Question 1 - Probability [Total: 13 marks] A prominent plastic recycling centre in Adelaide have made some recent frustrating observations regarding the behaviour associated with disposal of plastic bottles. Some people are failing to remove lids, caps and rings from plastic bottles prior to disposal, although this is clearly stated as a requirement prior to disposal on their website. So they conducted a survey, noting the bottle's "resin identification code" and whether the lid/cap/ring had been correctly removed. The 2 plastic bottle types recorded were: CODE 1 Polyethylene Terephthalate (PET) and CODE 2 High-Density Polyethylene (HDPE). Of 2036 PET and HDPE bottles, 936 of these bottles had their lid/cap/ring correctly removed prior to disposal, with 70% of the 890 HDPE bottles having their lid/cap/ring correctly removed. What is the probability of the following events? Show all working. No marks awarded for the correct answer without working out. Hint: A cross tabulation or contingency table can help.

I. What is the probability that a randomly selected bottle has a lid/cap/ring still attached? [2 marks]

II. What is the probability that a randomly selected bottle is a PET bottle or has the lid/cap/ring still attached? [2 marks]

III. What is the probability that a randomly selected bottle is a HDPE bottle and has the lid/cap/ring removed? [2 marks]

IV. What is the probability that a randomly selected bottle is a PET bottle given that the lid/cap/ring is removed? [2 marks]

V. Are the events "lid/cap/ring removed" and "HDPE plastic bottle type" independent? Demonstrate this appropriately. Then provide a brief description of what independence refers to in this specific case. [5 marks]

Thankyou in advanced