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# Home Free General Contractor Game Show

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Game Overview
● There are 15 toolboxes, each holding a prize of a set value.
● As the contestant, you must begin the game by randomly select one toolbox. The value hidden in this toolbox is what you will win if you choose to keep it throughout the game, but you will have multiple chances to trade it in for a guaranteed prize.
● The game show host, the General Contractor (GC), has two roles:
○ She narrates the game.
○ She offers you the options to accept prizes of a guaranteed value at set points in the game.
■ NOTE: When the GC makes you an offer, she determines the amount of the offer by considering the likelihood that you are holding a high value prize in your toolbox. If your chances of having a big prize in your toolbox are not very high, she will offer you a smaller prize value.
● The image below is what the game show will look like to you as the contestant. As you play the game, this screen image will adjust with the necessary changes.
● This first screen shows that you randomly selected Toolbox #9, but you don't know what prize it contains.

The GC exclaims: Let Round 1 begin! Contestant, open four toolboxes!
The screen below shows the toolboxes you open and the corresponding values you eliminate.

You opened and eliminated toolboxes 3, 7, 11, and 14. They held the following values:
\$500
\$5,000
\$25,000
\$75,000 (Bummer! This would have been enough to renovate a kitchen and bathroom!)

This is where your work begins:
1.) Knowing that you just eliminated four prizes, calculate the average value of all the unopened toolboxes. Show your work.
2.) What is the median value? Show your work. Why is this number important?

3.) What is the mode of the values of the remaining toolboxes?
The GC says: I will make you an offer of \$7,000 and you can walk away with that money right now. Put it toward a new home, a renovation, whatever! It's yours!
4.) Is this a good deal? Calculate the percentage of probability that you will win a bigger prize. Use math to explain your rationale.
You decide to keep playing—how often are you on a game show, right!?
The GC says: Okay, let Round 2 begin! Open four MORE toolboxes!!
After the second round, the screen looks like this:

You open toolboxes 1, 4, 10, and 13. The values you just eliminated include:
\$100 (Great, you ruled out the possibility of getting the lowest value home item!)
\$1,000
\$5,000
\$300,000 (Wow, that would have been enough to purchase a home.)

5.) Based on the values that remain, what is the probability that toolbox 9 holds the \$500,000 prize?
6.) You realize that you have eliminated over half of the prizes already! As you study the remaining prize values, you consider four different scenarios:
a. What is the probability that toolbox 9 contains a prize greater than \$10,000?
b. What is the probability that toolbox 9 contains a prize that has a 5 as one of the
digits?
c. What is the probability that the value of toolbox 9 is greater than \$10,000 AND has
a 5 as one of the digits? Show your work.
d. What is the probability that the value of toolbox 9 is greater than \$10,000 OR has a 5
as one of the digits? Show your work.
At the end of Round 2, the GC offers you \$20,000 to end the game.
7.) Why does the GC offer you a bigger prize than in the first round? Use math to explain.
8.) As you prepare to start the third and final round, you wonder: what is the probability that the next two boxes I open each have values of less than \$10,000?

The GC exclaims: Third and final round!! This is the BONUS CHALLENGE!
He wheels out a giant spinner with four colors on it and explains: If you spin two times in a row and land on the color red both times, you can take your toolbox AND automatically win another \$10,000. If you land on a different color, the value of the prize in your toolbox gets cut in half. You can either take this challenge or stop playing now and win whatever amount is in your toolbox.
9.) What is the probability that you spin and land on the color red two times in a row?
10.) What are the pros and cons of taking this bonus challenge? Would you prefer to take the challenge or call it quits and walk away with the full value of the toolbox 9? Why?
In the end, you decide to play it safe, decline the challenge offer, and keep toolbox 9, the box you originally chose. The value of that toolbox is \$100,000. Congratulations! You have won enough money for a large down payment on a house! Thank you for playing Home Free!

https://brainmass.com/statistics/probability/home-free-general-contractor-game-show-612101

#### Solution Preview

Game Overview
● There are 15 toolboxes, each holding a prize of a set value.
● As the contestant, you must begin the game by randomly select one toolbox. The value hidden in this toolbox is what you will win if you choose to keep it throughout the game, but you will have multiple chances to trade it in for a guaranteed prize.
● The game show host, the General Contractor (GC), has two roles:
○ She narrates the game.
○ She offers you the options to accept prizes of a guaranteed value at set points in the game.
■ NOTE: When the GC makes you an offer, she determines the amount of the offer by considering the likelihood that you are holding a high value prize in your toolbox. If your chances of having a big prize in your toolbox are not very high, she will offer you a smaller prize value.
● The image below is what the game show will look like to you as the contestant. As you play the game, this screen image will adjust with the necessary changes.
● This first screen shows that you randomly selected Toolbox #9, but you don't know what prize it contains.

The GC exclaims: Let Round 1 begin! Contestant, open four toolboxes!
The screen below shows the toolboxes you open and the corresponding values you eliminate.

You opened and eliminated toolboxes 3, 7, 11, and 14. They held the following values:
\$500
\$5,000
\$25,000
\$75,000 (Bummer! This would have been enough to renovate a kitchen and bathroom!)

This is where your work begins:
1.) Knowing that you just eliminated four prizes, calculate the average value of all the unopened toolboxes. Show your work.
Add all the dollar amounts in the unopened boxes, then divide by the number of unopened boxes.

Total of the dollar amounts in the remaining boxes
=\$100 + \$750 + \$1,000 + \$5,000 + \$5,000 + \$10,000 + \$25,000 + \$50,000 + \$100,000 + \$300,000 + 500,000
=\$996,850

Number of boxes remaining = 11
\$996,850/11=\$90,622.73

The average value of the unopened boxes is \$90,622.73.

2.) What is the median value? Show your work. Why is this number important?
There are 11 unopened boxes. List the amounts corresponding to these unopened boxes, from low to high.

\$100, \$750, \$1,000, \$5,000, \$5,000, \$10,000, \$25,000, \$50,000, \$100,000, \$300,000, \$500,000

Identify the middle number. Note that there are 5 numbers lower than the middle number, and 5 numbers higher than the middle number.

The median value is \$10,000. The median is important as this value is in the center of the data set.

3.) What is the mode of the values of the ...

#### Solution Summary

Explanations and step-by step computations are shown in a Word file.

\$2.19