclusion about any two circles.
10
l.
Draw circle C and circle D.
5
m. State a sequence of dilations and translations.
4
10
Geometry
Unit Activity
Unit: Circles With and Without Coordinates
This Unit Activity will help you meet these educational goals:
• Mathematical Practices—You will use mathematics to model real-world situations.
• Inquiry—You will conduct online research.
• STEM—You will apply mathematical and technology tools and knowledge to grow in
your understanding of mathematics as a creative human activity.
Introduction
In this Unit Activity, you will apply and extend your knowledge of circle properties and related
angle measurements.
__________________________________________________________________________
Directions and Analysis
Task 1: Satellite Placement
A communications company is planning to launch a new satellite into orbit to improve the
operation of its network. The longest distance that the satellite can send a reliable signal is
4,500 miles. The company is trying to determine the optimal height to operate the satellite so
that it has maximum coverage of Earth within its range.
You will use the GeoGebra geometry tool to model and solve this problem. Open GeoGebra,
and complete each step below. If you need help, follow these instructions for using
GeoGebra.
a. To achieve maximum coverage of the signal across Earth, what type of line should be
formed between the point where the satellite is located and the point on Earth’s surface
where the signal is received?
Type your response here:
b. Create a cross-sectional diagram of this situation in GeoGebra, with the circumference of
Earth depicted as a circle (your diagram need not be drawn to scale):
1. Create point C for the center of Earth, point S for the position of the satellite, and
points P1 and P2 for the farthest points on Earth where the signal will reach.
2. Draw a radius from C to P1 and a line segment from S to P1. Display the measure of
the angle formed at their intersection. In the space provided below, note how CP1 and
SP1 are related.
3. Use the Internet or another resource to find the approximate radius of Earth, and note
it in the space provided below. Then label the lengths of CP1 and SP1
Paste a screenshot of your work following your answers.
1
© 2013 EDMENTUM, INC.
Type your response here:
c. With the satellite directly overhead, how far above the surface of Earth does the satellite
need to be so it provides the maximum coverage of its signal? Explain how you arrived at
the answer.
Type your response here:
Task 2: Gears
The gear is one of the oldest mechanical devices. It has been used since ancient times for its
ability to increase or decrease rotational velocity. When one gear turns another, the outer
portions of the gears are turning at the same linear speed. However, when one gear is
smaller or larger than the other, the speed of rotation will be different for each gear.
Let’s look at an application of gears that you’ve seen before. On a bicycle, the pedals are
attached to a large gear at the center of the bike, and a chain connects the large gear to a
small gear attached to the rear wheel. The chain does not change the speed of rotation; it
serves only to separate the centers of rotation.
Consider a bicycle where the large central gear has a radius of 4 inches and the small back
gear has a radius of 2 inches.
a. How many inches would a point on the outer edge of the large gear travel in a 150°
rotation?
Type your response here:
b. What degree of rotation will the small gear undergo when a point on its outer edge travels
the same linear distance determined in part a?
Type your response here:
c. If the center of the small gear is rigidly attached to the center of a bicycle wheel with a
radius of 10 inches, how many inches will the bicycle travel with a single rotation of the
large gear?
Type your response here:
d. How does the distance you determined in part c change if the radius of the small gear is
1.5 inches and the radius of the large gear is 4.5 inches?
Type your response here:
2
Task 3: Circle Theorems
In this unit, you learned about some of the important properties of circles. Let’s take a look at
a property of circles that might be new to you.
You will use GeoGebra to demonstrate a theorem about chords and radii. Open GeoGebra,
and complete each step below.
a. Construct a circle of any radius, and draw a chord on it. Then construct the radius of the
circle that bisects the chord. Measure the angle between the chord and the radius. What
can you conclude about the intersection of a chord and the radius that bisects it? Paste a
screenshot below to support your answer.
Type your response here:
b. Write a paragraph proof of your conclusion in part a. To begin your proof, draw radii OA
and OC .
Type your response here:
c. In this part of the activity, you will investigate the converse of the theorem stated in part a.
To get started, reopen GeoGebra.
Draw a circle of any radius, and draw a chord on it. Construct the radius of the circle that
is perpendicular to the chord. Measure the line segments into which the radius divides the
chords. How are the line segments related? What can you conclude about the intersection
of a chord and a radius that is perpendicular to it? Paste a screenshot below to support
your answer.
Type your response here:
d. Write a paragraph proof of your conclusion in part c. To begin your proof, draw radii OA
and OC .
Type your response here:
__________________________________________________________________________
Resources
Document any references you used for this project below. At minimum, include a title and
URL for any Internet resource:
__________________________________________________________________________
3
Evaluation
This project will be evaluated on a rubric that is based on the completeness, clarity, and
thinking you exhibit in the Directions and Analysis section above.
Total Points: 10
Task 1: Satellite Placement
a. Identify the line for maximum satellite signal coverage.
Task points: 2
0.5
b. Draw a diagram depicting the satellite location.
0.5
c. Calculate the height of the satellite.
1
Task 2: Gears
a. Calculate the linear movement of the large gear.
Task points: 3
0.5
b. Calculate the angular movement of the small gear.
0.5
c. Calculate the linear movement of the bicycle (case I).
1
d. Calculate the linear movement of the bicycle (case II).
1
Task 3: Circle Theorems
a. Construct a circle that depicts the theorem.
Task points: 5
0.5
b. Write a paragraph proof for the theorem.
2
c. Construct a circle that depicts the converse of the theorem.
0.5
d. Write a paragraph proof for the converse of the theorem.
2
4
Probability
and Statistics
Unit Activity
,
Unit: Independent and Conditional Probability
The Lesson Activities will help you meet these educational goals:
• Mathematical Practices—You will use mathematics to model real-world situations,
use appropriate tools strategically, and look for and make use of structure.
• Inquiry—You will perform an investigation in which you will make observations,
analyze results, and communicate your results in written form.
• STEM—You will use mathematical processes and analysis in scientific investigation,
analyze real-world situations, and gain insight into careers in science, technology,
engineering, and math.
• 21st Century Skills—You will apply creativity and innovation, use critical-thinking and
problem-solving skills, communicate effectively, assess and validate information,
perform large-scale data analysis, and carry out technology-assisted modeling.
