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Goals
You will be faced with many situations involving uncertainty
throughout your life, and many of those situations will involve money. Whether
you are choosing to buy an insurance policy, a warranty on a printer, or a
chance to win a stuffed teddy bear in a spinner game, it is important that you
understand the mathematics behind these costs.
During this unit, you will design two spinner games.
First you’ll learn about spinner games and choose the
prizes. Then you’ll use expected values to design your games to make a profit
and simulate playing those games to see if they make the profit amount you
expected.
Project Files
You will use the following documents and tools for this
project:
·
Project
Overview: PRM_A_06_project_overview.doc
·
Sample
Project: PRM_A_06_sample_project.ppt
·
Project
Template: PRM_A_06_project_template.ppt
·
·
Spinner
Tool: Access the Spinner Tool though Student
Template
template and rename the file as GamePresentation_YourName.
This file will become your presentation.
Project Research
1.  Decide on the prize for your first spinner.
·
Look on the Internet to
find an item that sells for less than \$5. Prize
ideas include, but are not limited to, small stuffed animals, posters,
T-shirts, goldfish, and small puzzles. Assume that the site you find is the
site where the prizes would be purchased from. If the prizes are sold in bulk
(such as a dozen to a pack), figure out the cost per single prize. Do not use the same item that the sample
project uses.
2.  Decide on the prize for your second spinner.
·
Look on the Internet to
find an item that sells for between \$10 and \$30.
Prize ideas include, but are not limited to, large stuffed animals, sports
caps, board games, and backpacks. Assume that the site you find is the site
where the prizes would be purchased from. Do
not use the same item that the sample project uses.
3.  Open your presentation. On slide 1, type your name. On slide 2,
complete the table with prize information.
Project Writing
1.  Complete Lesson Checkpoint: Expected Value, an online, ungraded
assessment. You’ll practice finding and interpreting expected value for a game
of chance—a skill essential to completing your project. Reach out to your
teacher with any questions you have after taking this assessment.
2.  Design a draft of your first spinner game (the game with the
less expensive prize.)
·
Determine how many sectors
the spinner will have. Choose between 2 and 12 sectors.
·
Choose the cost to play the
game.
·
Tip: You want to make your game appealing to the public. You can
achieve this with a greater probability of winning and/or a lower cost to play.

3.  Open your spreadsheet and use the first table in the Expected Values tab to find the expected value of your
game. (Tabs are located at the bottom of the page.)
·
Use the value of your prize
to determine the outcomes. Type the outcomes in cells B5 and C5.
·
Use the number of sectors
in your spinner to determine the probabilities. Type the probabilities in cells
B6 and C6.
·
The expected value will
appear below the table in cell B8. Note: There
is a formula in cell B8. Do not type over it.
4.  Think about the expected value shown. Remember, the goal is to
make a profitable game that people will want to play. Modify your game until
you think you have achieved this. You can change the cost to play the game
and/or you can change the number of sectors in your spinner. If you choose to
change your prize, be sure to update the information on slide 2 in your
presentation. Update the values in cells B5, B6, C5, and C6 as needed. Save your work when you are finished.
5.  Open your presentation. On slide 3, type in the information about
6.  Design a draft of your second spinner game (the game with the
more expensive prize.)
·
Determine how many sectors
the spinner will have. Choose between 2 and 12 sectors.
·
Choose the cost to play the
game.
·
Tip: This game has a greater prize value than your first game. To
make the game profitable, you may need to charge more to play the game, and/or
decrease the probability of winning.
7.  Open your spreadsheet and use the second table in the Expected Values tab to find the
·
Use the value of your prize
to determine the outcomes. Type the outcomes in cells B13 and C13.
·
Use the number of sectors
in your spinner to determine the probabilities. Type the probabilities in cells
B14 and C14.
·
The expected value will appear
below the table in cell B16. Note: There is a
formula in cell B16. Do not type over it.
8.  Think about the expected value shown. Remember, the goal is to
make a profitable game that people will want to play. Modify your game until
you think you have achieved this. You can change the cost to play the game
and/or you can change the number of sectors in your spinner. If you choose to
change your prize, be sure to update the information on slide 2 in your
presentation. Update the values in cells B13, B14, C13, and C14 as needed. Save your work when you are finished.
9.  Open your presentation. On slide 4, type in the information
10.  Complete slide 5 by predicting how much money the game owner,
will make after each spinner game is played 100, 500, and 1000 times.
11.  Go to the online spinner. Set it up to model your first spinner.
·
Click Change Spinner. Click
the up and down arrows to adjust the number of sectors to equal the number of
sectors in your first spinner. When finished, click Apply.
12.  Simulate playing the game 100, 500, and 1000 times.
·
Predict where the arrow
will land. Consider this sector the winning sector for all plays.
·
With the Number of Spins
set to 100, click Go. Find the number of times, f, the arrow landed in your chosen sector. Record this as the
number of wins in the table below. Calculate the number of losses by subtracting
the number of wins from 100.
·
Change the number of spins
to 400 and click Go. There are now a total of 500 spins. Find the number of
times, f, the arrow landed in your
chosen sector. Record this as the number of wins in the table below. Calculate the
number of losses by subtracting the number of wins from 500.
·
Change the number of spins
to 500 and click Go. There are now a total of 1000 spins. Find the number of
times, f, the arrow landed in your
chosen sector. Record this as the number of wins in the table below. Calculate
the number of losses by subtracting the number of wins from 1000.

