a) Compute the indicated quantity.

P(A | B) = .1, P(B) = .4. Find P(A ∩ B).

P(A ∩ B) =

b)Compute the indicated quantity.

P(A) = .1, P(B) = .2. A and B are independent. Find P(A ∩ B).

P(A ∩ B) =

c)Find the conditional probability of the indicated event when two fair dice (one red and one green) are rolled. HINT [See Example 1.]

The red one is 1, given that the sum is 7.

Problem

You have developed an application for the task of registering crop information and have now been given crop data for analysis. The crop data includes the (x, y) coordinates of a robotic weed scanner (or unmanned ground vehicle (UGV) ) as it moves through a crop. Your task is to develop a console based representation of the movement patterns of the robotic scanner. Along side coordinates, the robot has also reported weed classification results for each (x, y) coordinate. The weed identification algorithms is using a lot of resources and for this reason graphics will be avoided in place of a console representation.

The robot is programmed to move on a grid and has an algorithm that converts GPS coordinates into a grid based system. In this scenario the robot will start moving at location (0, 0) and can move 1 space at a time in the x and y dimensions. That is, it can only increment x and/or y by 1 at each step. You will not need to implement this part, a text based data file will be provided to you. Location (0, 0) will be set to the top left hand corner and the grid will be set to a fixed width and height of 10 x 10 units. The text file will contain coordinates on each line with a 3 integer classification result (x, y, R1, R2, R3).  The result will require processing through a single artificial neuron based classifier, the neuron will contain 3 weights to match the 3 integers of the result.

This calculation through the neuron can be expressed:

Class = Round(R₁ X W₁ + R₂ X W₂ + R₃ X W₃)

The calculation should produce a result between 0 and 1, rounding will then produce 1 when the number is greater than 0.5 and 0 if it is below, for this you can use pythons inbuilt round function. Class 1 in this case will be a weed and class 0 will be no weed detected. The round function is being used in place of a step based activation function, typically depicted as g(.).

To complete this task, you will need to develop three functions that process and display the results in a console based grid. The first function will take a file name and return a 10x10 list of lists containing and representing the values from each line of the text file (x, y, R₁, R₂, R₃). This function will be called read_file (described below) and will take 1 parameter (the file name). The result values (R₁, R₂, R₃) will need to be stored as a list as well, this creates what is essentially a list of lists of lists. While this does sound complex, it can be created by appending the list results at each 2 dimensional index of a 2 dimensional list, which can be initialised as an empty list for each potential location in a 10x10 grid. The index will be the x and y values, that is, the first 2 values from each line of the text file.

Example, for a single set of values (1, 1, 4, 9, 1):

coord_results[x][y] = [R1, R2, R3]

or

coord_results[1][1] = [4,9,1]

Where coord_results is a 10x10 list of lists (or 2 dimensional list).

The second function will be called classify and will take the 3 result values and multiply them by their associated weight value. Using 3 iterations of a loop the values can be added together and rounded to produce a classification result. This function will take two parameters, a 3 float weight list and a 3 integer result. After the calculation the rounded result can be returned as an integer.

The third function will be called display_results and this function will display the text based grid in the console. A text data file and the expected output is depicted below. The function will print 1 line of the grid at a time in a loop and will require a nested loop to loop through all possible locations in the grid (all x and all y). Inside the nested loops you will need to check a set of conditions. If the index (x,y) has an empty list (meaning the robotic scanner did not pass over it) two spaces will be drawn. If the classify function returns 1 on the result for the coordinate an ' x' will be drawn, otherwise a full stop will be drawn (' .').

The three weight values that you will need to store in your application are provided below, these weights have been determined using a machine learning method to produce accurate classification.

An example data file is provided below with results displayed.

Note that program specifications are not always clear. If you are uncertain about any aspect, you are typically better off asking than making assumptions. Please use the appropriate discussion forum to ask for clarification, if required.

