Question

constraints, also written in terms of decision variables (some using both decision supplement. In addition, the Lecture notes page in this website gives you access Then you complete Task 1, you should have two decision variables, the ng constraints. We simply list all conditions that must objective function written in terms of those decision variables, and five 2. The second step is to find a solution. Task 2 tells you specifically to use Excel Solver to find this solution. The key here is to translate all mathematical expressions from Task 1 into Excel format. Instructions for doing so can be found under Topic 5 in the Excel booklet, as well as in the Linear Programming hot trying to solve the problem or HURSDAY variables, others just one of them). to Excel spreadsheets used to generate Excel output shown in lecture slides for Week 5. It may be worthwhile examining them before attempting Task 2. 3. The final step is interpreting the solution that has been found, which is Task 3. 4. The summary in Task 4 is a summary of the results from linear programming and sensitivity analysis in Tasks 2 and 3. Pat 3. A simple random sample of 220 university students were asked what pasta they usually order and with which sauce. The preferences of these respondents are summarised below: Total 60 Sauce Pasta Bolognese Carbonara Pesto Spaghetti 30 20 10 Gnocchi 40 30 25 Ravioli 22 25 25 Total 92 75 60 95 72 227

Answer 1

(b) Please note

Probability = Favorable Outcomes / Total Outcomes

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(i) P(Gnocchi)

Favorable Outcomes = 95

Total Outcomes = 227

The required probability = 95 / 227 = 0.4185

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(ii) P(Spaghetti and Bolognese)

Favorable Outcomes = Intersection value of Spaghetti and Bolognese = 30

Total Outcomes = 227

The required probability = 30 / 227 = 0.1322

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(iii) P(Ravioli or Carbonara sauce)

P(A or B) = P(A) + P(B) - P(A And B)

P(Ravioli) = Total of Ravioli / Overall Total = 72 / 227

P(Carbonara) = Total of Carbonara / Overall Total = 75 / 227

P(Ravioli and Carbonara) = Intersection of Ravioli and Carbonara / Overall Total = 25 / 227

The required probability = 72/227 + 75/227 - 25/227 = 122 / 227 = 0.5374

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(iv) P(Pasta is Ravioli, given that the sauce is Pesto)

By Bayes Theorem, P(A given B) = P(A and B) / P(B)

Therefore (Ravioli given Pesto) = P(Ravioli and Pesto) / P(Pesto)

P(Ravioli and Pesto) = 25 / 227

P(Pesto) = 60 / 227

Therefore the required probability = (25 / 227) / (60 / 227) = 25 / 60 = 5 / 12 = 0.4167

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(v) P(Sauce is Bolognese, given that the Pasta is Spaghetti)

By Bayes Theorem, P(A given B) = P(A and B) / P(B)

Therefore (Bolognese given Spaghetti) = P(Bolognese and Spaghetti) / P(Spaghetti)

P(Bolognese and Spaghetti) = 30 / 227

P(Spaghetti) = 60 / 227

Therefore the required probability = (30 / 227) / (60 / 227) = 30 / 60 = 1 / 2 = 0.5

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(c) 2 events A and B are independent when the probabilities of both events occurring is equal to the product of the probabilities of the individual events. i.e P(A And B) = P(A) x P(B)

P(Gnocchi) = 95 / 227

P(Sauce) = 227 / 227 = 1

P(Gnocchi) x P(Sauce) = 95/227 * 1 = 95/227

P(Gnocchi and Sauce) = P(Gnocchi and Bolognese) + P(Gnocchi and Carbonara) + P(Gnocchi and Pesto) = 40/227 + 30/227 + 25/227 = 95/227

Since P(Gnocchi) x P(Sauce) = P(Gnocchi and Sauce), the 2 events are independent.