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

·

Spreadsheet

Template: PRM_A_06_spreadsheet_template.xls

·

Spinner

Tool: Access the Spinner Tool though Student

Resources: Links.

Template

1. Download the project

template and rename the file as GamePresentation_YourName.

This file will become your presentation.

2. Download the spreadsheet template and rename the file as Spreadsheet_YourName. You will use this file to help

you create 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

your first spinner game.

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

expected value of your game.

·

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

about your second spinner game.

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

about your simulations and compare them to your predictions.

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

about your simulations and compare them to your predictions.

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?

·

What do you think about

consumers buying short-term warranties for brand-new items? Would you be likely

to buy one? Explain.

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

Confirm that your presentation contains all your work:

·

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)

Submit your project to your teacher.I ADDED TWO REAL EXAMPLES YOU CAN JUST PARAPHRASE THEMprm_a_06_project_template.pptprm_a_06_sample_project.pptprm_a_06_spreadhseet_template.xlsproject_3_n.pptxproject3.pptxSpinner Games

by

[Your Name]

The Prizes

Spinner 1 Prize

Name

Picture

Cost

Link

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

Ladies Retro Oversized Sunglasses

$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.

Some values on this page are

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.

Some values on this page are

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

Some values on this page are

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

Some values on this page are

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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