Extra Review DayDay 10 Spring 2018

Warm-UpA certain kind of bug lives only four weeks. 70% of the bugs survive from the first week of life to the second. 65% of those who make it to the second week also survive into the third. 50 % survive to the fourth week. No bugs live beyond four weeks. On average, 12 newborn bugs are produced by these bugs who make it to the fourth week. The population of bugs that you receive in the lab has 16 one week old, 10 two week old, 9 three week old, and 4 four week old bugs. (Hint: Create a table from the info.)

1. Construct a Leslie Matrix for the life cycle of this bug.

2. What is the initial distribution matrix for this scenario?

3. What is the new population distribution after 8 cycles?

4. What is the total bug population after 14 cycles? 2

Warm-Up ANSWERSA certain kind of bug lives only four weeks. 70% of the bugs survive from the first week of life to the second. 65% of those who make it to the second week also survive into the third. 50 % survive to the fourth week. No bugs live beyond four weeks. On average, 12 newborn bugs are produced by these bugs who make it to the fourth week. The population of bugs that you receive in the lab has 16 one week old, 10 two week old, 9 three week old, and 4 four week old bugs. (Hint: Create a table from the info.)

1. Construct a Leslie Matrix for the life cycle of this bug.

0 0.70 0 0

0 0 0.65 0

0 0 0 0.50

12 0 0 0

L

3

Weeks Birth Rate Survival Rate

1 0 .70

2 0 .65

3 0 .50

4 12 0

Warm-Up ANSWERSThe population of bugs that you receive in the lab has 16 one week old, 10 two week old, 9 three week old, and 4 four week old bugs.

2. What is the initial distribution matrix for this scenario?

3. What is the new population distribution after 8 cycles?

4. What is the total bug population after 14 cycles?

0 0.70 0 0

0 0 0.65 0

0 0 0 0.50

12 0 0 0

L

4

0 16 10 9 4P

1996.59 bugs

8

8 0 119.25 74.53 67.08 29.81P P L

Questions fromLast Night’s Homework

Packet p. 12AND Complete

Quiz Corrections(on NEW paper, show work with matrices and

formulas to complete the problems you missed)

Tonight’s Homework

Test Review SheetAND

Complete at least 2 Parts of today’s Review Activity

(if not finished in class today)

Unit 2 Matrix Test Topics

1. Strictly Determined Games: Find maximin, minimax and saddle points - be able to interpret each based on the context.

2. Non-Strictly Determined Games: Find payoff matrix, best strategy for row and column player, and expected value for row player.

3. Markov Chains: Create transition matrix and initial-state matrix and interpret values after a certain number of cycles.

4. Leslie Matrix: Create Leslie matrix to find population distribution, total population, long term growth rate, and time when a maximum population or future minimum population is reached.

5. Matrix Operations: Perform calculations and interpret properly, especially Matrix multiplication and Scalar multiplication.

6. Matrix Applications: Be able to use matrix operations to solve word problem applications and systems of equations.

Review ActivityShow your Work using matrices and/or

proper notation to receive credit.

Matrix Applications, Leslie Matrices, Markov Chains, and Game Theory Unit

Review Part ASuppose that Sol and Tina change their game. If Sol displays heads and Tina shows tails, Sol wins 3 cents. If Sol shows tails and Tina displays heads, then Sol wins 2 cents. If they both show heads, then Tina wins 1 cent. If they both show tails, then Tina wins 4 cents.

1. What is the payoff matrix for this game with Sol as the row player?

2. Is the game strictly determined? Explain.

3. Use the row matrix to find Sol’s best strategy for this game.

4. Use the column matrix to find Tina’s best strategy for this game.

5. What is Sol’s expected payoff matrix equation for this game?

6. Solve problem 5 and interpret the results.

1p p

1

q

q

Review Part B

2. Use the concept of dominance to solve each of the following games. Give the best row and column strategies and the saddle point of each game.

d. e. f.

1. Each of the following matrices represents a payoff matrix for a game. Determine if the game is strictly determined or not. If it is, find the best strategies for the row and column players, and the saddle point of the game.

