THE INTERNATIONAL UNIVERSITY (IU) Course: Probability Models in Operations Research Department of Industrial System Engineering
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1 MIDTERM EXAMINATION PROBABILITY MODELS IN OPERATIONS RESEARCH Head o ISE Department Lecturer: Student ID: Name: Duration: minutes Date: 9 th March, Assoc Pro. Ho Thanh Phong INSTRUCTIONS: Truong Ba Huy. This is an open book examination. No laptors, PDA,. Use o calculator is allowed; discussion and material transer are strictly prohibited. Total pages: (including this page) Question : ( pts) A gambler has in his pocket a air coin and a two-headed coin. He selects one o the coins at random, and when he lips it, it shows heads. / / F.C H H T / / / a. (pts) What is the probability that it is the air coin? P Fair. C H H H H T Probability Models in Operations Research Midterm Exam
2 b. (pts) Suppose that he lips the same coin a second time and again it shows heads. Now what is the probability that it is the air coin? P Fair. C HH c. (pts) Suppose that he lips the same coin a third time and it shows tails. Now what is the probability that it is the air coin? P Fair C H H T. Question : ( pts) A student has to sell books rom a collection o 6 math, 7 science, and economics books. How many choices are possible i a. (pts) Both books are to be on the same subject? C 6 7 C C b. (pts) The books are to be on dierent subjects? C C C C 6 Or : 7 C 7 7 C C C C C 9 6 Question : ( pts) A rat is trapped in a maze. Initially it has to choose one o two directions. I it goes to the right, then it will wander around in the maze or three minutes and will then return to its initial position. I it goes to the let, then with probability / it will depart the maze ater two minutes o traveling, and with probability / it will return to its initial position ater ive minutes o traveling. Assuming that the rat is at all times equally likely to go to the let or the right. What is the expected number o minutes that it will be trapped in the maze? / E[T] / R +E[T] L / / +E[T] Probability Models in Operations Research Midterm Exam
3 ET E T ET ET Question : ( pts) The joint probability density unction o random variables X and Y is given: x, y x cy Where c is a constant. a. (pts) Find the value o c. x cy dydx x c x c dx c c x, y otherwise b. (pts) Are X and Y independent? X Y x y x y dy x y dx xy X Y Not independen t! x x y y x y x, y y x c. (pts) Find P X Y Probability Models in Operations Research Midterm Exam
4 P X Y PY X X x X x PY x x dx x y dx dy x dx x x x dx x 7x 6x 9 dx Question : ( pts) A machine in a actory can be ailed because o two types o ailures. The time (in hours) required to repair a machine with type ailure is an exponentially distributed random variable with parameter λ =. The time required to recover a machine rom type ailure ollows the normal distribution with mean o. and standard deviation o.. The probabilities o type and type ailures are.6 and., respectively..6. Ex() N (.;. ) e -.9 RT > RT > a. (pts) What is the probability that a repair time exceeds hours? P ( RT ).6e.9*.. b. (pts) What is the expected repair time o a ailed machine?.6*+.*.=. Probability Models in Operations Research Midterm Exam
5 Question 6: ( pts) The number o times that a person get a cold in a given year is a Poisson random variable with parameter λ =. Suppose that a new wonder drug has just been marketed that reduces the Poisson parameter to λ = or only 7% o the population. For the other percent o the population, the drug has no eect. a. (pts) I an individual tries the drug or a year and has ewer than colds in that time, how likely is it that the drug is beneicial or him or her?.7. (E) P() (NE) P().. < <.*.7 P ( E ). 9.*.7.*. b. (pts) Determine the expected number o colds per year that a drug usercan get. E C *. *.7. GOOD LUCKS! Probability Models in Operations Research Midterm Exam
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