Question 2 (a)

Upon arrival at a bank's enquiry counter, potential customers are categorized into four groups according to the nature of the financial services they are enquiring, namely, (A) insurance, (B) stocks and securities, (C) loan and mortgages, and (D) others.

In the past year, 20% enquires are classified as (A), 10% as (B), 30% as (C), and 40% as (D). An enquiry is referred as a "successful" one if the enquirer enters a new business deal with the bank before leaving the counter. Each of the four categories has different success rate. The success rate of an (A) enquiry is 5%, (B) is 50%, (C) is 60%, and (D) is 40%. Given that an enquiry is NOT successful, what is the probability that the enquiry was categorized as (C) "loan and mortgages" upon arrival? [6 marks]

(b) A company is contemplating the introduction of a new product. The marketing office estimates that the sales volume is normally distributed approximately with a mean of 5,000 units and a standard deviation of 1,000 units.

(i) Calculate the probability that the sales volume will be within the interval 6,500 units to 7,000 units. [4 marks]

(ii) What is the sales volume that the company has 10% chance to achieve or underachieve? [3 marks]

(c) The number of typing errors per page in a company annual report follows the Poisson distribution with an average of 0.25 error per page.

(i) Find the probability that there are 2 or more typing errors on a randomly selected page of the report. [4 marks]

(ii) For 5 randomly selected pages of the report, what is the probability that there are 2 pages each with 2 or more typing errors? [3 marks]