Example Interactions

Given the following list of weights:

0.03
0.04
0.03

Given the following text file (attached below - crop1.dat):

0 0 5 4 7
1 1 4 9 1
1 2 3 2 7
1 3 6 4 6
2 4 8 2 1
3 5 3 7 2 
4 6 5 7 1 
4 7 6 2 1
5 8 7 2 7
6 9 9 8 6
7 9 1 1 1
8 9 6 2 4
9 9 1 2 8
9 8 6 8 6
8 7 4 5 0
7 6 5 8 6
7 5 8 2 1
7 4 3 2 5
8 3 8 1 3
9 2 1 7 1

Your display output should display the following output.

 x                   
   x . x             
         .           
           .         
             . .     
                 .   
                   x 
         . . x     . 
       .       .   . 
     .           x . 

(Scenario analysis) Family Security is considering introducing tiny GPS trackers that can be inserted in the sole of a​ child's shoe, which would then allow for the tracking of that child if he or she was ever lost or abducted. The​ estimates, that might be off by 8 percent​ (either above or​ below), associated with this new product are shown ​ here:  

Unit price:   $121
Variable costs:   $73
Fixed costs:   $255,000 per year
Expected sales:   10,600 per year

Since this is a new product​ line, you are not confident in your estimates and would like to know how well you will fare if your estimates on the items listed above are 8 percent higher or 8 percent lower than expected. Assume that this new product line will require an initial outlay of ​$1.03 ​million, with no working capital​ investment, and will last for 10 ​years, being depreciated down to zero using​ straight-line depreciation. In​ addition, the​ firm's required rate of return or cost of capital is 10.5 ​percent, and the​ firm's marginal tax rate is 34 percent. Calculate the​ project's NPV under the​ "best-case scenario"​ (that is, use the high estimates- unit price 8 percent above​ expected, variable costs 8 percent less than​ expected, fixed costs 8 percent less than​ expected, and expected sales 8 percent more than​ expected). Calculate the​ project's NPV under the​ "worst-case scenario."

The NPV for the​ best-case scenario will be ​$

The NPV for the​ worst-case scenario will be ​$

SmartAuto Manufacturing is engaged in the production of replacement parts for automobiles. One plant specializes in the production of two parts: Part #127 and Part #234. Part #127 produced the highest volume of activity, and for many years it was the only part produced by the plant. Five years ago, Part #234 was added. Part #234 was more difficult to manufacture and required special tooling and setups. Profits increased for the first three years after the addition of the new product. In the last two years, however, the plant faced intense competition, and its sales of Part #127 dropped. In fact, the plant showed a small loss in the most recent reporting period.

Much of the competition was from foreign sources, and the plant manager was convinced that the foreign producers were guilty of selling the part below the cost of producing it. The following conversation between Patricia Wang, plant manager, and James Tin, divisional marketing manager, reflects the concerns of the division about the future of the plant and its products.

JAMES:           You know, Patricia, the divisional manager is real concerned about the plant's trend. He indicated that in this budgetary environment, we can't afford to carry plants that don't show a profit. We shut one down just last month because it couldn't handle the competition.

PATRICIA:      James, you and I both know that Part #127 has a reputation for quality and value. It has been a mainstay for years. I don't understand what's happening.

JAMES:           I just received a call from one of our major customers concerning Part #127. He said that a sales representative from another firm offered the part at $20 per unit – $11 less than what we charge. It's hard to compete with a price like that. Perhaps the plant is simply obsolete.

PATRICIA:      No. I don't buy that. From my sources, I know we have good technology. We are efficient.

And it's costing a little more than $21 to produce that part. I don't see how these companies can afford to sell it so cheaply. I'm not convinced that we should meet the price. Perhaps a better strategy is to emphasize producing and selling more of Part #234. Our margin is high on this product, and we have virtually no competition for it.

JAMES:           You may be right. I think we can increase the price significantly and not lose business. I called a few customers to see how they would react to a 25 percent increase in price, and they all said that they would still purchase the same quantity as before.

PATRICIA:      It sounds promising. However, before we make a major commitment to Part #234, I think we had better explore other possible explanations. I want to know how our production costs compare to those of our competitors. Perhaps we could be more efficient and find a way to earn our normal return on Part #127. The market is so much bigger for this part. I'm not sure we can survive with only Part #234. Besides, my production people hate that part. It's very difficult to produce.