Review Part CThere are 30 students in the Math Club and every week they bring snacks to their meeting. This week 13 brought chips, 9 brought drinks and 8 brought dessert. 16% of those who brought chips to the first meeting brought chips again and 38% brought drinks. Of those that brought drinks, 30% brought drinks again and the rest brought dessert to the next meeting. And of those that brought dessert to the first meeting, 24% brought dessert again and 40% brought chips.

1. What is the initial distribution matrix for the math club?

2. What is the transition matrix for the student council?

3. Approximately how many students will bring desserts to the 4th meeting??

4. In the long run, how many of these students will bring each item to a meeting?

Review Part D: Matrix Applications PracticeA local designer clothes manufacturer tries to keep up with her sales in Target, Old Navy, and Kohl’s. The inventories of her most popular items in all three storesare recorded in the table.

T-shirts Jackets CardigansTarget 15 20 23

Old Navy 18 23 21Kohl’s 17 26 19

Label your rows and columns in your work and your answer.1. The designer makes the T-shirts for $10, the jackets for $13, and the cardigans for $17. Use matrices to calculate the manufacturer’s cost of making these popular items for each store. Call the resulting cost matrix C.

2. The designer sells the T-shirts to the stores for $13, the jackets for $18, and the cardigans for $16. Use matrices to calculates the income, I, that the designer makes from selling these popular items to the stores.

3. Use matrices to calculate the profit that the designer makes on her sales to each store.

Review Part ASuppose that Sol and Tina change their game. If Sol displays heads and Tina shows tails, Sol wins 3 cents. If Sol shows tails and Tina displays heads, then Sol wins 2 cents. If they both show heads, then Tina wins 1 cent. If they both show tails, then Tina wins 4 cents.

1. What is the payoff matrix for this game with Sol as the row player?

2. Is the game strictly determined? Explain.

3. Use the row matrix to find Sol’s best strategy for this game.

4. Use the column matrix to find Tina’s best strategy for this game.

5. What is Sol’s expected payoff matrix equation for this game?

6. Solve problem 5 and interpret the results.

1p p

1

q

q

Review Part A ANSWERSSuppose that Sol and Tina change their game. If Sol displays heads and Tina shows tails, Sol wins 3 cents. If Sol shows tails and Tina displays heads, then Sol wins 2 cents. If they both show heads, then Tina wins 1 cent. If they both show tails, then Tina wins 4 cents.

1. What is the payoff matrix for this game with Sol as the row player?

1 3

2 4

H T

H

T

Review Part A Answers

3. Use the row matrix to find Sol’s best strategy for this game.

4. Use the column matrix to find Tina’s best strategy for this game.

1p p

1

q

q

Sol should play heads 6 of the 10 times and tails 4 of the 10 times.

Tina should play heads 7 out of 10 times and tails 3 out of 10 times.

Suppose that Sol and Tina change their game so that the payoffs to Sol are

2. Is the game strictly determined? Explain.

No. The game is NOT strictly determine because maximin does not equal minimax (there is not a saddle point).

1 3

2 4

H T

H

T

Review Part A Answers

1 3 .7

.6 .42 4 .3

.2

Sol should expect to win 2 cents every 10 times.

(or 0.2 cents each game.

Suppose that Sol and Tina change their game so that the payoffs to Sol are

5. What is Sol’s expected payoff matrix equation for this game?

6. Solve part d and interpret the results.

1 3 .7

.6 .42 4 .3

'Sol s Expected Payoff

1 3

2 4

H T

H

T

Review Part B Answers

Not a Strictly Determined Game because maximin does not equal minimax.

Saddle Point at 4.Best strategies are row 3 and column 3.

Saddle Point at 0.Best strategies are row 1 and column 1.

0-2

3 1 2

-6-44

6 8 4

0-3-4

0 3 2

1. Each of the following matrices represents a payoff matrix for a game. Determine if the game is strictly determined or not. If it is, find the best strategies for the row and column players, and the saddle point of the game.

2. Use the concept of dominance to solve each of the following games. Give the best row and column strategies and the saddle point of each game.

1031

4 3 7

-20-22

4 2 5

Saddle point at 2.Best strategy for the Row Player is option D, and for the Column Player is option F.

Saddle Point at 3.Best strategy for the Row Player is option C, and for the Column Player is option F.

Review Part B Answers (cont.)