Mathematical Probability (STAT2003/STAT7003)
Assignment 1
The due date/time is given on Blackboard. STAT7003 students have additional question(s),
marked with a star (*).
1. Let A and B be two events, where P(A) = 1
2
, P(B) = 1
3
, and P(Ac ∩Bc) = 1
4
.
(a) Find P(A ∩B). [3 marks]
(b) Find P(A |B) and P(B |A). [3 marks]
(c) Find P(A ∩Bc) and P(A ∪Bc). [4 marks]
(d) (*) Let C be an event such that P(C) = 3
4
. Show that 1
4
⩽ P(A ∩ C) ⩽ 1
2
.
[3 marks]
(e) (*) Let D and E be two events such that P(D ∪ E) = 1
2
and P(D ∪ Ec) = 7
10
.
Find P(D). [5 marks]
2. From a full deck of 52 bridge cards you receive 6 cards.
(a) What is the probability that they are all of the same suit? [2 marks]
(b) What is the probability that they contain only one pair? [2 marks]
3. A traveller bought travel insurance for a five day trip. The policy will pay a benefit
of $5000 for accidents on the first day, and the benefit will decrease by $1000 each
subsequent day. Suppose the probability of having accidents during a day is 0.4 and
is independent of other days. Find the expected benefit for this trip. [3 marks]
4. In the game of Oz Lotto, 10 balls are drawn randomly without replacement from a
pool of 47 balls numbered from 1 to 47. The first seven are ‘winning numbers’ and
the last three are ‘supplementary numbers’. An entry involves selecting 7 of the 47
numbers. There are 5 prize divisions. Division 1 requires the entry to include all 7
winning numbers. Division 2 requires any 6 winning numbers plus one supplementary
number. Division 3 requires any 6 winning numbers. Division 4 requires any 5
winning numbers plus one supplementary number. Division 5 requires any 5 winning
numbers.
(a) Find the probability of an entry winning each of these divisions. [7 marks]
(b) What is the probability of an entry winning in any division? [1 mark]
(c) An entry costs $1.50. The prizes are as follows: Division 1 $1,000,000; Division
2 $25,000; Division 3 $3,500; Division 4 $300; and Division 5 $50. What is the
expected gain of an entry? [2 marks]
1
5. The waiting time (in weeks) for an appointment to see a psychologist can be modeled
by a random variable X with pdf
f(x) =
{
x3
2500
if 0 ⩽ x ⩽ 10,
0 otherwise.
(a) Find the time (in days) within which 90% of patients can get an appointment.
[2 marks]
(b) Find the cdf F of X. [2 marks]
(c) Use matplotlib.pyplot from Python to plot the pdf and cdf of X. [2 marks]
(d) Calculate the expectation and variance of X. [2 marks]
(e) What is the probability that a patient needs to wait at least another two weeks,
given that the patient had already waited for two weeks? [2 marks]
6. To be granted a licence to operate certain machinery, a candidate must pass a test.
Only two attempts are permitted. Suppose that the probability of passing the test
in the first attempt is 0.65, and the probability of passing the test is 0.7. Given that
a candidate passed the test, what is the probability that this candidate only passed
at the second attempt? [4 marks]
7. Let X be a random variable with pdf
f(x) =
{
0.2 e−0.2x if x > 0,
0 otherwise.
(a) Find the moment generating function of X, remembering to state the range for
which it holds. [3 marks]
(b) Hence find the expectation and variance of X. [2 marks]
8. Before 2020, the number plates on Queensland’s vehicles have the format of three
digits followed by three letters. How many vehicles should I expect to see before the
letter U on a number plate occurred? [4 marks]
9. (*) While replacing some LED bulbs, an apprentice accidentally dropped two dead
bulbs into a box of three new bulbs. The five bulbs are now mixed up and needs to
be tested individually to find out which one is dead. Let X be the number of tests
needed in order to identify the two dead bulbs. Determine the pmf of X. [7 marks]