# Assignment: Applied Probability and Statistics

Assignment: Applied Probability and Statistics ORDER NOW FOR CUSTOMIZED AND ORIGINAL ESSAY PAPERS ON Assignment: Applied Probability and Statistics MATH 331 sec003 Homework 4 Instructor: Dr. Brown 1. We have an urn with three green balls and two yellow balls. We pick a sample of two without replacement and put these two balls in a second urn that was previously empty. Next we sample two balls from the second urn with replacement. (a) What is the probability that the first sample had two balls of the same color? Would you guess the answer if you took the sample without order? (b) What is the probability that the second sample had two balls of the same color? (c) Given that the last two balls have the same color, what is the probability that the second urn contains two balls of the same color? 2. Show that if Pr(A) = 0 or Pr(A) = 1 then any event B is independent of A. 3. We have a system that has 2 independent components. Both components must function in order for the system to function. The first component has 8 independent elements that each work with probability 0.95. If at least 6 of the elements are working then the first component will function. The second component has 4 independent elements that work with probability 0.90. If at least 3 of the elements are working then the second component will function. a. What is the probability that the system functions? b. Suppose the system is not functioning. Given that information, what is the probability that the second component is not functioning? 4. Suppose an urn has 3 green balls and 4 red balls. a. 9 draws are made with replacement. Let X be the number of times a green ball appeared. Identify by name the probability distribution of X. Find the probabilities Pr(X ? 1) and Pr(X ? 5). Assignment: Applied Probability and Statistics b. Draws with replacement are made until the first green ball appears. Let N be the number of draws that were needed. Identify by name the probability distribution of N . Find the probability Pr(N ? 9). (Give a simplified expression, not a sum.) c. Compare Pr(X ? 1) and Pr(N ? 9). Is there a reason these should be the same? 5. On a test there are 20 true or false questions. For each problem the student knows the answer with probability p, thinks he knows the correct answer, but actually he does not with probability q, is aware of the fact that he does not know the answer with probability r. We assume that these happen independently for each question, and that p + q + r = 1. If the student does not know what to answer he will guess by choosing true or false with probability 1/2 ? 1/2. What is the probability that he will get the correct answer for at least 19 questions? 6. Jane must get at least 3 of the four problems on the exam correct to get an A. She has been able to do 80% of the problems on old exams, so she assumes that the probability she gets any problem correct is 0.8. She also assumes that the results on different problems are independent. a. What is the probability she gets an A? b. If she gets the first problem correct, what is the probability she gets an A? 7. Show that if X ? Geom(p) then Pr(X = n + k|X > n) = Pr(X = k) for n, k ? 1. 8. Flip a coin three times. Assume the probability of tails is p and that successive flips are independent. Let A be the event that we have exactly one tails among the first two coin flips and B the event that we have exactly one tails among the last two coin flips. For which values of p are events A and B independent? 9. We play a card game where we get 13 cards at the beginning out of the deck of 52. We play 50 games one evening. For each of the following random variables identify the name and the parameters of the distribution. a. The number of aces I get in the first game. b. The number of times I received at least one ace during the evening. c. The number of times all my cards were from the same suit. d. The number of spades I received in the 5th game. 10. A Martian year is 669 Martian days long. Solve the birthday problem for Mars. That is: for each n find the probability that out of n randomly chosen Martians there will be at least two with the same birthday. Estimate the value of n where this probability becomes bigger than 90%. Assignment: Applied Probability and Statistics Get a 10 % discount on an order above $ 100 Use the following coupon code : NURSING10