Test Probability and Quantitative Reasoning

Name ________________________________

Math 131

Spring 04

Final Exam

You may use one sheet of notes (double sided) and manipulatives, computers, calculators, etc. For computer questions, you may submit a disk with your test. There are four questions, choose three to complete. Indicate clearly which questions you want graded.

1. A Probability Game

2,1,3,4

5,5,6,6

Player A’s spinner Player B’s Spinner

Player A and Player B play the following game:

Each player spins her spinner, shown above.

They multiply the two results.

If the answer is even, Player A wins; if the answer is odd, Player B wins.

For example, if Player A spins a 4 and player B spins a 5, then player A wins because is an even number.

a. List a sample space showing all possible combinations of A’s and B’s spins.

b. Are all the outcomes listed in your sample space equally likely? Explain.

c. What is the probability that Player A will win the game? That Player B will win the game? Explain.

d. Change the numbers on the spinners so as to make this game fair, i.e. select 4 natural numbers (perhaps with repeats) for each spinner, so that when the players spin the two spinners and multiply the results, the probability that the product is even is . Explain your reasoning.

e. Set up both the original game and your new game on fathom (using randomPick( ) and even( ) ). Try at least 100 cases of each, hitting CTRL Y several times to resample. Do these experimental results seem to match your theoretical results? Explain. (Turn in your fathom work on a disk; disks are available at the front of the room).

Scoring Guide:

Level 5 Answers all parts of the question correctly, sets up fathom correctly, and includes very thoughtful work in part e.

Level 4 Answers parts a-d correctly and does some good work on fathom.

Level 3 Answers to parts a-d are mostly correct or equivalent.

Level 2 Answers to parts a-c are mostly correct.

Level 1 Does some good work on the problem.

Score ________________

2. Some Percent Problems.

Below is data taken from the U.S. Census Bureau.

Countries Ranked by Population: 1954

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Rank Country                                  Population

--------------------------------------------------------

   1 China                                   594,724,844

   2 India                                   397,016,000

   3 United States                           163,026,000

   4 Russia                                  109,208,917

   5 Japan                                    88,753,892

   6 Indonesia                                88,521,458

   7 Germany                                  69,868,115

   8 Brazil                                   59,989,219

   9 United Kingdom                           50,765,000

  10 Bangladesh                               48,603,271

Countries Ranked by Population: 2004

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Rank Country                                  Population

--------------------------------------------------------

   1 China                                 1,298,847,624

   2 India                                 1,065,070,607

   3 United States                           293,027,571

   4 Indonesia                               238,452,952

   5 Brazil                                  184,101,109

   6 Pakistan                                159,196,336

   7 Russia                                  143,782,338

   8 Bangladesh                              141,340,476

   9 Nigeria                                 137,253,133

  10 Japan                                   127,333,002

a. Using only simple estimates, explain how you can tell that the U.S. population grew by less than 100% from 1954 to 2004.

b. Between China and India, explain how you can tell which country’s population grew by a larger percentage between 1954 and 2004, using only simple estimates.

c. Now calculate the percentage growth in the populations of China, India, and the U.S. between 1954 and 2004. Show your work.

d. Suppose the populations of China, India, and the U.S. all grew by the same percentage in the next 50 years that they did in the last 50 years. What would each of their populations be in 2054? Does anything surprise you about these results?

e. Also using data from the U.S. Census Bureau, in 1954 about 21.8% of all people in the world lived in China. What was the world population in that year? Show your work.

f. Make up one other percent problem using the population data and solve it.

Scoring Guide:

Level 5 Answers all parts of the question correctly, chooses a challenging problem in part f.

Level 4 Uses good reasoning in parts a and b, and calculations in the rest of the problem are correct, except for very minor mistakes..

Level 3 Answers three of c, d, e, and f correctly, reasoning on a and b is generally good.

Level 2 Answers over half the questions correctly.

Level 1 Does some good work on the problem.

Score ________________

3. Manipulative Problems:

1. Use the clues below to find the colors of four Cuisinaire Rods (numbered 1 to 4), such that the rods satisfy ALL the clues.

A) The rods are four different colors.

B) Rods 1, 2, and 3 together are the same length as Rod 4.

C) Rod 4 is 200% longer than Rod 1.

D) Rod 2 is longer than Rod 3.

a. Find one possible solution to the problem. Write the color of each rod in the designated space below. Explain how you found your solution and why it is correct. Note: if you cannot find a solution, explain what results you have found:

Rod 1: ___________ Rod 2: ___________ Rod 3: ___________ Rod 4: ___________

b. Find ALL possible solutions to the problem. A solution is a list of the 4 rods. Solutions are identical only when values for all rods are the same (i.e. if there is at least one rod that is different, you have a new solution, even if some rods are still the same).

c. How could you convince a skeptic that your list in part b is complete?

There are some tiles on the table. The tiles are red, yellow, blue, and green:

The number of blue tiles is 2/3 the number of red tiles.

There are 2 yellow tiles on the table.

40% of the tiles on the table are red.

There are fewer than 25 tiles on the table.

d. How many tiles of each color are on the table?

Red _____ Blue _____ Green ____ Yellow _____

e. Give an argument that could convince a skeptic that your answer is correct, and that it is the only possible solution.

f. Suppose you were allowed to use up to 50 tiles. What other solutions would there be to this problem? Give an argument that could convince a skeptic that you have found all additional solutions.

Scoring Guide:

Level 5 Answers all parts of the question correctly with excellent justification.

Level 4 Solves parts a and d correctly and one of parts b and d, with excellent justification, or equivalent.

Level 3 Solves parts a and d correctly with good justification or equivalent.

Level 2 Solves part a or part d correctly or equivalent.

Level 1 Does some good work on the problem.

Score ________________

4. Credit Card Problem

Jaekwon owes $5,000 in student loans, at 8% annual interest.

a. To three decimal places, what is the monthly interest rate on the student loan?

b. Jaekwon got a great new job, and he decided to pay $1,000 per month until his student loan was paid off. Write an amortization table below (or make one yourself on Excel and turn it in) that describes his month-to-month payments.

c. Explain how you calculated the values in each column.

d. How much interest would Jaekwon pay?

e. Suppose Jaekwon paid $500 per month instead of $1000 per month. Would it take him twice as long to pay off his loan? Would he pay twice as much interest? Explain how you can answer these questions without making a new amortization table.

f. Suppose Jaekwon paid $50 per month. How long would it take him to pay off the loan? Explain how you can answer this question without constructing a new amortization table.

Scoring Guide:

Level 5 Answers all parts of the question correctly with excellent justification.

Level 4 Answers all parts of the question correctly with mostly good justification or equivalent.

Level 3 Answers a-d correctly and shows some good thinking on e-f.

Level 2 Sets up the amortization table correctly or equivalent.

Level 1 Does some good work on the problem.

Score ________________

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