Introduction
Suppose you’re an actuary working for a life insurance company and are evaluating eight
potential clients for life insurance policies:
• Jacob is a newborn non-Hispanic white male (30-year policy)
• Carol is a 44-year-old non-Hispanic black female (20-year policy)
• Geraldo is a 25-year-old Hispanic male (30-year policy)
• Meg is a 39-year-old Asian female (20-year policy)
• Earvin is a 68-year-old non-Hispanic black male (10-year policy)
• Dora is a 53-year-old Hispanic female (20-year policy)
• Adam is an 18-year-old Asian male (30-year policy)
• Sally is an 80-year-old non-Hispanic white female (10-year policy)
__________________________________________________________________________
Directions and Analysis
Task 1: Finding the Likelihood of Survival of Different Clients
In this task, you will refer to these life tables published by the US government and find the
probabilities of your eight potential clients surviving to the end of their policy periods. Make
sure you use the appropriate life table for each individual.
a. What is the probability that Jacob, the newborn you’re considering for a 30-year policy,
lives to be 30 years old? Jacob is a non-Hispanic white male, so look in table 14.
Type your response here:
b. What is the probability that Carol, the 44-year-old you’re considering for a 20-year policy,
lives to be 64 years old? Note that this is a conditional probability. Another way of saying
Copyright © 2012 PLATO Learning, Inc. All rights reserved.
1
this is, “What is the probability that Carol turns 64 years old given that she turns 44 years
old?” Carol is a non-Hispanic black female, so look in table 18.
Type your response here:
c. What is the probability that Geraldo, the 25-year-old you’re considering for a 30-year
policy, lives to be 55 years old? Remember that Geraldo is a Hispanic male.
Type your response here:
d. What is the probability that Meg, the 39-year-old you’re considering for a 20-year policy,
lives to be 59 years old? Meg is an Asian female, but there is no specific life table for
Asian females; look in table 3, which is a general table for females.
Type your response here:
e. What is the probability the Earvin, the 68-year-old you’re considering for a 10-year policy,
lives to be 78 years old? Remember that Earvin is a non-Hispanic black male.
Type your response here:
f. What is the probability that Dora, the 53-year-old you’re considering for a 20-year policy,
lives to be 73 years old? Remember that Dora is a Hispanic female.
Type your response here:
g. What is the probability that Adam, the 18-year-old you’re considering for a 30-year policy,
lives to be 48 years old? Adam is an Asian male. There is no specific data for Asian
males; look at data in table 2, which is a general table for all males.
Type your response here:
h. What is the probability that Sally, the 80-year old you’re considering for a 10-year policy,
lives to be 90 years old? Sally is a non-Hispanic white female, so look at the data in table
15.
Type your response here:
i.
What is the probability that all of the potential clients would be alive when their respective
policies end? Note that each potential client surviving to the end of his or her respective
policy is an independent event.
2
Type your response here:
Task 2: Simulating Client Lives
For this task, you will use the worksheet labeled Task 2a in this spreadsheet. It gives
abbreviated life tables for the eight potential clients. Look at the third column (Status) for each
potential client. For each client, the rest of his or her life has been randomly simulated based
on the probabilities that he or she lives through each year. Jacob, for example, is a newborn,
so during each of the age ranges listed, it’s possible that he’ll die, but the probabilities of
dying for the age ranges at the beginning of his life are small.
Each time you press F9 on your keyboard, you see an alternate life for Jacob, with his status
for each age range shown as either alive or dead. If the dead were first to appear for the age
range of 75 to 76, for example, this would mean that Jacob died between the ages of 75 and
76, or that he lived to be 75 years old.
a. Press F9 on your keyboard a few times and see how long Jacob lives in each of his
alternate lives. How long did Jacob live each time?
Type your response here:
The rest of the potential clients are similar to Jacob, but since they’ve already lived parts of
their lives, their status will always be alive for the age ranges that they’ve already lived.
For example, Carol is 44 years old, so no matter how many times you press F9 on your
keyboard, Carol’s status will always be alive for all the age ranges up to 43–44. Starting with
the age range of 44–45, however, there is the possibility that Carol’s status will be dead.
b. Press F9 on your keyboard five more times and see how long Carol lives in each of her
alternate lives. Remember that she will always live to be at least 44 years old, since she is
already 44 years old. How long did Carol live each time?
Type your response here:
Now you will find the percent survival of each of your eight clients to the end of his or her
policy using the simulation in the spreadsheet.
For each potential client, you will see whether he or she would be alive at the end of his or
her policy. The cells in the spreadsheet that you should look at to determine this are
highlighted in yellow.
c. Next, go to the worksheet labeled Task 2b and record either alive or dead for the first trial.
Once you do this, the All column will say yes if all the clients were alive at the end of their
policies or no if all the clients were not alive at the end of their policies. Were all the clients
alive at the end of their policies in the first trial?
3
Type your response here:
Next, go back to the Task 2a worksheet, press F9, and repeat this process until you have
recorded 20 trials in the Task 2b worksheet.
In the Percent Survived row at the bottom of the table on the Task 2b worksheet, it will show
the percentage of times each client survived to the end of his or her policy, and it will also
show the percentage of times that all of the clients survived to the end of their respective
policies. Check to see whether these percentages are in line with the probabilities that you
calculated in questions 1 through 9 in Task 1.
d. Are your probabilities from the simulation close to the probabilities you originally
calculated?
Type your response here:
Now save your spreadsheet and submit it to your teacher.
Task 3: Randomly Choosing Two Potential Clients
For this task, you’ll use the same set of eight potential clients: Jacob, Carol, Geraldo, Meg,
Earvin, Dora, Adam, and Sally. You have decided to just randomly choose two of them to
offer life insurance policies.
a. How many pairs of potential clients can be randomly chosen from the pool of eight
candidates?
Type your response here:
b. What is the probability of any particular pair being chosen?
Type your response here:
c. What is the probability that the pair chosen is Jacob and Meg or Geraldo and Sally?
Type your response here:
d. What is the probability that the pair chosen is Carol and Earvin, Earvin and Dora, or Dora
and Adam?
Type your response here:
4
e. What is the sample space of the pairs of potential clients that could be chosen?
Type your response here:
__________________________________________________________________________
Resources
life tables
__________________________________________________________________________
Evaluation
This project will be evaluated on a rubric that is based on the completeness, clarity, and
thinking you exhibit in the Directions and Analysis section above.