Number of plays

Win

Lose

100

500

1000

13.  Open your spreadsheet and use the first row of tables in the Simulations tab to calculate the game owner’s profits
from the simulation. Notice that cells for the outcomes are already populated.
·
Use the values from your
table in Step 12 for the frequencies of the outcomes. Type the frequencies in
cells B6 and C6, F6 and G6, and J6 and K6.
·
The profits for the
simulations will appear below the tables in cells B8, F8, and J8. Note: There are formulas in cells B8, F8, and J8. Do
not type over them.
14.  Open your presentation. On slide 6, type in the information
15.  Go back to the online spinner. Set it up to model your second
spinner.
·
Click Change Spinner. Click the up and down arrows to adjust
the number of sectors to equal the number of sectors in your first spinner.
When finished, click Apply.
16.  Simulate playing the game 100, 500, and 1000 times.
·
Predict where the arrow
will land. Consider this sector the winning sector for all plays.
·
With the Number of Spins
set to 100, click Go. Find the number of times, f, the arrow landed in your
chosen sector. Record this as the number of wins in the table below. Calculate
the number of losses by subtracting the number of wins from 100.
·
Change the number of spins
to 400 and click Go. There are now a total of 500 spins. Find the number of
times, f, the arrow landed in your chosen sector. Record this as the number of
wins in the table below. Calculate the number of losses by subtracting the
number of wins from 500.
·
Change the number of spins
to 500 and click Go. There are now a total of 1000 spins. Find the number of
times, f, the arrow landed in your chosen sector. Record this as the number of
wins in the table below. Calculate the number of losses by subtracting the
number of wins from 1000.

