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

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 of 1000 plastic bottles, noting their "resin identification code" and whether the lid/cap/ring had been correctly removed. The main plastic bottle types recorded were: CODE 1 Polyethylene Terephthalate (PET); CODE 2 High-Density Polyethylene (HDPE); and CODE 5 Polypropylene (PP). Of the 843 PET and HDPE bottles of particular interest, 702 of these bottles had their lid/cap/ring correctly removed prior to disposal, while 12.5% of the 472 PET bottles did not have their lid/cap/ring correctly removed. What is the probability of the following event? 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 is a PET bottle? [1.5 marks] II. What is the probability that a randomly selected bottle is a HDPE bottle or has the lid/cap/ring still attached? [1.5 marks] III. What is the probability that a randomly selected bottle is a PET bottle and has the lid/cap/ring still attached? [1.5 marks] IV. What is the probability that a randomly selected bottle is a HDPE bottle given that the lid/cap/ring is removed? V. [1.5 marks] Are the events "PET plastic bottle type" and "lid/cap/ring removed" independent? Show your working.