Total Points: 10
Task 1: Finding the Likelihood of Survival of Different
Clients
Use life tables to calculate the probability of each of eight
potential clients living to the end of his or her policy.
Task points: 4
• 4 points
Task 2: Simulating Client Lives
Compare the probabilities calculated in Task 1 with the
probabilities obtained from the simulation.
Task points: 4
• 4 points
Task 3: Randomly Choosing Two Potential Clients
Find the possible combinations and probabilities of randomly
choosing pairs.
Task points: 2
• 2 points
5
Probability
and Statistics
Unit Activity
Unit: Applying Probability
The Lesson Activities will help you meet these educational goals:
• Mathematical Practices—You will make sense of problems and solve them, reason
abstractly and quantitatively, attend to precision, and look for and make use of
structure.
• Inquiry—You will analyze data and communicate your results in written form.
• STEM—You will apply mathematical and technology tools and knowledge to analyze
real-world situations.
• 21st Century Skills—You will employ online tools for research, use critical-thinking
and problem-solving skills, and communicate effectively.
Introduction
Take a look at this report produced by the Aviation Consumer Protection Division of the
Office of Aviation Enforcement and Proceedings, which is a part of the US Department of
Transportation. Specifically, you’re interested in table 2, on-time flight arrivals, which starts on
page 6 and ends on page 9. The data in the table is from a typical month, so each
percentage in the table can be thought of as a probability. You can answer many probability
questions using this table.
__________________________________________________________________________
Directions and Analysis
Task 1: Questions You Can Answer with Table 2
Refer to table 2 of the Air Travel Consumer Report to answer these questions.
a. Given that a randomly chosen flight arrives in Atlanta (ATL), what is the probability that it
arrives on time?
Type your response here:
b. Given that a randomly chosen US Airways (US) flight arrives in Denver (DEN), what is the
probability that it arrives on time?
Type your response here:
c. Given that a randomly chosen flight arrives in Los Angeles (LAX), what is the probability
that the carrier is American Airlines (AA)?
Type your response here:
Copyright © 2012 PLATO Learning, Inc. All rights reserved.
1
d. Chicago has two airports (MDW and ORD). If you’re flying with Delta Airlines (DL) and
your primary concern is to arrive on time, is it better to fly to MDW or to ORD?
Type your response here:
e. New York City has two airports (JFK and LGA). If you’re flying with United Airlines (UA)
and your primary concern is to arrive on time, is it better to fly to JFK or to LGA?
Type your response here:
Task 2: Questions That Are Difficult or Nearly Impossible to Answer with Table 2
Many questions are difficult or nearly impossible to answer using table 2 in its current form.
Since computing the answers to these questions is difficult or close to impossible, just give
the formulas you would need to obtain the answers to these questions.
a. What is the probability that a randomly chosen flight arrives on time?
Type your response here:
b. Overall, which carrier has the highest probability of arriving on time?
Type your response here:
c. Given that a randomly chosen flight arrives on time, what is the probability that it arrives in
Philadelphia (PHL)?
Type your response here:
d. Given that the carrier is Hawaiian Airlines (HA), what is the probability that a randomly
chosen flight arrives on time?
Type your response here:
e. If you’re a frequent flyer who travels to many different airports, and you have the choice
between SkyWest Airlines (OO) and Southwest Airlines (WN) for all your flights, which
carrier should you choose if your primary concern is arriving on time?
Type your response here:
2
Task 3: Transforming Table 2
The reason it was difficult or nearly impossible to answer the questions in task 2 is that table
2 isn’t quite a two-way table. In other words, there is a total row at the bottom of the table, but
there is no total column at the right of the table. In this activity, you’ll transform the table into a
proper two-way table.
Open this spreadsheet. The worksheet labeled Original contains the data as it appears in the
report. Scroll through the data in this worksheet and do a spot check to verify that it is, in fact,
the same table. This will give you some practice in navigating the table.
Once you are comfortable navigating, click the Modified tab. In this worksheet, a few changes
have been made to the table:
• Cells that contained H/ in the original table now contain zeros. H/ indicated that a
particular carrier didn’t have a hub at a particular airport. In other words, it had 0
arriving flights with 0% arriving on time.
• For each airport, an additional column labeled # ON TIME has been added.
• On the far right, you will find a TOTAL area with three new columns: # OF ARR., # ON
TIME, and % ON TIME.
By filling in all of the blank columns, you’ll transform the table into a two-way table. Below
you’ll find instructions for filling in the blank columns to complete the two-way table.
Before you start, make a copy of the Modified worksheet in case you make a mistake. One
way to do this is to right-click the Modified tab, select Move or Copy on the shortcut menu,
select the Create a copy checkbox, and click OK. This procedure may vary slightly depending
on your version of Microsoft Excel. If you make a mistake, just delete the Modified tab you
created by right-clicking it and selecting Delete from the shortcut menu. Then make a new
copy of the Modified worksheet and start again.
To complete the # ON TIME column for each airport:
• Click in cells C8–C22 and look at the formula bar to get an understanding of the formulas
in these cells. Do the same with cell C23.
• Copy cells C8–C23, and paste them into the space for cells F8–F23.
• Verify that all the formulas are doing what you expect.
• Repeat this procedure to fill in the # ON TIME column for each individual airport.
To complete the TOTAL area on the right:
• Go to columns CK and CL at the far right of the table. Examine what the formulas do for
rows 8–22. Now examine the formula in cells CK23 and CL23.
• Click in Cell CM8.
• Enter a formula in the formula bar that will calculate what percentage of total arrivals are
on-time arrivals.
• Once you’re satisfied this formula works, copy it into all the cells in that column (CM8–
CM22, plus CM23).
3
Task 4: Questions That Can Now Be Answered by Table 2
Remember those questions that were difficult or nearly impossible to answer with table 2?
See if you can answer them now, using the spreadsheet you modified.
a. What is the probability that a randomly chosen flight arrives on time?
Type your response here:
b. Overall, which carrier has the highest probability of arriving on time?
Type your response here:
c. Given that a randomly chosen flight arrives on time, what is the probability that it arrives in
Philadelphia (PHL)?
Type your response here:
d. Given that the carrier is Hawaiian Airlines (HA), what is the probability that a randomly
chosen flight arrives on time?
Type your response here:
e. If you’re a frequent flyer who travels to many different airports, and you have a choice
between SkyWest Airlines (OO) and Southwest Airlines (WN) for all your flights, which
carrier should you choose if your primary concern is arriving on time?
