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midterm_1_sol

# midterm_1_sol - IE 4521 Midterm#1 Prof John Gunnar Carlsson...

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Unformatted text preview: IE 4521 Midterm #1 Prof. John Gunnar Carlsson March 2, 2010 Before you begin: This exam has 9 pages (including the normal distribution table) and a total of 8 problems. Make sure that all pages are present. To obtain credit for a problem, you must show all your work. if you use a formula to answer a problem, write the formula down. Do not open this exam until instructed to do so. 1 1. (10 points) Let E , F , and G be three events in a state space S . Write expressions for the events that of E , F , and G (a) only E occurs Solution E ∩ F c ∩ G c (b) both E and G but not F occur Solution E ∩ F c ∩ G (c) at least one of the events occurs Solution E ∪ F ∪ G (d) at least two of the events occur Solution ( E ∩ F ) ∪ ( E ∩ G ) ∪ ( F ∩ G ) (e) all three occur Solution E ∩ F ∩ G (f) none of the events occurs Solution E c ∩ F c ∩ G c (g) at most one of the events occurs Solution ( E ∩ F c ∩ G c ) ∪ ( E c ∩ F ∩ G c ) ∪ ( E c ∩ F c ∩ G ) (h) at most two of the events occur Solution ( E ∩ F ∩ G ) c (i) exactly two of the events occur Solution ( E ∩ F ∩ G c ) ∪ ( E ∩ F c ∩ G ) ∪ ( E c ∩ F ∩ G ) (j) Simplify the expression ( E ∪ F ) ∩ ( F ∪ G ) (hint: draw a Venn diagram) Solution F ∪ ( E ∩ G ) 2 2. (15 points) You ask your neighbor to water a sickly plant while you are on vacation. Without water it will die with probability . 8 ; with water it will die with probability . 15 . You are 90 percent certain that your neighbor will remember to water the plant. (a) What is the probability that the plant will be alive when you return? Solution Solution Let A be the event that your neighbor waters your plant and B the event that your plant is alive. By the law of total probability, Pr( B ) = Pr( B | A )Pr( A ) + Pr( B | A c )Pr( A c ) = 0 . 85 · . 9 + 0 . 2 · . 1 = 0 . 785 so there is a...
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