Assignment 1 Material from Modules 1 & 2 Questions Total of marks is 29. Question 1 - 6 marks To test the efficacy of admissions or release, an emergency room embarks on a controlled experiment. Each...

Assignment 1 Material from Modules 1 & 2 Questions Total of marks is 29. Question 1 - 6 marks To test the efficacy of admissions or release, an emergency room embarks on a controlled experiment. Each experiment consists of choosing 4 patients at randomon a particular night. The patients are selected from a group where the doctors are not certain whether they should be admitted or not. Name the patients P1, P2, P3, P4. Of these 4 patients, 2 are chosen at random. The first patient will be released and the second admitted. (a) Display a tree diagram of the possible outcomes. [2 marks] (b) Denote by A the event that at least one of the patients has an even numbered index (P2 and P4 have even numbered indices). Which outcomes are included in A? [2 marks] (c) Suppose that P1 and P2 are over 50 years old and P3 and P4 are less than 40 years old. Denote by B the event that exactly one of the patients selectesd is over 50 years old. Which outcomes are included in B? [2 marks] Data Source: Statistics and data with R. Y. Cohen & J. Cohen, Wiley (2008). Question 2 - 7 marks A patient is seen at a clinic. A recent epidemic in town shows that the probability that the patient suffers from the flu is 0.75. The probability that the patient suffers from walking pneumonia is 0.55. The probability that he/she suffers from both is 0.50. Denote by F the event suffers from the flu and by M that he/she suffers from pneumonia. (a) Interpret and compute P(F|M) [2 marks] (b) Interpret and compute P(M|F) [2 marks] (c) Are F and M independent? Explain. [3 marks] Data Source: Statistics and data with R. Y. Cohen & J. Cohen, Wiley (2008). Question 3 is on the next page 1 Question 3 - 8 marks Suppose that the Markov chain has 4 states s1, s2, s3, s4 and stationary transition probabilities as specified by the following transition matrix: ? ??? s1 s2 s3 s4 s1 1 4 1 4 0 1 2 s2 0 1 0 0 s3 1 2 0 1 2 0 s4 1 4 1 4 1 4 1 4 ? ??? (a) If the chain is in state s3 at a given time n, what is the probability that it will be in state s2 at time n + 2? [4 marks] (b) If the chain is in state s1 at a given time n, what is the probability that it will be in state s3 at time n + 3? [4 marks] (Hint: recommend Maxima) Data Source: Probability and Statistics M. DeGroot & M. Schervish, Addison-Wesley (2002). Question 4 - 8 marks Each time that a shopper purchases a tube of toothpaste, she chooses either brand A or brand B. Suppose that the probability is 1 3 that she will choose the same brand chosen on her previous purchase and the probability is 2 3 that she will change brands. (a) If her first purchase is brand A, what is the probability that her fifth purchase will be brand B? [4 marks] (b) If her first purchase is brand B, what is the probability that her fifth purchase will be brand B? [4 marks] Data Source: Probability and Statistics M. DeGroot & M. Schervish, Addison-Wesley (2002). 2