Type your response here:
Task 5: Applying Conditional Probability to a Real-World Scenario
Gary Foshee created a popular probability puzzle that goes like this: “I have two children.
One is a boy born on a Tuesday. What is the probability I have two boys?”
In this puzzle, knowing that one of Gary’s children was born on a Tuesday is as important as
knowing that he has one boy. Assuming that having boys and girls are equally likely and that
births are equally likely on every day of the week, what is the probability that Gary has two
boys, given the available information? (Hint: When Gary said that one is a boy born on a
Tuesday, he meant at least one child is a boy born on a Tuesday.)
Type your response here:
4
__________________________________________________________________________
Resources
Air Travel Consumer Report
Hype about Conditional Probability Puzzles
__________________________________________________________________________
Evaluation
This project will be evaluated on a rubric that is based on the completeness, clarity, and
thinking you exhibit in the Directions and Analysis section above.
Total Points: 10
Task 1: Questions You Can Answer with Table 2
Identify and derive the conditional probabilities.
Task points: 2
• 2 points
Task 2: Questions That Are Difficult or Nearly Impossible to Task points: 2
Answer with Table 2
Formulate solutions to conditional probability problems.
• 2 points
Task 3: Transforming Table 2
Transform the table into a two-way table.
Task points: 1
• 1 point
Task 4: Questions That Can Now Be Answered by Table 2
Derive the conditional probabilities.
Task points: 2
• 2 points
Task 5: Applying Conditional Probability to a Real-World
Scenario
Compute the boy-boy conditional probability.
Task points: 3
•
3 points
5
Probability
and Statistics
Unit Activity
Unit: Applying Probability
The Lesson Activities will help you meet these educational goals:
• Mathematical Practices—You will make sense of problems and solve them, reason
abstractly and quantitatively, attend to precision, and look for and make use of
structure.
• Inquiry—You will analyze data and communicate your results in written form.
• STEM—You will apply mathematical and technology tools and knowledge to analyze
real-world situations.
• 21st Century Skills—You will employ online tools for research, use critical-thinking
and problem-solving skills, and communicate effectively.
Introduction
Take a look at this report produced by the Aviation Consumer Protection Division of the
Office of Aviation Enforcement and Proceedings, which is a part of the US Department of
Transportation. Specifically, you’re interested in table 2, on-time flight arrivals, which starts on
page 6 and ends on page 9. The data in the table is from a typical month, so each
percentage in the table can be thought of as a probability. You can answer many probability
questions using this table.
__________________________________________________________________________
Directions and Analysis
Task 1: Questions You Can Answer with Table 2
Refer to table 2 of the Air Travel Consumer Report to answer these questions.
a. Given that a randomly chosen flight arrives in Atlanta (ATL), what is the probability that it
arrives on time?
Type your response here:
b. Given that a randomly chosen US Airways (US) flight arrives in Denver (DEN), what is the
probability that it arrives on time?
Type your response here:
c. Given that a randomly chosen flight arrives in Los Angeles (LAX), what is the probability
that the carrier is American Airlines (AA)?
Type your response here:
Copyright © 2012 PLATO Learning, Inc. All rights reserved.
1
d. Chicago has two airports (MDW and ORD). If you’re flying with Delta Airlines (DL) and
your primary concern is to arrive on time, is it better to fly to MDW or to ORD?
Type your response here:
e. New York City has two airports (JFK and LGA). If you’re flying with United Airlines (UA)
and your primary concern is to arrive on time, is it better to fly to JFK or to LGA?
Type your response here:
Task 2: Questions That Are Difficult or Nearly Impossible to Answer with Table 2
Many questions are difficult or nearly impossible to answer using table 2 in its current form.
Since computing the answers to these questions is difficult or close to impossible, just give
the formulas you would need to obtain the answers to these questions.
a. What is the probability that a randomly chosen flight arrives on time?
Type your response here:
b. Overall, which carrier has the highest probability of arriving on time?
Type your response here:
c. Given that a randomly chosen flight arrives on time, what is the probability that it arrives in
Philadelphia (PHL)?
Type your response here:
d. Given that the carrier is Hawaiian Airlines (HA), what is the probability that a randomly
chosen flight arrives on time?
Type your response here:
e. If you’re a frequent flyer who travels to many different airports, and you have the choice
between SkyWest Airlines (OO) and Southwest Airlines (WN) for all your flights, which
carrier should you choose if your primary concern is arriving on time?
Type your response here:
2
Task 3: Transforming Table 2
The reason it was difficult or nearly impossible to answer the questions in task 2 is that table
2 isn’t quite a two-way table. In other words, there is a total row at the bottom of the table, but
there is no total column at the right of the table. In this activity, you’ll transform the table into a
proper two-way table.
Open this spreadsheet. The worksheet labeled Original contains the data as it appears in the
report. Scroll through the data in this worksheet and do a spot check to verify that it is, in fact,
the same table. This will give you some practice in navigating the table.
Once you are comfortable navigating, click the Modified tab. In this worksheet, a few changes
have been made to the table:
• Cells that contained H/ in the original table now contain zeros. H/ indicated that a
particular carrier didn’t have a hub at a particular airport. In other words, it had 0
arriving flights with 0% arriving on time.
• For each airport, an additional column labeled # ON TIME has been added.
• On the far right, you will find a TOTAL area with three new columns: # OF ARR., # ON
TIME, and % ON TIME.
By filling in all of the blank columns, you’ll transform the table into a two-way table. Below
you’ll find instructions for filling in the blank columns to complete the two-way table.
Before you start, make a copy of the Modified worksheet in case you make a mistake. One
way to do this is to right-click the Modified tab, select Move or Copy on the shortcut menu,
select the Create a copy checkbox, and click OK. This procedure may vary slightly depending
on your version of Microsoft Excel. If you make a mistake, just delete the Modified tab you
created by right-clicking it and selecting Delete from the shortcut menu. Then make a new
copy of the Modified worksheet and start again.
To complete the # ON TIME column for each airport:
• Click in cells C8–C22 and look at the formula bar to get an understanding of the formulas
in these cells. Do the same with cell C23.
• Copy cells C8–C23, and paste them into the space for cells F8–F23.
• Verify that all the formulas are doing what you expect.
• Repeat this procedure to fill in the # ON TIME column for each individual airport.
