Analysis Sheet Game Overview There are 15 toolboxes, each containing a prize of a set value. As the contestant, you must begin the game by randomly selecting one toolbox. The value hidden in your chosen 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 during the game. NOTE: When the GC makes you an offer, she determines the amount of her 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 on the next page 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 One 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. Since boxes 3,7,11, and 14 are eliminated I must add up the value of all of the remaining boxes and divide that by the total number of tool boxes. So this would look like $100 + $750 + $1,000 + $5,000 + $5,000 + $10,000 + $25,000 + $50,000 + $100,000 + $300,000 + $500,000. This is equal to $991,850. Since there are 11 left we must now divide the value of the boxes which is $996,850 by 11. 996,850/ 11 = 90,622.72 So average value of the unopened toolboxes rounded to the nearest hundred would be $90,700. 2. What is the median value of the remaining toolboxes? Show your work. Explain why this number is important. To find the median you put all of the numbers in numerical order $100 + $750 + $1,000 + $5,000 + $5,000 + $10,000 + $25,000 + $50,000 + $100,000 + $300,000 + $500,000. Since there are odd number of results, the median is the middle number. Therefore, the median in this set of numbers is $10,000. It is important to know the median because it helps you know the middle number of your data set. 3. What is the mode of the values in the remaining toolboxes? The mode of the values in the remaining toolboxes is the number you see the most in which case the mode in this data set would be $5,000 because there are two cases that hold the value $5,000 remaining. 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. The GC offered me $7,000. I have a 6/11 chance that I pick a toolbox with a higher value than the amount the GC offered me. There is only a 5/11 chance that I will pick a case with a lesser value than the GC offered me. So if divide 6 by 11 I would get 0.54545454545 and I would multiply that by 100% which would leave me with 54.45%. So the percentage of probability that I will win a bigger prize is 54%. I don't believe this is a good offer because I little over half of a chance to win a bigger prize. You decide to keep playinghow often are you on a game show, right!? The GC says, \"Okay, let Round Two 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? There is only one tool box left with $500,000 prize and six more left with different prizes. So the probability that my tool box holds the $500,000 prize is 1/7. 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. A. The probability that toolbox 9 contains a prize greater than $10,000 is 4/7. B. The probability that tool box 9 contains a prize that has a 5 as one of the digits is 5/7. C. The probability that the value of toolbox 9 is greater than $10,000 and has a 5 as one of the digits is 3/7. D. The probability that the value of toolbox 9 is greater than 10,000 or has a 5 as one the digits is 6/7. At the end of Round Two, the GC offers you $20,000 to end the game. 7. Why does the GC offer you a larger prize than she did in the first round? Use math to explain. The GC offers me a larger prize than she did in the first round because the probability of me choosing a toolbox with a higher value is greater this round. Round 1: There was a 6/11 chance of me picking a larger prize than the GC offered which is a probability of 54%. Round 2: There is currently a 4/7 chance of me picking a larger prize than the GC offered me which is a probability of 57%. In Round 1: There was a 5/11 chance of me picking a smaller prize than the GC offered which is a probability of 45%. Round 2: There is currently a 3/7 chance of me picking a smaller prize than the GC offered which is a probability 43%. So overall, my chance of picking a prize of a smaller amount has decreased so the GC offer increased, as did the chance of me winning a prize of higher value. 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? Show your work. There are two boxes out of the seven remaining with values less than $10,000, so that's 2/7. So there is a 28.571429% probability that the next two boxes will be less than $10,000. Answer rounded is 29%. A = How are boxes are < $10,000 before you make the first pick? A=4 B = Population when you make first pick B=7 C = How many boxes are < $10,000 when you make the second pick? C=3 D = Population when you make second pick D=6 Probability that BOTH picks are < $10,000 = (A / B) x (C / D) =(4/7)x(3/6) 0.57142857142 x 0.5 =.028571428571 then I times that by 100% and I will get 28.571428571 On the second pick, one is eliminated so I only have six remaining. So, that's 1/6, which 16.67% probability. The GC exclaims, \"Third and final round! This is the BONUS CHALLENGE!\" She 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? Show your work. Since we know there are four colors, \"I'm assuming four different colors\