Question 2

a) In a regional city, two lawn companies fertilise lawns during the summer. Company A has 72% of the market. Thirty per cent of the lawns fertilised by Company A could be rated as very healthy one month after the service. Company B has the other 28% of the market. Twenty per cent of the lawns fertilised by Company B could be rated as very healthy one month after the service. A lawn that has been fertilised by one of these companies within the last month is selected randomly and is rated as very healthy.

i) What is the revised probability that Company A fertilised the lawn?

ii) What is the corresponding probability that Company B fertilised the lawn?

b) Of 250 employees of a company, a total of 130 are full-time employees. The remain-der are part-time employees. There are 150 males working for this company, 85 of whom are full-time employees. What is the probability that an employee chosen at random:

i) is a part-time employee?

ii) is female and a full-time employee?

iii) is a full-time employee, given that the employee is female?

iv) is a female, given that the employee is full-time?

v) Are the events \employee chosen at random is female" and \employee chosen at random is full-time" statistically independent?

c) Let P (A[B) = 0:9, P (A) = 0:5 and P (B j A) = 0:4. Calculate P (B) and P (A j B). Are A and B independent? Are A and B disjoint? Justify your answers.

d) Let P (A) = 0:2, P (A [ B) = 0:6, and suppose the events A and B are independent. Calculate P (B).

e) Let P (A) = 0:6, P (A j B) = 0:6 and P (A[B) = 0:8. Calculate P (B) and P (A\B). Are A and B independent? Justify your answer.

f) A toy manufacturer buys pre-assembled robotic arms from three different suppliers 50% of the total order comes from Supplier 1, 30% of the total order comes from Supplier 2, and the remaining 20% from Supplier 3. Past data shows that the quality control standards of the three suppliers are different. Two percent of the arms produced by Supplier 1 are defective, while Suppliers 2 and 3 produce defective arms at the rates of 3% and 4% respectively.

Let Si be the event that a given arm comes from Supplier i, i = 1; 2; 3 and let D be the event that a given arm is defective.

i) Draw a tree diagram that models this situation.

ii) What proportion of the arms in the manufacturer's inventory are non-defective?

iii) If an arm is found to be defective, what is the probability that it came from Supplier 1? Give your answer to 4 decimal places.