To complete the TOTAL area on the right:
• Go to columns CK and CL at the far right of the table. Examine what the formulas do for
rows 8–22. Now examine the formula in cells CK23 and CL23.
• Click in Cell CM8.
• Enter a formula in the formula bar that will calculate what percentage of total arrivals are
on-time arrivals.
• Once you’re satisfied this formula works, copy it into all the cells in that column (CM8–
CM22, plus CM23).
3
Task 4: Questions That Can Now Be Answered by Table 2
Remember those questions that were difficult or nearly impossible to answer with table 2?
See if you can answer them now, using the spreadsheet you modified.
a. What is the probability that a randomly chosen flight arrives on time?
Type your response here:
b. Overall, which carrier has the highest probability of arriving on time?
Type your response here:
c. Given that a randomly chosen flight arrives on time, what is the probability that it arrives in
Philadelphia (PHL)?
Type your response here:
d. Given that the carrier is Hawaiian Airlines (HA), what is the probability that a randomly
chosen flight arrives on time?
Type your response here:
e. If you’re a frequent flyer who travels to many different airports, and you have a choice
between SkyWest Airlines (OO) and Southwest Airlines (WN) for all your flights, which
carrier should you choose if your primary concern is arriving on time?
Type your response here:
Task 5: Applying Conditional Probability to a Real-World Scenario
Gary Foshee created a popular probability puzzle that goes like this: “I have two children.
One is a boy born on a Tuesday. What is the probability I have two boys?”
In this puzzle, knowing that one of Gary’s children was born on a Tuesday is as important as
knowing that he has one boy. Assuming that having boys and girls are equally likely and that
births are equally likely on every day of the week, what is the probability that Gary has two
boys, given the available information? (Hint: When Gary said that one is a boy born on a
Tuesday, he meant at least one child is a boy born on a Tuesday.)
Type your response here:
4
__________________________________________________________________________
Resources
Air Travel Consumer Report
Hype about Conditional Probability Puzzles
__________________________________________________________________________
Evaluation
This project will be evaluated on a rubric that is based on the completeness, clarity, and
thinking you exhibit in the Directions and Analysis section above.
Total Points: 10
Task 1: Questions You Can Answer with Table 2
Identify and derive the conditional probabilities.
Task points: 2
• 2 points
Task 2: Questions That Are Difficult or Nearly Impossible to Task points: 2
Answer with Table 2
Formulate solutions to conditional probability problems.
• 2 points
Task 3: Transforming Table 2
Transform the table into a two-way table.
Task points: 1
• 1 point
Task 4: Questions That Can Now Be Answered by Table 2
Derive the conditional probabilities.
Task points: 2
• 2 points
Task 5: Applying Conditional Probability to a Real-World
Scenario
Compute the boy-boy conditional probability.
Task points: 3
•
3 points
5
Probability
and Statistics
Unit Activity
Unit: Applying Probability
The Lesson Activities will help you meet these educational goals:
• Mathematical Practices—You will make sense of problems and solve them, reason
abstractly and quantitatively, attend to precision, and look for and make use of
structure.
• Inquiry—You will analyze data and communicate your results in written form.
• STEM—You will apply mathematical and technology tools and knowledge to analyze
real-world situations.
• 21st Century Skills—You will employ online tools for research, use critical-thinking
and problem-solving skills, and communicate effectively.
Introduction
Take a look at this report produced by the Aviation Consumer Protection Division of the
Office of Aviation Enforcement and Proceedings, which is a part of the US Department of
Transportation. Specifically, you’re interested in table 2, on-time flight arrivals, which starts on
page 6 and ends on page 9. The data in the table is from a typical month, so each
percentage in the table can be thought of as a probability. You can answer many probability
questions using this table.
__________________________________________________________________________
Directions and Analysis
Task 1: Questions You Can Answer with Table 2
Refer to table 2 of the Air Travel Consumer Report to answer these questions.
a. Given that a randomly chosen flight arrives in Atlanta (ATL), what is the probability that it
arrives on time?
Type your response here:
b. Given that a randomly chosen US Airways (US) flight arrives in Denver (DEN), what is the
probability that it arrives on time?
Type your response here:
c. Given that a randomly chosen flight arrives in Los Angeles (LAX), what is the probability
that the carrier is American Airlines (AA)?
Type your response here:
Copyright © 2012 PLATO Learning, Inc. All rights reserved.
1
d. Chicago has two airports (MDW and ORD). If you’re flying with Delta Airlines (DL) and
your primary concern is to arrive on time, is it better to fly to MDW or to ORD?
Type your response here:
e. New York City has two airports (JFK and LGA). If you’re flying with United Airlines (UA)
and your primary concern is to arrive on time, is it better to fly to JFK or to LGA?
Type your response here:
Task 2: Questions That Are Difficult or Nearly Impossible to Answer with Table 2
Many questions are difficult or nearly impossible to answer using table 2 in its current form.
Since computing the answers to these questions is difficult or close to impossible, just give
the formulas you would need to obtain the answers to these questions.
a. What is the probability that a randomly chosen flight arrives on time?
Type your response here:
b. Overall, which carrier has the highest probability of arriving on time?
Type your response here:
c. Given that a randomly chosen flight arrives on time, what is the probability that it arrives in
Philadelphia (PHL)?
Type your response here:
d. Given that the carrier is Hawaiian Airlines (HA), what is the probability that a randomly
chosen flight arrives on time?
Type your response here:
e. If you’re a frequent flyer who travels to many different airports, and you have the choice
between SkyWest Airlines (OO) and Southwest Airlines (WN) for all your flights, which
carrier should you choose if your primary concern is arriving on time?
Type your response here:
2
Task 3: Transforming Table 2
The reason it was difficult or nearly impossible to answer the questions in task 2 is that table
2 isn’t quite a two-way table. In other words, there is a total row at the bottom of the table, but
there is no total column at the right of the table. In this activity, you’ll transform the table into a
proper two-way table.
Open this spreadsheet. The worksheet labeled Original contains the data as it appears in the
report. Scroll through the data in this worksheet and do a spot check to verify that it is, in fact,
the same table. This will give you some practice in navigating the table.
Once you are comfortable navigating, click the Modified tab. In this worksheet, a few changes
have been made to the table:
• Cells that contained H/ in the original table now contain zeros. H/ indicated that a
particular carrier didn’t have a hub at a particular airport. In other words, it had 0
arriving flights with 0% arriving on time.
