Question 1. Please indciate what probability ceoncept is involved in each of the following questions and then calcuate each probability (your answer must include formula to show you calcaute): 5 points for each, 45 points in total.

1) A manufacturer of computer disks has a historical defective rate of .003. What is the probability that in a batch of 1000 disks, no more than 3 would be defective? (round to 0.001)

2) After a severe winter, potholes develop in a state highway at the rate of 6.5 per mile. What is the probability of finding 10 potholes in the two miles of highway? (round to 0.001)

3) The weight of a 1 cubic yard bag of landscape mulch in uniformly distributed over the interval from 70.5 to 90.5 pounds. What is the probability that a bag will weigh less than 80 pounds?

4) The time it takes to travel from Canyon to Amarillo is normally distributed with mean = 20 minutes and standard deviation = 7 minutes. What is the probability the trip takes more than 24 minutes?

5) Assume that the time required to download a music from the Internet is exponentially distributed with mean equal to 3 minutes. What is the probability that a download will require at least 2 but not more than 4 minutes?

6) Global Airlines operates two types of jet planes: jumbo and ordinary. On jumbo jets, 35% of the passengers are on business while on ordinary jets 40% of the passengers are on business. Of Global's air fleet, 40% of its capacity is provided on jumbo jets. What is the probability a randomly chosen business customer flying with Global is on a jumbo jet?

7) A mail order company tracks the number of returns it receives each day. Information for the last 100 days shows in the right:

In this question, how many sample points are there?

What is the probability of 100-299 returns?

8) "A market study taken at a local sporting goods store showed that of 20 people questioned, 6 owned tents, 10 owned sleeping bags, 8 owned camping stoves, 4 owned both tents and camping stoves, and 4 owned both sleeping bags and camping stoves.

Let: Event A = owns a tent, Event B = owns a sleeping bag, Event C = owns a camping stove. "

Please find P(B∩C)
Please find P(B∩A)
Are the events B and C independent events? Why?

Question 2. An appliance dealer must decide how many (if any) new microwave ovens to order for next month. The ovens cost $250 and sell for $320. Because the oven company is coming out with a new product line in two months, any ovens not sold next month will have to be sold at the dealer's half price clearance sale. Additionally, the appliance dealer feels he suffers a loss of $25 for every oven demanded when he is out of stock. On the basis of past months' sales data, the dealer estimates the probabilities of monthly demand (D) for 0, 1, 2, or 3 ovens to be 0.2, 0.35, 0.25, and 0.2, respectively.

The dealer is considering conducting a telephone survey on the customers' attitudes towards microwave ovens. The results of the survey will either be favorable (F), unfavorable (U) or no opinion (N). The dealer's probability estimates for the survey results based on the number of units demanded are:

P(F|D = 0) = .1, P(F|D = 2)=0.3, P(U|D = 0) = 0.8, P(U|D = 2) = 0.1;
P(F|D = 1) = .2, P(F|D = 3)=0 9, P(U|D = 1) = 0.3, P(U|D = 3) =0.1.

Please answer the following questions and fill out the cells

a. What is the dealer's optimal decision without conducting the survey?
b. What is the EVPI?
c. Based on the survey results what is the optimal decision strategy for the dealer?
d. What is the maximum amount he should pay for this survey?
e. Is the the efficiency of the survey high or low? Try to explain why?

Question 3. Eastern Chemicals manufactures three chemicals: X, Y, and Z. The chemicals are produced via two production process: 1 and 2. Running process 1 for an hour costs $400 and yields 300 units of X, 100 units of Y, and 50 units of Z. Running process 2 for an hour costs $160 and yields 100 units of X and 100 units of Y. In addition, running processes 1 and 2 per hour will generate 8 and 6 units of wastes, respectively. In order to meet the this comany's environmental policy, the total units of wastes generated daily must not exceed 300 units. To meet customer demands, at least 9000 units of X, 3500 units of Y, and 1200 units of Z must be produced daily. Please follow the five-step instruction and use Solver to determine a daily production plan (that is how many hours to run processes 1 and 2 respectively) that minimize the cost of meeting the company's daily demands. Please make sure that your plan is feasible.

Question 4. Sports of All Sorts produces, distributes, and sells high-quality skateboards. Its supply chain consists of four factories (located in Detroit, Los Angeles, Louisville, and Austin) that produce skateboards. The Detroit and Louisville facilities can produce 600 skateboards per week, but the Austin and Los Angeles plants are larger and can produce up to 800 skateboards per week. Skateboards produced in these factories must be shipped to the three major US retailers: Just Sports, Sports 'N Stuff, and The Sports Dude. The weekly demands are 500 skateboards at Just Sports, 800 skateboards at Sports 'N Stuff, and 1000 skateboards at The Sports Dude. The following table displays the per-unit costs for shipping skateboards between the factories and the retailers.

1) Please draw the network representation for this problem, including nodes, arcs, and functions (cost, demand, capacity) assosiated with arcs and/or nodes in the right.

2) Build a linear programming model to minimize the total transportation cost and use Excel Solver to find the optimal solution.

Please follow the example in our class. You must list your decision variables, objective function, all the contraints.

3) "Sports of All Sorts plan to run a distribution center (DC) in one of the following locations: Iowa, Maryland, Arksansas, and Idaho. Under this plan, all the skateboards will be delivered to the distribution center and then shipped to retailers via the distribution center. The following tables display the estimated per-unit costs for shipping skateboards between the factories and DCs and for shipping between the DCs and the retailers. The only additional cost for running a distribution center is $ 10,000 per week. As a business analyst, please help Sports of All Sorts make a right decision:

a) Should they run a distribution center or not? Why? [Hint: which one is smaller, the minimum transportation cost of running a distributon center plus the running cost or the minimum transportation cost without a distribution center?

b) If your answer is yes, which location they should choose to run such a distribution center? Why?

You need to use another worksheet (Question 4b) to answer this question."

Question 4b. "Sports of All Sorts plan to run a distribution center (DC) in one of the following locations: Iowa, Maryland, Arksansas, and Idaho. Under this plan, all the skateboards will be delivered to the distribution center and then shipped to retailers via the distribution center. The following tables display the estimated per-unit costs for shipping skateboards between the factories and DCs and for shipping between the DCs and the retailers. The only additional cost for running a distribution center is $ 10,000 per week. As a business analyst, please help Sports of All Sorts make a right decision:

a) Should they run a distribution center or not? Why? [Hint: which one is smaller, the minimum transportation cost of running a distributon center plus the running cost or the minimum transportation cost without a distribution center?

b) If your answer is yes, which location they should choose to run such a distribution center? Why?

Attachment:- Assignment.xlsx