Question

To get full marks for the following questions you need to convert the question from words to a mathematical expression (i.e. use mathematical notation), defining your events where necessary, and using correct probability statements.

Suppose the University of Newcastle (UON) service area consists of the three Main Statistical Areas from which most students from the University of Newcastle live: the Central Coast (CC), Hunter excluding Newcastle (HEN), and Lake Macquarie and Newcastle (LMN) areas. According to the Australian Bureau of Statistics, 14.6% of residents in the CC area were born overseas, 8.4% of residents in the HEN area were born overseas, and 11.7% of residents in the LMN area were born overseas. Across the UON service area, 34.6% live in the CC area, 27.0% live in the HEN area, and 38.4% live in the LMN area.

Let B be the event that a resident was born overseas, CC be the event that a resident lives in the Central Coast, HEN be the event that a resident lives in the Hunter excluding Newcastle area, and LMN be the event that a resident lives in the Lake Macquarie and Newcastle area.

(a) Construct a tree diagram that summarises the given probability information.
(b) What is the probability that a randomly selected resident in the UON service area is a resident of the Central Coast and was born overseas?
(c) What is the probability that a randomly selected resident in the UON service area was born overseas?
(d) Are the events B and CC independent? Why or why not?
(e) If a randomly selected resident in the UON service area was born overseas, what is the probability that he or she is a resident of the Hunter excluding Newcastle area?
(f) In part (c), you found the probability that a randomly selected resident in the UON service area was born overseas. Can you infer that this probability is the same as the probability that a UON student was born overseas? Why or why not?

Answer 1

(a)

The tree diagram about overseas resident probabilities is as follows.

(b)

Required probability is given by

(c)

Required probability is given by

(d)

So, the events B and CC are not independent.

(e)

Required conditional probability is given by

(f)

The given information is about residents of the said areas. So, we found the probabilities corresponding to all residents. Probabilities corresponding to students may differ from that of residents, So we can not infer the same as the probability that a UON student was born overseas.