• For each airport, an additional column labeled # ON TIME has been added.
• On the far right, you will find a TOTAL area with three new columns: # OF ARR., # ON
TIME, and % ON TIME.
By filling in all of the blank columns, you’ll transform the table into a two-way table. Below
you’ll find instructions for filling in the blank columns to complete the two-way table.
Before you start, make a copy of the Modified worksheet in case you make a mistake. One
way to do this is to right-click the Modified tab, select Move or Copy on the shortcut menu,
select the Create a copy checkbox, and click OK. This procedure may vary slightly depending
on your version of Microsoft Excel. If you make a mistake, just delete the Modified tab you
created by right-clicking it and selecting Delete from the shortcut menu. Then make a new
copy of the Modified worksheet and start again.
To complete the # ON TIME column for each airport:
• Click in cells C8–C22 and look at the formula bar to get an understanding of the formulas
in these cells. Do the same with cell C23.
• Copy cells C8–C23, and paste them into the space for cells F8–F23.
• Verify that all the formulas are doing what you expect.
• Repeat this procedure to fill in the # ON TIME column for each individual airport.
To complete the TOTAL area on the right:
• Go to columns CK and CL at the far right of the table. Examine what the formulas do for
rows 8–22. Now examine the formula in cells CK23 and CL23.
• Click in Cell CM8.
• Enter a formula in the formula bar that will calculate what percentage of total arrivals are
on-time arrivals.
• Once you’re satisfied this formula works, copy it into all the cells in that column (CM8–
CM22, plus CM23).
3
Task 4: Questions That Can Now Be Answered by Table 2
Remember those questions that were difficult or nearly impossible to answer with table 2?
See if you can answer them now, using the spreadsheet you modified.
a. What is the probability that a randomly chosen flight arrives on time?
Type your response here:
b. Overall, which carrier has the highest probability of arriving on time?
Type your response here:
c. Given that a randomly chosen flight arrives on time, what is the probability that it arrives in
Philadelphia (PHL)?
Type your response here:
d. Given that the carrier is Hawaiian Airlines (HA), what is the probability that a randomly
chosen flight arrives on time?
Type your response here:
e. If you’re a frequent flyer who travels to many different airports, and you have a choice
between SkyWest Airlines (OO) and Southwest Airlines (WN) for all your flights, which
carrier should you choose if your primary concern is arriving on time?
Type your response here:
Task 5: Applying Conditional Probability to a Real-World Scenario
Gary Foshee created a popular probability puzzle that goes like this: “I have two children.
One is a boy born on a Tuesday. What is the probability I have two boys?”
In this puzzle, knowing that one of Gary’s children was born on a Tuesday is as important as
knowing that he has one boy. Assuming that having boys and girls are equally likely and that
births are equally likely on every day of the week, what is the probability that Gary has two
boys, given the available information? (Hint: When Gary said that one is a boy born on a
Tuesday, he meant at least one child is a boy born on a Tuesday.)
Type your response here:
4
__________________________________________________________________________
Resources
Air Travel Consumer Report
Hype about Conditional Probability Puzzles
__________________________________________________________________________
Evaluation
This project will be evaluated on a rubric that is based on the completeness, clarity, and
thinking you exhibit in the Directions and Analysis section above.
Total Points: 10
Task 1: Questions You Can Answer with Table 2
Identify and derive the conditional probabilities.
Task points: 2
• 2 points
Task 2: Questions That Are Difficult or Nearly Impossible to Task points: 2
Answer with Table 2
Formulate solutions to conditional probability problems.
• 2 points
Task 3: Transforming Table 2
Transform the table into a two-way table.
Task points: 1
• 1 point
Task 4: Questions That Can Now Be Answered by Table 2
Derive the conditional probabilities.
Task points: 2
• 2 points
Task 5: Applying Conditional Probability to a Real-World
Scenario
Compute the boy-boy conditional probability.
Task points: 3
•
3 points
5
Probability
and Statistics
Unit Activity
,
Unit: Independent and Conditional Probability
The Lesson Activities will help you meet these educational goals:
• Mathematical Practices—You will use mathematics to model real-world situations,
use appropriate tools strategically, and look for and make use of structure.
• Inquiry—You will perform an investigation in which you will make observations,
analyze results, and communicate your results in written form.
• STEM—You will use mathematical processes and analysis in scientific investigation,
analyze real-world situations, and gain insight into careers in science, technology,
engineering, and math.
• 21st Century Skills—You will apply creativity and innovation, use critical-thinking and
problem-solving skills, communicate effectively, assess and validate information,
perform large-scale data analysis, and carry out technology-assisted modeling.
Introduction
Suppose you’re an actuary working for a life insurance company and are evaluating eight
potential clients for life insurance policies:
• Jacob is a newborn non-Hispanic white male (30-year policy)
• Carol is a 44-year-old non-Hispanic black female (20-year policy)
• Geraldo is a 25-year-old Hispanic male (30-year policy)
• Meg is a 39-year-old Asian female (20-year policy)
• Earvin is a 68-year-old non-Hispanic black male (10-year policy)
• Dora is a 53-year-old Hispanic female (20-year policy)
• Adam is an 18-year-old Asian male (30-year policy)
• Sally is an 80-year-old non-Hispanic white female (10-year policy)
__________________________________________________________________________
Directions and Analysis
Task 1: Finding the Likelihood of Survival of Different Clients
In this task, you will refer to these life tables published by the US government and find the
probabilities of your eight potential clients surviving to the end of their policy periods. Make
sure you use the appropriate life table for each individual.
a. What is the probability that Jacob, the newborn you’re considering for a 30-year policy,
lives to be 30 years old? Jacob is a non-Hispanic white male, so look in table 14.
Type your response here:
b. What is the probability that Carol, the 44-year-old you’re considering for a 20-year policy,
lives to be 64 years old? Note that this is a conditional probability. Another way of saying
Copyright © 2012 PLATO Learning, Inc. All rights reserved.
1
this is, “What is the probability that Carol turns 64 years old given that she turns 44 years
old?” Carol is a non-Hispanic black female, so look in table 18.
Type your response here:
c. What is the probability that Geraldo, the 25-year-old you’re considering for a 30-year
policy, lives to be 55 years old? Remember that Geraldo is a Hispanic male.
