MTH210 End-of-Course Assessment

Learning Outcome

• Calculate the probability of the occurrence of an event.
• Compute conditional probability and identify independent events.
• Determine expectation and variance of random variables.
• Solve probability distributions.
• Comment on results of Normal/Poisson approximation to Binomial distribution.
• Apply probability models in practical settings.

Answer ALL questions. (Full marks 100)

Question 1

(a) The probability that a company will hold a sales promotion for product A is 0.7. The probability that it will hold a sales promotion for product B is 0.4. The probability that it will hold a sales promotion for either product A or product B or both is 0.8. Calculate the probability that the company will hold a sales promotion for neither product

(4 marks)

(b) Table Q1 below shows the data of 100 individuals collected from a survey where respondents were classified based on their smoking status and whether they had lung disease.

Table Q1

If an individual is randomly selected, solve the probability that this person has lung disease, given that the person is a smoker

(6 marks)

(c) The probability that a student passes Maths (A) is 0.7. The probability to pass Physics (B) is 0.6 and the probability to pass Chemistry (C) is 0.8. It is also given that the probability that a student passes both Maths and Physics is 0.5, and the probability to pass at least one of the three subjects is 0.90. Solve P(A|A ∪ B ∪ C)

(6 marks)

(d) Of all the women who submit pregnancy tests, 75% of them are pregnant. A certain pregnancy test kit has a probability of 0.02 of giving a false positive result (i.e. predict that a woman is pregnant when in fact she is not pregnant). The test kit has a probability 0.99 of giving a valid positive result (i.e. predict positive when the woman is really pregnant). Compute the probability that a woman is not pregnant given that the test is negative (i.e. test predicts that she is not pregnant).

(9 marks)

Question 2

(a) There are six telephone lines at a company reception counter. Let X be the number of lines in used at a certain time of the day. The probability mass function (pmf) of X is given in Table Q2 below.

Table Q2

(i) Compute the probability that between one and four lines, inclusive, are in use.

(b) The probability that an applicant will pass a pre-requisite assessment for a course in each attempt is 0.7.

(i) Determine the probability that an applicant will pass the assessment in at most 3 attempts.

(5 marks)

(ii) In a certain round of the course application, there are 8 applicants taking the pre-requisite assessment. Analyse the problem and determine what probability model can be applied to solve for the probability that at least four applicants will pass the assessment in the first attempt.

(10 marks)

Question 3

(a) Determine the value of the constants b and c in the probabilities

P(Z ≤ b) = 0.121 and P(−c ≤ Z ≤ c) = 0.668

where Z ~ N(0, 1).

(10 marks)

(b) In a supermarket, the mass in gram of a packet of sweets has the distribution X ~ N(120, 15²), and the mass of a packet of chocolate has the distribution Y ~ N(180, 20²). X and Y are independent. Calculate the probability that a randomly selected packet of sweets weighed less than 130 grams. In addition, compute the probability that a randomly selected packet of sweets and a randomly selected packet of chocolate have a total mass of at least 310 grams.

(15 marks)