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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 1 Review Combinatorial Analysis (a) Multiplication Principle (b) Selection of r from n distinct objects: With replacement Without replacement Ordered n r n P r = n ! ( n- r )! Unordered ( n + r- 1 r ) = ( n + r- 1)! r ! × ( n- 1)! ( n r ) = n ! r ! × ( n- r )! (c) Arrangement of n objects, with r distinct types. In other words, there are n 1 objects of type 1, n 2 objects of type 2, ..., n r objects of type r . (e.g. number of different letter arrangements can be formed using the letters STATISTICS.) n ! n 1 ! × n 2 ! × ··· × n r ! (d) Partition of n distinct objects into r distinct groups with specified size n 1 ,n 2 , ··· ,n r . (e.g. number of ways in dividing a class of 40 into groups of 10, 10 and 20.) n n 1 ,n 2 , ··· ,n r = n ! n 1 ! × n 2 ! × ··· × n r ! (e) Partition of n indistinguishable objects into r distinct groups (i.e. we only concern the number of objects in each group). n + r- 1 n = ( n + r- 1)! n ! × ( r- 1)! Problems Problem 1. A product code of manufacturer consists of 2 letters followed by 3 digits. The man- ufacturer only uses the letters A, B, C and D. For example, AA032 and BD369 are valid product codes. If a product is selected at random, what is the probability thatcodes....
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