Type your response here:
d. What is the probability that Meg, the 39-year-old you’re considering for a 20-year policy,
lives to be 59 years old? Meg is an Asian female, but there is no specific life table for
Asian females; look in table 3, which is a general table for females.
Type your response here:
e. What is the probability the Earvin, the 68-year-old you’re considering for a 10-year policy,
lives to be 78 years old? Remember that Earvin is a non-Hispanic black male.
Type your response here:
f. What is the probability that Dora, the 53-year-old you’re considering for a 20-year policy,
lives to be 73 years old? Remember that Dora is a Hispanic female.
Type your response here:
g. What is the probability that Adam, the 18-year-old you’re considering for a 30-year policy,
lives to be 48 years old? Adam is an Asian male. There is no specific data for Asian
males; look at data in table 2, which is a general table for all males.
Type your response here:
h. What is the probability that Sally, the 80-year old you’re considering for a 10-year policy,
lives to be 90 years old? Sally is a non-Hispanic white female, so look at the data in table
15.
Type your response here:
i.
What is the probability that all of the potential clients would be alive when their respective
policies end? Note that each potential client surviving to the end of his or her respective
policy is an independent event.
2
Type your response here:
Task 2: Simulating Client Lives
For this task, you will use the worksheet labeled Task 2a in this spreadsheet. It gives
abbreviated life tables for the eight potential clients. Look at the third column (Status) for each
potential client. For each client, the rest of his or her life has been randomly simulated based
on the probabilities that he or she lives through each year. Jacob, for example, is a newborn,
so during each of the age ranges listed, it’s possible that he’ll die, but the probabilities of
dying for the age ranges at the beginning of his life are small.
Each time you press F9 on your keyboard, you see an alternate life for Jacob, with his status
for each age range shown as either alive or dead. If the dead were first to appear for the age
range of 75 to 76, for example, this would mean that Jacob died between the ages of 75 and
76, or that he lived to be 75 years old.
a. Press F9 on your keyboard a few times and see how long Jacob lives in each of his
alternate lives. How long did Jacob live each time?
Type your response here:
The rest of the potential clients are similar to Jacob, but since they’ve already lived parts of
their lives, their status will always be alive for the age ranges that they’ve already lived.
For example, Carol is 44 years old, so no matter how many times you press F9 on your
keyboard, Carol’s status will always be alive for all the age ranges up to 43–44. Starting with
the age range of 44–45, however, there is the possibility that Carol’s status will be dead.
b. Press F9 on your keyboard five more times and see how long Carol lives in each of her
alternate lives. Remember that she will always live to be at least 44 years old, since she is
already 44 years old. How long did Carol live each time?
Type your response here:
Now you will find the percent survival of each of your eight clients to the end of his or her
policy using the simulation in the spreadsheet.
For each potential client, you will see whether he or she would be alive at the end of his or
her policy. The cells in the spreadsheet that you should look at to determine this are
highlighted in yellow.
c. Next, go to the worksheet labeled Task 2b and record either alive or dead for the first trial.
Once you do this, the All column will say yes if all the clients were alive at the end of their
policies or no if all the clients were not alive at the end of their policies. Were all the clients
alive at the end of their policies in the first trial?
3
Type your response here:
Next, go back to the Task 2a worksheet, press F9, and repeat this process until you have
recorded 20 trials in the Task 2b worksheet.
In the Percent Survived row at the bottom of the table on the Task 2b worksheet, it will show
the percentage of times each client survived to the end of his or her policy, and it will also
show the percentage of times that all of the clients survived to the end of their respective
policies. Check to see whether these percentages are in line with the probabilities that you
calculated in questions 1 through 9 in Task 1.
d. Are your probabilities from the simulation close to the probabilities you originally
calculated?
Type your response here:
Now save your spreadsheet and submit it to your teacher.
Task 3: Randomly Choosing Two Potential Clients
For this task, you’ll use the same set of eight potential clients: Jacob, Carol, Geraldo, Meg,
Earvin, Dora, Adam, and Sally. You have decided to just randomly choose two of them to
offer life insurance policies.
a. How many pairs of potential clients can be randomly chosen from the pool of eight
candidates?
Type your response here:
b. What is the probability of any particular pair being chosen?
Type your response here:
c. What is the probability that the pair chosen is Jacob and Meg or Geraldo and Sally?
Type your response here:
d. What is the probability that the pair chosen is Carol and Earvin, Earvin and Dora, or Dora
and Adam?
Type your response here:
4
e. What is the sample space of the pairs of potential clients that could be chosen?
Type your response here:
__________________________________________________________________________
Resources
life tables
__________________________________________________________________________
Evaluation
This project will be evaluated on a rubric that is based on the completeness, clarity, and
thinking you exhibit in the Directions and Analysis section above.
Total Points: 10
Task 1: Finding the Likelihood of Survival of Different
Clients
Use life tables to calculate the probability of each of eight
potential clients living to the end of his or her policy.
Task points: 4
• 4 points
Task 2: Simulating Client Lives
Compare the probabilities calculated in Task 1 with the
probabilities obtained from the simulation.
Task points: 4
• 4 points
Task 3: Randomly Choosing Two Potential Clients
Find the possible combinations and probabilities of randomly
choosing pairs.
Task points: 2
• 2 points
5
Probability
and Statistics
Unit Activity
Unit: Applying Probability
The Lesson Activities will help you meet these educational goals:
• Mathematical Practices—You will make sense of problems and solve them, reason
abstractly and quantitatively, attend to precision, and look for and make use of
structure.
• Inquiry—You will analyze data and communicate your results in written form.
• STEM—You will apply mathematical and technology tools and knowledge to analyze
real-world situations.
• 21st Century Skills—You will employ online tools for research, use critical-thinking
and problem-solving skills, and communicate effectively.
Introduction
Take a look at this report produced by the Aviation Consumer Protection Division of the
Office of Aviation Enforcement and Proceedings, which is a part of the US Department of
Transportation. Specifically, you’re interested in table 2, on-time flight arrivals, which starts on
page 6 and ends on page 9. The data in the table is from a typical month, so each
percentage in the table can be thought of as a probability. You can answer many probability
questions using this table.
__________________________________________________________________________
Directions and Analysis
Task 1: Questions You Can Answer with Table 2
Refer to table 2 of the Air Travel Consumer Report to answer these questions.
a. Given that a randomly chosen flight arrives in Atlanta (ATL), what is the probability that it
arrives on time?
