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Revision Exercise - SMSL/07 THE UNIVERSITY OF HONG KONG...

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SMSL/07 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability and Statistics I Revision Exercises A Chapters 1-5 A1. In an experiment k balls are drawn randomly without replacement from an urn containing 1 red ball, 2 blue balls and 17 yellow balls. All the balls are identical apart from their colours. (a) Suppose k 17. Show how you may obtain the number of distinct combinations of the k balls by considering the expression (1 + x )(1 + x + x 2 )(1 - x ) - 1 (1 - x 2 )(1 - x 3 )(1 - x ) - 3 . (b) Show that there are six distinct ways of drawing three balls randomly without replacement from the urn. A2. A Democrat, a Republican, a Communist and a Socialist run for President of the Cabinet. A ballot is held for this matter and twelve ministers cast their votes anonymously. (a) Calculate the number of distinct vote distributions among the four candidates. (b) What is the probability that the Democrat secures all the twelve votes, (i) if each distinct vote distribution is equally likely? (ii) if the ministers cast their votes at random independently? A3. A total of r identical robots are being lifted from the ground floor to the n upper floors of a building. (a) Show that there are n + r - 1 r distinct exit patterns. (b) If each robot is independently and equally probably allocated to each of the n floors, then the probability that all robots end up on the first floor is given by Number of ways of sending all robots to 1/F Number of distinct exit patterns = n + r - 1 r - 1 . Comment on the above statement. What should the true answer be if you think it is wrong? 1
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A4. A basket contains 4 bananas, 4 oranges and 4 apples, among which four items are to be selected without replacement. Objects of the same type are assumed to be indistinguishable . Let m be the number of different ways of selecting the four items. (a) Explain why m is equal to the number of distinct unordered selections of 4 items with replacement from a basket containing 1 banana, 1 orange and 1 apple. (b) Determine m . (c) Suppose that all different combinations of the 4 selected items are equally likely. Calculate the probability that at least 1 banana, 1 orange and 1 apple are selected. A5. (a) Let B 1 B 2 ⊃ · · · be a decreasing sequence of events, and B = T n =1 B n . Prove that lim n →∞ P ( B n ) = P ( B ) . [ Hint: You may assume the probabilistic axiom that for any increasing sequence of events A 1 A 2 ⊂ · · · , we have lim n →∞ P ( A n ) = P ( [ n =1 A n ) . ] (b) Consider the following experiment. On day n , a number is drawn from the set { 1 , 2 , . . . , n } , independent of results from previous days, such that P ( n is drawn on day n ) = exp ( - 1 /n !) , P ( j is drawn on day n ) = 1 n - 1 { 1 - exp ( - 1 /n !) } , j = 1 , 2 , . . . , n - 1 , ) for n 2, and P (1 is drawn on day 1) = 1.
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