02_02 - Example (The Matching Problem). Applications: (a)...

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p. 2-11 Example (The Matching Problem). Applications: (a) Taste Testing. (b) Gift Exchange. Let be all permutations = ( i 1 , …, i n ) of 1, 2, …, n . Thus, # = n !. Let A j = { : i j = j } and , Q : P ( A )=? (What would you expect when n is large?) By symmetry, Here, σ k = n k P ( A 1 ··· A k ) , A = n i =1 A i for k = 1, …, n . p. 2-12 Proportion: If A 1 , A 2 , …, is a partition of , i.e., 1. 2. A 1 , A 2 , …, are mutually exclusive, then, for any event A , P ( A )= i =1 P ( A A i ) . i =1 A i = , So, Note : approximation accurate to 3 decimal places if n 6. P ( A σ 1 σ 2 + +( 1) n +1 σ n = n k =1 ( 1) k +1 1 k ! , P ( A )=1 n k =0 ( 1) k 1 k ! 1 1 e P ( A c ) e 1
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p. 2-13 • Monotone Sequences Q : How to define probability in a continuous sample space? Definition: A sequence of events A 1 , A 2 , …, is called increasing if and decreasing if The limit of an increasing sequence is defined as and the limit of an decreasing sequence is A 1 A 2 ··· A n A n +1 A
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02_02 - Example (The Matching Problem). Applications: (a)...

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