Type your response here:
b. Given that a randomly chosen US Airways (US) flight arrives in Denver (DEN), what is the
probability that it arrives on time?
Type your response here:
c. Given that a randomly chosen flight arrives in Los Angeles (LAX), what is the probability
that the carrier is American Airlines (AA)?
Type your response here:
Copyright © 2012 PLATO Learning, Inc. All rights reserved.
1
d. Chicago has two airports (MDW and ORD). If you’re flying with Delta Airlines (DL) and
your primary concern is to arrive on time, is it better to fly to MDW or to ORD?
Type your response here:
e. New York City has two airports (JFK and LGA). If you’re flying with United Airlines (UA)
and your primary concern is to arrive on time, is it better to fly to JFK or to LGA?
Type your response here:
Task 2: Questions That Are Difficult or Nearly Impossible to Answer with Table 2
Many questions are difficult or nearly impossible to answer using table 2 in its current form.
Since computing the answers to these questions is difficult or close to impossible, just give
the formulas you would need to obtain the answers to these questions.
a. What is the probability that a randomly chosen flight arrives on time?
Type your response here:
b. Overall, which carrier has the highest probability of arriving on time?
Type your response here:
c. Given that a randomly chosen flight arrives on time, what is the probability that it arrives in
Philadelphia (PHL)?
Type your response here:
d. Given that the carrier is Hawaiian Airlines (HA), what is the probability that a randomly
chosen flight arrives on time?
Type your response here:
e. If you’re a frequent flyer who travels to many different airports, and you have the choice
between SkyWest Airlines (OO) and Southwest Airlines (WN) for all your flights, which
carrier should you choose if your primary concern is arriving on time?
Type your response here:
2
Task 3: Transforming Table 2
The reason it was difficult or nearly impossible to answer the questions in task 2 is that table
2 isn’t quite a two-way table. In other words, there is a total row at the bottom of the table, but
there is no total column at the right of the table. In this activity, you’ll transform the table into a
proper two-way table.
Open this spreadsheet. The worksheet labeled Original contains the data as it appears in the
report. Scroll through the data in this worksheet and do a spot check to verify that it is, in fact,
the same table. This will give you some practice in navigating the table.
Once you are comfortable navigating, click the Modified tab. In this worksheet, a few changes
have been made to the table:
• Cells that contained H/ in the original table now contain zeros. H/ indicated that a
particular carrier didn’t have a hub at a particular airport. In other words, it had 0
arriving flights with 0% arriving on time.
• For each airport, an additional column labeled # ON TIME has been added.
• On the far right, you will find a TOTAL area with three new columns: # OF ARR., # ON
TIME, and % ON TIME.
By filling in all of the blank columns, you’ll transform the table into a two-way table. Below
you’ll find instructions for filling in the blank columns to complete the two-way table.
Before you start, make a copy of the Modified worksheet in case you make a mistake. One
way to do this is to right-click the Modified tab, select Move or Copy on the shortcut menu,
select the Create a copy checkbox, and click OK. This procedure may vary slightly depending
on your version of Microsoft Excel. If you make a mistake, just delete the Modified tab you
created by right-clicking it and selecting Delete from the shortcut menu. Then make a new
copy of the Modified worksheet and start again.
To complete the # ON TIME column for each airport:
• Click in cells C8–C22 and look at the formula bar to get an understanding of the formulas
in these cells. Do the same with cell C23.
• Copy cells C8–C23, and paste them into the space for cells F8–F23.
• Verify that all the formulas are doing what you expect.
• Repeat this procedure to fill in the # ON TIME column for each individual airport.
To complete the TOTAL area on the right:
• Go to columns CK and CL at the far right of the table. Examine what the formulas do for
rows 8–22. Now examine the formula in cells CK23 and CL23.
• Click in Cell CM8.
• Enter a formula in the formula bar that will calculate what percentage of total arrivals are
on-time arrivals.
• Once you’re satisfied this formula works, copy it into all the cells in that column (CM8–
CM22, plus CM23).
3
Task 4: Questions That Can Now Be Answered by Table 2
Remember those questions that were difficult or nearly impossible to answer with table 2?
See if you can answer them now, using the spreadsheet you modified.
a. What is the probability that a randomly chosen flight arrives on time?
Type your response here:
b. Overall, which carrier has the highest probability of arriving on time?
Type your response here:
c. Given that a randomly chosen flight arrives on time, what is the probability that it arrives in
Philadelphia (PHL)?
Type your response here:
d. Given that the carrier is Hawaiian Airlines (HA), what is the probability that a randomly
chosen flight arrives on time?
Type your response here:
e. If you’re a frequent flyer who travels to many different airports, and you have a choice
between SkyWest Airlines (OO) and Southwest Airlines (WN) for all your flights, which
carrier should you choose if your primary concern is arriving on time?
Type your response here:
Task 5: Applying Conditional Probability to a Real-World Scenario
Gary Foshee created a popular probability puzzle that goes like this: “I have two children.
One is a boy born on a Tuesday. What is the probability I have two boys?”
In this puzzle, knowing that one of Gary’s children was born on a Tuesday is as important as
knowing that he has one boy. Assuming that having boys and girls are equally likely and that
births are equally likely on every day of the week, what is the probability that Gary has two
boys, given the available information? (Hint: When Gary said that one is a boy born on a
Tuesday, he meant at least one child is a boy born on a Tuesday.)
Type your response here:
4
__________________________________________________________________________
Resources
Air Travel Consumer Report
Hype about Conditional Probability Puzzles
__________________________________________________________________________
Evaluation
This project will be evaluated on a rubric that is based on the completeness, clarity, and
thinking you exhibit in the Directions and Analysis section above.
Total Points: 10
Task 1: Questions You Can Answer with Table 2
Identify and derive the conditional probabilities.
Task points: 2
• 2 points
Task 2: Questions That Are Difficult or Nearly Impossible to Task points: 2
Answer with Table 2
Formulate solutions to conditional probability problems.
• 2 points
Task 3: Transforming Table 2
Transform the table into a two-way table.
Task points: 1
• 1 point
Task 4: Questions That Can Now Be Answered by Table 2
Derive the conditional probabilities.
Task points: 2
• 2 points
Task 5: Applying Conditional Probability to a Real-World
Scenario
Compute the boy-boy conditional probability.
Task points: 3
•
3 points
5