After her meeting with James, Patricia requested an investigation of the production costs and comparative efficiency. She received approval to hire a consulting group to make an independent investigation. After a three-month assessment, the consulting group provided the following information on the plant's production activities and costs associated with the two products:

* Calculated using a plantwide rate based on direct labor hours. This is the current way of assigning the plant's overhead to its products.

The consulting group recommended switching the overhead assignment to an activity-based approach. It maintained that activity-based cost assignment is more accurate and will provide better information for decision making. To facilitate this recommendation, it grouped the plant's activities into homogeneous sets with the following costs:

Please formulate and solve each of the following problems. For each problem, you should include the final SOLVER printout (either your final spreadsheet or an answer report), as well as (1) clear and precise definitions for all decision variable; (2) your objective function indicating whether it is to be maximized and minimized; (3) all constraints, including non-negativity and integrality (if necessary); and (4) what the optimal decision is (in words) and what outcome will be produced.

1. A manufacturer of stereos has plants in Atlanta, New Haven and Dallas, and distributions centers in San Francisco, Boston, Washington, D.C., and Cleveland. The tables show weekly production capacities, demand requirements, and unit transportation costs (in dollars).

                                                                                Cleveland                                       65

The goal is to minimize total transportation costs.

3. A company has five jobs, each of which must be assigned to a single machine. The table shows the dollar costs for each possible job-machine assignment:

JOB                                                    MACHINE

                                    A                     B                     C                     D                     E

1                                  138                  127                  118                  121                  143

2                                  157                  138                  129                  132                  160

3                                  143                  129                  131                  130                  172

4                                  111                  119                  123                  107                  120

5                                  102                  120                  100                  119                  100     

Find the set of assignments with the lowest possible total cost.

A firm is evaluating two projects for this year’s capital budget. Its WACC is 14%. Project A costs $6,000 and its expected cash inflows would be $2,000 per year for 5 years. Project B costs $18,000 and its expected cash inflows would be $5,600 per year for 5 years. Calculate NPV and IRR for each project.

Refer to problem 7. Calculate the MIRR and payback for each project.

Refer to Problem 7. Calculate the discounted payback for each project.

Refer to Problem 7. If the projects were mutually exclusive, which one would you recommend? If the projects were independent, which one(s) would you recommend? Explain.

A firm is evaluating two projects for this year’s capital budget. Its WACC is 14%. Project A costs $6,000 and its expected cash inflows would be $2,000 per year for 5 years. Project B costs $18,000 and its expected cash inflows would be $5,600 per year for 5 years. Calculate NPV and IRR for each project.

Refer to problem 7. Calculate the MIRR and payback for each project.

Refer to Problem 7. Calculate the discounted payback for each project.

Refer to Problem 7. If the projects were mutually exclusive, which one would you recommend? If the projects were independent, which one(s) would you recommend? Explain.

A manufacturer produces both a deluxe and a standard model of automatic sander designed for home use. Selling prices obtained from a sample of retail outlets follow.

What is the 95% confidence interval for the difference between the mean prices of the two models (to 2 decimals)?

(X , X)

A stock's returns have the following distribution: Demand for the Company's Products Probability of This Demand Occurring Rate of Return If This Demand Occurs Weak 0.2 (22%) Below average 0.1 (7) Average 0.5 15 Above average 0.1 40 Strong 0.1 68 Assume the risk-free rate is 3%. Calculate the stock's expected return, standard deviation, coefficient of variation, and Sharpe ratio. Do not round intermediate calculations. Round your answers to two decimal places. Stock's expected return: Standard deviation: Coefficient of variation: Sharpe ratio:

Consider the following two​ projects: Project Year 0 ​C/F Year 1 ​C/F Year 2 ​C/F Year 3 ​C/F Year 4 ​C/F Year 5 ​C/F Year 6 ​C/F Year 7 ​C/F Discount Rate Alpha minus79 20 25 30 35 40 ​N/A ​N/A 16​% Beta minus80 25 25 25 25 25 25 25 17​% The net present value​ (NPV) for project beta is closest​ to: