XIAMEN UNIVERSITY MALAYSIA
ONLINE MIDTERM EXAMINATION
Course Code: BSC126
Course Name: Probability and Statistics B
Question paper setter: Gloria Teng Ai Hui, Goh Chien Yong
Academic Session: 2021-04 Question paper: A B
Total No. of Pages: 5 Time Allocated: 2 hours
Additional Materials: Statistical tables
Apparatus Allowed: Calculator
INSTRUCTIONS TO CANDIDATES
1. This paper consists of 5 questions. Answer all questions.
2. Please follow the requirement of each question and give the corresponding
answers in PDF format.
3. Show your working. If you do not show any working, only partial marks
will be awarded even if your answer is correct.
4. Unless specified otherwise, you may leave your answers either in terms of
fractions or decimal places.
5. Communication between candidates in any means is forbidden. Answers
must be entirely individual candidate’s independent effort. If you are found
sharing your solutions with other candidates, or suspected of doing so, you
would be penalized accordingly.
(Student ID: Full Name: )
CONFIDENTIAL 202104/BSC126
Question 1 (15 marks)
(a) A number is randomly picked from a set of integers, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. List the
outcomes in the following events for the experiment above.
(i) The number is divisible by 3. [1]
(ii) The number is an even number. [1]
(iii) The number is positive. [1]
(iv) The number is less than or equal to 5. [1]
(v) The number is 11. [1]
(b) It is found that out of 100 students of a school, 65 students go to school by bus, 15 students
walk to school and the remaining 20 students go to school by other transportations. If a
student is randomly selected from this group of 100 students, compute the probability that
the student selected either goes to school by bus or walks to school. [2]
(c) A student in a school can join the English Language Society (E), Malay Language Society
(M) and Science Society (S). Information regarding the students joining these societies are
as follows:
P (E) = 0.4, P (E ∪ M) = 0.65, P (E ∪ S) = 0.8, P (E ∩ M) = 0.15, P (E ∩ S) =
0.2, P (M ∩ S) = 0.2 and P (E ∪M ∪ S) = 0.95
A student is selected at random from the school. Find the probability that the student is
(i) a member of Malay Language Society, [2]
(ii) a member of all the three societies, [4]
(iii) not a member of any society. [2]
Page 1 of 5
CONFIDENTIAL 202104/BSC126
Question 2 (20 marks)
(a) In an examination, Teng has to choose two correct statements from a list of five statements
for a particular question. He does not know which two statements are correct and decides to
guess. He randomly selects a statement from the list of five statements, and then randomly
selects another statement from the remaining four statements.
(i) Let C1 be the event that the first guess is correct, and C2 be the event that the second
guess is correct. Draw a tree diagram for these events. For each branch, clearly
indicate the event, probability labels and their respective probabilities. [4]
(ii) Compute the probability that only one of the guesses is correct. [2]
(iii) If Teng definitely knows the first statement but has to guess the second one, find the
probability that both statements are correct. [2]
(b) In a factory, three machines A, B and C are operated to make certain parts of a device. The
percentages of the parts manufactured by the machines A, B and C are 35%, 50% and 15%
respectively. It is known that 8%, 5% and 16% of the parts produced by the machines A, B
and C respectively are defective. If a part is randomly picked, calculate the probability that
the part is from the machine A given that it is defective. [6]
(c) A telemarketing company makes a total of 100 phone calls to its customers, with 60 calls
in the morning and 40 calls in the afternoon. The success rates of selling its products in the
morning and in the afternoon are 0.25 and 0.16 respectively. Find the total success rate of
the company in selling its products. [6]
Page 2 of 5
CONFIDENTIAL 202104/BSC126
Question 3 (25 marks)
(a) Classify each random variable as either discrete or continuous.
(i) The number of patrons arriving at a restaurant between 5:00pm and 6:00pm. [1]
(ii) The number of new cases of influenza in a particular county in a coming month. [1]
(iii) The air pressure of a tire on an automobile. [1]
(b) A discrete random variable X has the following probability distribution:
x 0 1 2 3
P (X = x) c c2 c2 + c 3c2 + 2c
Find the value of the constant c. [4]
(c) The discrete random variable X has the following probability mass function:
p(x) =

1
8
x = 0, 3
3
8
x = 1, 2
0 otherwise
(i) Find the cumulative distribution function F (x). [6]
(ii) Use the results from Part (i) to determine P (X < 2) and P (X ≤ 2.5). [4]
(d) A continuous random variable X can take only values in the interval [0, 4]. In this interval,
the probability that X takes a value greater than x is equal to ax2 + b.
(i) Determine the values of a and b. [4]
(ii) Find the cumulative distribution function F (x). [4]
Page 3 of 5
CONFIDENTIAL 202104/BSC126
Question 4 (20 marks)
(a) Every working day, Alan reverses his car from his driveway on to the road in such a way
that there is a very small probability p that his car will be involved in a collision.
(i) Show that the probability that there will be no collision in a five-day week is (1−p)5.
[2]
(ii) State one assumption that is made in your answer above. [2]
(iii) Let p = 0.001. Check that the Poisson approximation can be used, and find the
approximate probability that Alan will avoid a collision in 500 working days. Round
your answer to 4 decimal places. [4]
(b) Organisms are present in ballast water discharged from a ship according to a Poisson pro-
cess with a concentration of 4 organisms per cubic meter. Find the probability that the
number of organisms in 1.5 cubic meter of discharge exceeds its mean value by more than
one standard deviation. Round your answer to 4 decimal places. [5]
(c) The number of times that a person gets a cold in a given year is a Poisson random variable
with parameter λ = 5. Suppose that a new supplement which has just been marketed
reduces the Poisson parameter to λ = 3 for 75% of the population. For the other 25% of
the population, the supplement gives no significant reduction on the number of colds. If an
individual tries the supplement for a year and has 2 colds in that time, find the conditional
probability that the supplement is beneficial for him or her. Round your answer to 4 decimal
places. [7]
Page 4 of 5
CONFIDENTIAL 202104/BSC126
Question 5 (20 marks)
(a) A machine fills coke into bottles automatically. Actual volumes are normally distributed
with µ = 500 ml and σ = 10 ml.
(i) Determine the probability that the bottles are filled within the tolerance limit of 485
ml to 515 ml. [4]
(ii) Suppose the mean remains the same. Determine the change needed in the standard
deviation if 95% of the bottles must be within the tolerance limit of 485 ml to 515 ml.
[4]
(b) It was found that an exponential distribution with a mean of 5 minutes was suitable to
model the duration of a visitor at a website.
(i) Find the probability that the website is viewed for at least 3 minutes. [3]
(ii) If the website has already been viewed for 1.5 minutes, find the probability that the
home page is viewed for an additional 0.5 minutes. [3]
(c) Jones figures that the total number of thousands of miles that an auto can be driven before
it would need to be junked is uniformly distributed over [0, 40]. Smith has a used car that
he claims has been driven only 10,000 miles. If Jones purchases the car, determine the
probability that she would get at least 20,000 additional miles out of it. [6]
-END OF PAPER-
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