POLYTECHNIC OF NAMIBIA SCHOOL OF HEALTH AND APPLIED SCIENCES DEPARTMENT OF MATHEMATICS AND STATISTICS QUALIFICATION: Bachelor of Science Applied Mathematics and Statistics QUALIFICATION CODE: 35BAMS COURSE NAME: PROBABILITY THEORY 1 COURSE CODE: PBT501 S DATE: DURATION: MARKS: JUNE 2015 3 HOURS 100 FIRST OPPORTUNITY EXAMINATION EXAMINER: MODERATOR: MR. D. Ntirampeba Mr. A. Roux INSTRUCTIONS: a. Answer all the questions in the booklet provided b. Show clearly all the steps used in the calculations c. All written work MUST be done in blue or black ink and sketches must be done in pencils. PERMISSIBLE MATERIAL 1. Calculator APPENDIX: STATISTICAL-TABLE This paper consists of 4 pages excluding this cover and Z- table
Question 1 [30 marks] 1.1 Define the following terminologies as they are applied in set theory and probability theory [10] 1.1.1 A sample space [2] 1.1.2 An event [2] 1.1.2 Complement of a set A [2] 1.1.3 Mutually exclusive events [2] 1.1.4 Independent events (say A and B) [2] 1.2 Consider the following subsets of the sample spaces= { 1, 2, 3, 4, 5, 6}: Rt = {1, 2, 5},R2 ={3, 4, 5, 6}, R3 ={2, 4, 6}, R4 ={1, 3, 6}, R 5 ={1, 2, 5}. Find: 1.2.1 RI U R 2 and P{ R 1 U R 2 ) [2] 1.2.2 R 4 n Rs and P( R 4 n Rs) [2] 1.2.3 R 5 and P( R 5 ) [2] 1.2.4 (RI U(R4 nrs)) and P( RI U{ R4 n Rs)) [2] 1.3 [2] Indicate which of the following random variables are d = discrete, and which are c =continuous: (one mark to each correct answer) 1.3.1 The time required to answer this question. 1.3.2 The number of words in a book chosen at random from the library. 1.3.3 The number of goals scored by African Stars in their weekend league matches. 1.3.4 The maximum temperature recorded at Ho sea Kutako International Airport. 1.3.5 The length of time you have to wait for a taxi at Wernhill Park after work. 1.4 Indicate which of the following variables are a = quantitative and which are b = qualitative. (one mark to each correct answer) 1.4.1 Number of children under 18 years of age in a family. 1.4.2 Colour of cars in the Polytechnic car park. 1.4.3 Age of students in a first year Mathematics class. 1.4.4 Time to commute from home to Polytechnic ofnamibia. 1.4.5 Number of errors in a student's report. ] 1
Question 2 [35 marks] 2.1. A mutual fund salesperson has arranged to call on three persons tomorrow. Based on the experience, the salesperson knows that there is a 50% chance of closing a sale on each call. Let X be the number of sales. 2.1.1. Develop a tree diagram for this experiment. And what is the sample space? (use S=sale and S = no sale ) [ 4] 2.1.2 Determine all possible values that X can take on 2.1.2 Find the probability of each value of X [2] [2] 2.2. Suppose that 18 red beads, 12 yellow beads, 8 blue beads and 12 black beads are to be strung in a row. How many arrangements of beads can be made? [2] 2.3 A fast-food restaurant chain has 600 outlets in Namibia. The following table categorizes cities by size and location, and presents the number of restaurants in the cities of each category. A restaurant is to be chosen at random from 600 to test market a new style of chicken. Region Population of city North East South East South West North West Under 50000 30 35 15 5 50000-500000 60 90 70 30 Over 500000 150 25 30 60 2.3.1 What is the probability the restaurant is in a city with a population under 50000 and is located in the North East? [2] 2.3.2 What is the probability the restaurant is in a city with a population over 50000 or is located in the North West? [ 4] 2.3.3 If the restaurant is in the city with a population over 500000, what is the probability that is located in the South East? [4] 2.4. Three airlines serve the northern part of Namibia. Airline A has 50% of all scheduled flights, airline B has 30%, and Airline C has the remaining 20%. Their on-time rates are 80%, 65%, and 40%, respectively. A plane has just left on time. What is the probability that it was airline A? 2
2.5. Suppose a discrete random variable X has a probability mass function defined by the table below. X=x P(X = x) Work out the following. 2 0.18 3 6 8 0.23 0.40 0.19 2.5.1 mean of X 2.5.2 variance of X 2.5.3 coefficient of variation [4] Question 3 [35 marks] 3.1 Assume that セ L@ r;, r;, and セ @ are independent random variables, with I セHe@ = 2 ]Iセ Hv@4 E{r;)=-1 v(r;)=6 E(I;) = 4, eh セ I@ = -2 V{l;)=8, v サセ I]Y@ Let U = セ @ +r; -21; MT セ @ and w] セ @ -2r; -5Y 4 Find: 3.1.1 E(W) 3.1.2 Var(U) 3.2 A Harris Interactive survey for InterContinental Hotels & Resorts asked respondents, "When traveling internationally, do you generally venture out on your own to experience culture, or stick with your tour group and itineraries?" The survey found that 23% of the respondents stick with their tour group (USA Today, January 21, 2004). In a sample of six international travellers, what is the probability that at least two will stick with their tour group? 3.3 Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways. 3.3.1 Compute the probability of receiving no calls in one hour interval of time. 3.3.2 Compute the probability of receiving at most two calls in 15 minutes. 3
3.4 In an article about the cost of health care, Money magazine reported that a visit to a hospital emergency room for something as simple as a sore throat has a mean cost of N$328 (Money, January 2009). Assume that the cost for this type of hospital emergency room visit is normally distributed with a standard deviation of N$92. Answer the following questions about the cost of a hospital emergency room visit for this medical service. 3.4.1 What is the probability that the cost will be between N$300 and N$400? 3.4.2 If the cost to a patient is in the lower 8% of charges for this medical service, what was the cost of this patient's emergency room visit? 3.5 A Myrtle Beach resort hotel has 120 rooms. In the spring months, hotel room occupancy is approximately 85%. What is the probability that 100 or more rooms are occupied on a given day? [4] END OF EXAM PAPER 4
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