GEM2900, 2014/2015 Semester I Exercise sheet 04

Exercise 1.There are 500 students taking GEM2900 this semester. Someone taking GEM2900 is trying toestimate the proportion of 1st year students (freshmen) in the class. A random sample of 60students was drawn and 12 students were freshmen.

(a) Based on this sample, what is your best estimate of the proportion of freshmen takingGEM2900?

(b) Compute and explain a measure of the reliability of this estimate.

Exercise 2.Assume the midterm exam for GEM2900 includes 100 multiple choice questions with five pos-sible responses to each question. Suppose a student did not study and decided to guess theanswer for all questions in a completely random fashion.

(a) What is the expected number of questions that the student guesses correctly?

(b) What is the standard deviation of the number of questions that the student guesses cor-rectly?

Exercise 3.A friend of yours claims to have the ability to read your mind. He asks you to flip a fair coin 5times without showing him the result, recording the sequence of coin flips on a piece of paper(again without showing him the sequence). Then he asks you to concentrate on the sequenceon the piece of paper and he will be able to tell you what the sequence is by reading your mind.

(a) If we assume that your friend has no mind reading ability and that he is just guessing,what is the distribution of the number of correct guesses out of 5?

(b) What is the probability that all 5 guesses will be correct?

Exercise 4.After playing the game in exercise 3 above with you your friend failed to correctly predict all 5coin tosses. He continues to insist that he can read your mind, claiming that he wasn’t “warmedup” yet. He suggests that you play the game with him 10 more times (that is, you generate 10more sequences of 5 coin tosses where for each sequence he must guess the result of all 5 tossescorrectly). What is the chance that he will guess all 5 tosses correctly on at least one of the 10occasions?

Exercise 5.A lecturer administers an exam to his class of 500 GEM2900 students. He drops all the examson the floor so that the answer booklets and the cover sheets identifying the students areseparated, so that he has no way of knowing which student submitted which answer booklet.He decides to pair up the answer books and cover sheets randomly. What is the (approximate)probability that at least one student gets assigned their correct mark (that is, what is thechance that at least one cover sheet and answer booklet are correctly matched)?

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