What is a Conditional Probability? or What does it Mean that two Events are Statistically Independent: Mathematics Coursework, BC, Singapore

Task 1 – Probability and Tablets

Part A

Explain in your own words:-

• What is conditional probability?
• What does it mean that two events are statistically independent?
• How are these two concepts connected?

You need to give a full explanation with the appropriate use of examples and mathematical formulae.

Part B

The marketing director of a famous electronics company is studying the purchase intentions of consumers towards the tablets they are selling over a period of six months.

She wants to know the following:-

• The consumers who were interested in buying a tablet are effectively buying it within the considered time frame?
• Did the consumers buy a Wi-Fi model or a Wi-Fi + 3G model?
• Are the consumers satisfied with their purchase?

In order to answer these questions, a survey of 2000 consumers was conducted. After 6 months the same consumers were interviewed to check if they effectively bought the tablet, which model they have bought, and if they were satisfied with their purchase. The results are shown in the tables below:-

The marketing director would like to answer the following questions:-

• What is the probability that a consumer decides to buy a tablet and will effectively buy it?
• If we know that a consumer decides to buy a tablet, what is the probability that they will do so?
• What is the probability that a consumer, after buying a tablet, is satisfied with their purchase?
• Would the last probability be affected by the fact that the consumer bought a Wi-Fi+3G model?

Learning Outcomes

Knowledge and Understanding

KU1) demonstrate mathematical literacy by consistently using correct Mathematical notation, terminology, conventions, and relevant diagrams and graphs;

KU2) identify whether a real-life situation is best solved exactly, numerically, or probabilistically;

Subject-specific Skills

SS3) identify, simplify and abstract the underlying mathematical structure of a situation allowing a problem to be modeled and solve;

SS4) choose and apply appropriate techniques, including numerical techniques, in algebra, co-ordinate geometry, trigonometry, and calculus to find a solution to a problem;

SS5) make predictions about real-life problems subject to chance using probabilistic and statistical models;

Key and Employability Skills

KE6) use a range of IT packages, including calculators, spreadsheets, word processors, and skills to analyze and solve problems;

KE7) think critically to assess suitable approaches to problems and communicate their work effectively;

KE8) study independently and collaboratively, undertaking appropriate research both under the direction and independently

Task 2 – Opening a Pizzeria

Carl would like to open a pizzeria and you are to support him analyzing costs, revenues and overall profit.

He would like to sell just ‘Margherita’ pizzas and he has the following costs:-

• Variable costs for buying the ingredients: flour, tomatoes, yeast, etc.
• Fixed costs for renting a place, for paying the salaries, etc.

To simplify the problem, we assume that the fixed costs are £1.50 for each sold pizza and the variable costs are on average £450 per day. The revenues depend mainly on the number of pizzas sold. The idea is to consider a selling price that is not too big to attract people but not too small since Carl needs to earn money.

We know that if the cost of one pizza is 7.50, or even more, then the demand is zero and if we reduce the cost of each pizza by 15p we have an increase of roughly 10 sold pizzas per day.

You are to write a report to advise Carl on how many pizzas he needs to sell each day to cover his costs. You also need to analyze the number of daily pizzas Carl needs to sell if he wants to start earning money. How much can we decrease the cost of each pizza without losing money?

You can start considering the following variables:-
𝑥 = number of pizzas sold in one day
𝑐 = total daily cost in terms of the number of pizzas sold
𝑝 = cost of each pizza in terms of the number of pizzas sold
𝑟 = total daily revenue in terms of the number of pizzas sold
𝑃 = daily profit which is given by the revenue take away the costs i.e. 𝑔 = 𝑟 − 𝑐
Then you can analyze the profit in terms of the number of pizzas sold.

You need to give full explanations in your own words, using appropriate diagrams and mathematical formulae. You need to comment on the profitability of the project and answer the questions to advise Carl.

Learning Outcomes

Knowledge and Understanding

KU1) demonstrate mathematical literacy by consistently using correct Mathematical notation, terminology, conventions, and relevant diagrams and graphs;

Subject-specific Skills

SS3) identify, simplify and abstract the underlying mathematical structure of a situation allowing a problem to be modeled and solve;

SS4) choose and apply appropriate techniques, including numerical techniques, in algebra, co-ordinate geometry, trigonometry and calculus to find a solution to a problem;

Key and Employability Skills

KE6) use a range of IT packages, including calculators, spreadsheets, word processors, and skills to analyse and solve problems;

KE7) think critically to assess suitable approaches to problems and communicate their work effectively;

CODE:
KE8) study independently and collaboratively, undertaking appropriate research both under the direction and independently.

Task 3 – Rainfalls

The Ministry of Agriculture is launching a project to encourage farmers to grow rice. Initially, two locations were chosen and their daily rainfall data were collected over a year:-

When checking the data it is found that

• the highest value for both Regions is 188 mm, and
• the lowest value is 5 for Region A, and 8 for Region B

Write a short report comparing the data for both Regions. You need to:

– Calculate appropriate measures of location and dispersion.

– Discuss the advantages and disadvantages of each measure, with reference to your data.

Include any appropriate diagrams to illustrate your report explaining how it facilitates comparison of the data in both Regions.

Learning Outcomes

Knowledge and Understanding

KU1) demonstrate mathematical literacy by consistently using correct Mathematical notation, terminology, conventions, and relevant diagrams and graphs;

KU2) identify whether a real-life situation is best solved exactly, numerically, or probabilistically;

Subject-specific Skills

SS3) identify, simplify and abstract the underlying mathematical structure of a situation allowing a problem to be modelled and solve;

SS4) choose and apply appropriate techniques, including numerical techniques, in algebra, co-ordinate geometry, trigonometry, and calculus to find a solution to a problem;

Key and Employability Skills

KE6) use a range of IT packages, including calculators, spreadsheets, word processors, and skills to analyse and solve problems;

KE7) think critically to assess suitable approaches to problems and communicate their work effectively;

KE8) study independently and collaboratively, undertaking appropriate research both under the direction and independently.

Task 4 – Circles

A circle is given by the equation (𝑥 − 𝑎) 2 + (𝑦 − 𝑏) 2 = 𝑟 2 . Changing the values of 𝑎, 𝑏, and 𝑟 will result in a transformation of the circle.

a) Describe with the aid of appropriate graphs and using correct terminology, the transformation corresponding to a separate change in each of the values of 𝑎, 𝑏 or 𝑟?

b) Given a circle diameter 𝐴(−4, 2) and 𝐵(8, −6), find the equation of the perpendicular to its centre and of its tangent at (6, 4).

The diameter is extended from 𝐴(−4, 2) through point 𝐵(8, −6), to point 𝐶(𝑥, 𝑦) whereby 𝐵(8, −6), becomes its midpoint.

c) Find the equation of the circle at point 𝐶 which has an area of the original circle in part b) multiplied by the last non-zero digit of your student ID number.

Learning Outcomes

Knowledge and Understanding

KU1) demonstrate mathematical literacy by consistently using correct Mathematical notation, terminology, conventions, and relevant diagrams and graphs;

Subject-specific Skills

SS4) choose and apply appropriate techniques, including numerical techniques, in algebra, co-ordinate geometry, trigonometry, and calculus to find a solution to a problem;

Key and Employability Skills

KE6) use a range of IT packages, including calculators, spreadsheets, word processors, and skills to analyze and solve problems;