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]
Marking Criteria for Question 1:
You need to demonstrate understanding of the probability concepts involved in each part of this
question and how to make appropriate interpretations of each numerical answer. Thus, for full marks
you need to show all workings (including the equation used with correct notation) AND provide a very
brief relevant written statement interpreting the answer to the question (so place the numerical answer
into a correct full statement showing you understand what it means).

The probability that a randomly selected bottle will have its ring attached is the ratio of such bottles to the total number of bottles: 1100/2036 ≈ 0.540.

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II. P(PET∨ring) = 1 -P(HDPE∧no ring) = 1 -623/2036 = 1413/2036 ≈ 0.694

The probability that a random bottle is PET or has the ring attached is the ratio of such bottles to the total: 1413/2036 ≈ 0.694.

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III. P(HDPE∧no ring) = 623/2036 ≈ 0.306

The probability that a random bottle is HDPE and has the ring removed is the ratio of such bottles to the total number of bottles: 623/2036 ≈ 0.306.

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IV. P(PET | no ring) = P(PET∧no ring)/P(no ring) = 313/936 ≈ 0.334

The probability that a random bottle is PET given that it has no ring is the ratio of ringless PET bottles to the total number of ringless bottles: 313/936 ≈ 0.334.

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V. P(ringless) = 936/2036 ≈ 0.460

P(HDPE) = 890/2036 ≈ 0.386

P(ringless)×P(HDPE) = 0.460×0.386 ≈ 0.177 ≠ P(HDPE∧no ring) = 0.306

If events A and B are independent, the probability of the joint event A∧B is the product of the probabilities of the individual events. Here the probability of an HDPE bottle having no ring is about 0.306, while the product of probabilities that a bottle is HDPE and that a bottle has no ring is about 0.177. "Ring removed" and "HDPE bottle type" are not independent.

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Attached is the recommended cross tabulation table. The numbers in blue are given in the problem statement (623 = 70%×890). The numbers in black are the numbers required in order to achieve the given totals.