Number of plays

Win

Lose

100

500

1000

17.  Open your spreadsheet and use the second row of tables in the Simulations tab to calculate the game owner’s profits
from the simulation. Notice that cells for the outcomes are already populated.
·
Use the values from your
table in Step 16 for the frequencies of the outcomes. Type the frequencies in
cells B14 and C14, F14 and G14, and J14 and K14.
·
The profits for the
simulations will appear below the tables in cells B16, F16, and J16. Note: There are formulas in cells B16, F16, and J16.
Do not type over them.
18.  Open your presentation. On slide 7, type in the information
Project Reflection
1.  Join the online session set up for you and your classmates.
2.  Write a response to any of the following.
·
Would you be more likely to
play a game with a greater chance of winning a small prize amount or a game
with a lesser chance of winning a large prize amount? Explain.
·
Knowing that games of
chance are designed for the customers to lose, on average, would you save your
money and never play these types of games? Why or why not?
·
consumers buying short-term warranties for brand-new items? Would you be likely
3.  Comment on at least one other student’s post.
Alternate Reflection Assignment
If your teacher excuses you from the online discussion, then
add a slide (slide 8) to the end of your presentation. On the slide, explain
whether, if given the opportunity in the real world, you would play either of
the two games you created, or if you would save your money and walk away. Also,
if you had to choose between the two games, which one would you play and why?
Submission
·
Prize information for each
spinner game
·
Facts, probability distribution
tables, expected values, and expected value explanations for each spinner game
·
Predicted profits after
100, 500, and 1000 plays of each game
·
Simulation results
(frequency distribution tables and profits) for each game with accompanying
discussion
·
Alternate reflection
assignment slide (ONLY if you did not participate in the online discussion)
by
The Prizes
Spinner 1 Prize
Name
Picture
Cost
Spinner 2 Prize
Spinner 1 Game
Game Facts
Prize value
Cost to play game
Number of sectors in spinner
Probability Distribution Table
Event
Win
Lose
Outcome (X)
Probability P(X)
E(X)
The expected value means that, on average:
• Players will lose [amount] per game.
• The game owner will make [amount] per game.
Spinner 2 Game
Game Facts
Prize value
Cost to play game
Number of sectors in spinner
Probability Distribution Table
Event
Win
Lose
Outcome (X)
Probability P(X)
E(X)
The expected value means that, on average:
• Players will lose [amount] per game.
• The game owner will make [amount] per game.
Predictions
My first game has an expected value of [expected value].
Therefore, I predict that the owner will make a profit of
• about [profit] after 100 plays
• about [profit] after 500 plays
• about [profit] after 1000 plays
My second game has an expected value of [expected value].
Therefore, I predict that the owner will make a profit of
• about [profit] after 100 plays
• about [profit] after 500 plays
• about [profit] after 1000 plays
Simulation Results: Game 1
100 plays
Event
Win
500 plays
Lose
Event
Win
1000 plays
Lose
Event
Win
Outcome (X)
Outcome (X)
Outcome (X)
Frequency f
Frequency f
Frequency f
Profit
[Discussion]
Profit
Profit
Lose
Simulation Results: Game 2
100 plays
Event
Win
500 plays
Lose
Event
Win
1000 plays
Lose
Event
Win
Outcome (X)
Outcome (X)
Outcome (X)
Frequency f
Frequency f
Frequency f
Profit
[Discussion]
Profit
Profit
Lose
Spinner Games
by
S. Student
The Prizes
Spinner 1 Prize
Name
Spinner 2 Prize
Goldfish
\$3.52
\$12.99
Picture
Cost
Spinner 1 Game
E(X)
–\$0.41
The expected value means that, on average:
• Players will lose \$0.41 per game.
• The game owner will make \$0.41 per game.
purposely hidden.
Spinner 2 Game
The expected value means that, on average:
• Players will lose \$0.70 per game.
• The game owner will make \$0.70 per game.
purposely hidden.
Predictions
My first game has an expected value of –\$0.41. Therefore, I
predict that the owner will make a profit of
• about \$41 after 100 plays
• about \$205 after 500 plays
• about \$410 after 1000 plays
My second game has an expected value of –\$0.70. Therefore, I
predict that the owner will make a profit of
• about \$70 after 100 plays
• about \$350 after 500 plays
• about \$700 after 1000 plays
Simulation Results: Game 1
purposely hidden.
The profits for 500 and 1000 plays are very close to the predictions. The best
prediction was for 1000 spins, which was only about \$2 off. This is not surprising
because experimental probabilities get closer to the theoretical probabilities as
the number of trials increases. The profit for 500 plays was only about \$6 off,
which is only about 3% off. The worst prediction was for 100 plays, which is not
surprising because it is the smallest number of plays. There were three more
wins than expected, which made the profit about 20% less than the predicted
value.
Simulation Results: Game 2
purposely hidden.
These profits are somewhat close to what was predicted, with the predictions being
the closest to the actual profit amounts for the greatest number of plays. This profit
was only about \$27, or about 4%, more than expected for 1000 plays. The profit for
500 plays was almost \$40 more than expected, because it had three fewer wins than
expected. This may not sound like much, but each of those losses saved the owner
from giving out three prizes, each worth more than \$10. The profit after 100 plays
was about \$13 off from what was expected. Interestingly, the number of wins was
only one more than expected, but with a high-value prize, the profit loss is more
noticeable after a smaller number of plays.
Expected Values
Spinner 1
Win
Event
Outcome (X )
Probability P (X )
E(X) =
\$0.00
Spinner 2
Win
Event
Outcome (X )
Probability P (X )
E(X) =
Lose
\$0.00
Lose
Simulations
Spinner 1: 100 plays
Event
Win
Lose
Outcome (X )
\$0.00
\$0.00
Frequency f
Profit:
\$0.00
Spinner 2: 100 plays
Event
Win
Lose
Outcome (X )
\$0.00
\$0.00
Frequency f
Profit:
\$0.00
Spinner 1: 500 plays
Event
Win
Lose
Outcome (X )
\$0.00
\$0.00
Frequency f
Profit:
\$0.00
Spinner 2: 500 plays
Event
Win
Lose
Outcome (X )
\$0.00
\$0.00
Frequency f
Profit:
\$0.00
Spinner 1: 1000 plays
Event
Outcome (X )
Frequency f
Profit:
Spinner 2: 1000 plays
Event
Outcome (X )
Frequency f
Profit:
Spinner 1: 1000 plays
Win
Lose
\$0.00
\$0.00
\$0.00
Spinner 2: 1000 plays
Win
Lose
\$0.00
\$0.00
\$0.00
The expected value means that, on average:
• Players will lose \$2.012 per game.
• The game owner will make \$2.012 per game.

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