LEC_EAS305_F11_0909

LEC_EAS305_F11_0909 - Review Permutations and Combinations Some applications of counting techniques Fall 2011 EAS 305 Lecture Notes Prof Jun Zhuang

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Unformatted text preview: Review Permutations and Combinations Some applications of counting techniques Fall 2011 EAS 305 Lecture Notes Prof. Jun Zhuang University at Bu↵alo, State University of New York September 9 ... 2011 Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 1 of 16 Review Permutations and Combinations Some applications of counting techniques Agenda for Today 1 Review Permutations and Combinations 2 Some applications of counting techniques Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 2 of 16 Review Permutations and Combinations Some applications of counting techniques Di↵erence between permutations and combinations Combinations — not concerned w/order: (a, b , c ) = (b , a, c ). Permutations — concerned w/order: (a, b , c ) 6= (b , a, c ). The number of permutations of n things taken r -at-a-time is always as least as large as the number of combinations. In fact,. . . Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 3 of 16 Review Permutations and Combinations Some applications of counting techniques Remark Choosing a permutation is the same as first choosing a combination and then putting the elements in order, i.e., ✓◆ n! n = r! r (n r )! So ✓◆ n! n . = r (n r )!r ! ✓◆ ✓ ◆✓◆ ✓◆ ✓◆ ✓ ◆ n n n n n n = , = = 1, = = n. r nr 0 n 1 n1 Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 4 of 16 Review Permutations and Combinations Some applications of counting techniques Examples for Combination Example: An NBA team has 12 players. How many ways can the coach choose the starting 5? ✓◆ 12! 12 = 792. = 5 5!7! Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 5 of 16 Review Permutations and Combinations Some applications of counting techniques Examples for Combination Example: An NBA team has 12 players. How many ways can the coach choose the starting 5? ✓◆ 12! 12 = 792. = 5 5!7! Example: Smith is one of the players on the team. How many of the 792 starting line-ups include him? ✓◆ 11! 11 = = 330. 4 4!7! (Smith gets one of the five positions for free; there are now 4 left to be filled by the remaining 11 players.) Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 6 of 16 Review Permutations and Combinations Some applications of counting techniques More Example Example: 7 red shoes, 5 blues. Find the number of arrangements. R B R R B B R R R B R B I.e., how ✓ ◆ ways to put 7 reds in 12 slots? many 12 Answer: . 7 Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 7 of 16 Review Permutations and Combinations Some applications of counting techniques Hypergeometric Distribution You have a objects of type 1 and b objects of type 2. Select n objects w/o replacement from the a + b . Pr(k type 1’s were picked) (# ways to choose k 1’s)(choose n k 2’s) = # ways to choose n out of a + b ✓ ◆✓ ◆ a b k nk ✓ ◆ = (the hypergeometric distr’n). a+b n Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 8 of 16 Review Permutations and Combinations Some applications of counting techniques Example Example: 25 sox in a box. 15 red, 10 blue. Pick 7 w/o replacement. ✓ ◆✓ ◆ 15 10 3 4 ✓◆ Pr(exactly 3 reds are picked) = 25 7 Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 9 of 16 Review Permutations and Combinations Some applications of counting techniques Permutations vs. Combinations Permutations vs. Combinations — It’s all how you approach the problem! Example: 4 red marbles, 2 whites. Put them in a row in random order. Find. . . (a) Pr(2 end marbles are W) (b) Pr(2 end marbles aren’t both W) (c) Pr(2 W’s are side by side) Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 10 of 16 Review Permutations and Combinations Some applications of counting techniques Method 1 (using permutations): Let the sample space S = {every random ordering of the 6 marbles}. (a) A: 2 end marbles are W — WRRRRW. |A| = 2!4! = 48 ) Pr(A) = ¯ (b) Pr(A) = 1 |A| 48 1 = = . |S | 720 15 Pr(A) = 14/15. Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 11 of 16 Review Permutations and Combinations Some applications of counting techniques (c) B : 2 W’s side by side — WWRRRR or RWWRRR or . . . or RRRRWW |B | = (# ways to select pair of slots for 2 W’s) ⇥(# ways to insert W’s into pair of slots) ⇥(# ways to insert R’s into remaining slots) = 5 ⇥ 2! ⇥ 4! = 240. Pr(B ) = |B | 240 1 = =. |S | 720 3 But — The above method took too much time! Here’s an easier way. . . Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 12 of 16 Review Permutations and Combinations Some applications of counting techniques Method 2 (using combinations): Which 2 positions do the W’s occupy? Now let S = {possible pairs of slots that the W’s occupy}. ✓◆ 6 Clearly, |S | = = 15. 2 (a) Since the W’s must occupy the end slots in order for A to occur, |A| = 1 ) Pr(A) = |A|/|S | = 1/15. ¯ (b) Pr(A) = 14/15. (c) |B | = 5 ) Pr(B ) = 5/15 = 1/3. Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 13 of 16 Review Permutations and Combinations Some applications of counting techniques Birthday Problem n people in a room. Find the prob that at least two have the same birthday. (Ignore Feb. 29, and assume that all 365 days have equal prob.) A: All birthdays are di↵erent. S = {(x1 , . . . , xn ) : xi = 1, 2, . . . , 365} (xi is person i ’s birthday), and note that |S | = (365)n . |A| = P365,n = (365)(364) · · · (365 n + 1) (365)(364) · · · (365 n + 1) (365)n 364 363 365 n + 1 = 1· · ··· 365 365 365 Pr(A) = Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 14 of 16 Review Permutations and Combinations Some applications of counting techniques We want ¯ Pr(A) = 1 ✓ 364 363 365 n + 1 1· · ··· 365 365 365 ¯ Notes: When n = 366, Pr(A) = 1. ¯ For Pr(A) to be > 1/2, n must be ¯ When n = 50, Pr(A) = 0.97. Prof. Jun Zhuang ◆ 23. (surprising) Fall 2011 EAS 305 Lecture Notes Page 15 of 16 Review Permutations and Combinations Some applications of counting techniques Multinomial Coe cients Example: n1 blue sox, n2 reds. # of assortments is ✓ n1 + n2 n1 (binomial coe cients). P Generalization (for k types of objects): n = k=1 ni i # of arrangements is n1 !n2n!!···nk ! . Example: How many ways can “Mississippi” be arranged? ◆ # perm’s of 11 letters 11! = = 34, 650. (# m’s)!(# p ’s)!(# i ’s)!(# s ’s)! 1!2!4!4! Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 16 of 16 ...
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This note was uploaded on 09/25/2011 for the course ECON 101 taught by Professor Wang during the Spring '11 term at SUNY Buffalo.

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