3/15/2010 ENGR 200. Counting

Transcription

1 ENGR 200 Counting 1

2 Are these events conditionally independent? Blue coin: P(H = 0.99 Red coin: P(H = 0.01 Pick a random coin, toss it twice. H1 = { 1 st toss is heads } H2 = { 2 nd toss is heads } given B = { the blue coin was selected } Conditional Independence does not imply Independence P(H1 H2 B = P(H1 BP(H2 B = 0.99 x 0.99 (implied by cond. independence P(H1 H2 B = P(H1 H2 BP(H2 B =.99x.99 (computed using the chain rule Now lets see if H1 and H2 are independent P(H1=P(BP(H1 B+P(RP(H1 R=½ P(H2=P(BP(H2 B+P(RP(H2 R=½ P(H1 H2 =.99x.99x.5+.01x.01x.5 P(H1*P(H2=¼ 2

3 Independence of Multiple Events A1, A2,..., A P( A is i n is are independent if P( A for every subset S of {1,2,...,n} i Example: Three Events P( A 1 P( A 1 P( A 2 P( A 1 A A 2 3 A A 3 2 P( A A 3 1 P( A 1 P( A P( A P( A 2 P( A P( A P( A Pairwise Independence 2 P( A 3 3

4 Are these independent? Consider two fair coin tosses: H1 = { 1 st toss is a head } H2 = { 2 nd toss is a head } D = { the two tosses are different } Pairwise Independence does not Imply Independence Consider two fair coin tosses: H1 = { 1 st toss is a head } H2 = { 2 nd toss is a head } D = { the two tosses are different } P(D H1 = P(H1 D/P(H1 = ½ = P(D P(D H2 = P(H2 D/P(H2 = ½ = P(D P(H1 H2 = P(H1 P(H2 => Pairwise independence is satisfied P(H1 H2 D = 0 P(H1P(H2P(D = ½ x ½ x ½ = 1/8 => They are not independent 4

5 Are these independent? Consider two rolls of a die. A = { 1 st roll is 1, 2, or 3 } B = { 1 st roll is 3, 4, or 5 } C = { the sum of the two rolls is 9 } P(A B C=P(AP(BP(C does not show independence A = { 1 st roll is 1, 2, or 3 } P(A = ½ B = { 1 st roll is 3, 4, or 5 } P(B = ½ C = { the sum is 9 } P(C = 4/36 P(A B C=P(AP(BP(C But they are not independent!!! (Check the pairwise conditions 5

6 Summary P(A B = P(A means independence. Disjoint events are not independent. conditional independence P(A B C = P(A C P(B C P(A B C = P(A C P(B A C = P(B C Conditional independence does not imply independence and vice versa. For multiple events independence means: Any number of events carry no information about the others. Counting When the sample space W has a finite number of equally likely outcomes: P(A # of # of elements in A elements in 6

7 The counting principle Tree diagram If there are n k, possible results at stage k, k=1,...,r. Then the total number of possible results in the r-stage process is: n 1 n 2... n r. Example: # of telephone numbers Phone number: 7-digit sequence First digit cannot be a 1 or 0 How many distinct phone numbers? 7

8 Example: # of subsets of n-element set S={s 1, s 2,... s n } How many subsets does it have (including itself and empty set? Cheat Sheet Permutations of n objects: n! k-permutations of n objects: n!/(n-k! Combinations of k out of n objects: n!/(k!(n-k! Partitions of n objects into r groups, with the i th group having n i objects: n!/(n 1!n 2!...n r! 8

9 Permutations We have n distinct objects, where one of each kind of objects is available. How many different sequences of k of these can be formed? P kn = n (n-1 (n-2.. (n-k+2 (n-k+1 For P kn to be useful the order of objects should be of interest. 4 Letter Words How many words can you make up that consist of four distinct letters? ABCD ABCE ZYXW 26*25*24*23 9

10 Example n 1 : # of classical CDs n 2 : # of rock CDs n 3 : # of folk music CDs You keep all CDs of one kind together, otherwise you have no constraints. In how many ways can you arrange them? n 1!*n 2!*n 3!*3! Combinations We have n distinct objects. In how many different ways can you select k of these objects? Note that the sequence of objects is no longer important. C k P n k, n n!, n k k! k!( n k! In other words, any set of k objects out of n objects counts as a combination. 10

11 Grade of service An ISP has c modems and n customers. At a given time each customer needs a connection with probability p. What is the Pr he will get one? How many possible relevant outcomes are there? C(n,0 + C(n,1 +. +C(n,c What are their probabilities? C(n,0*p^(0*(1-p^(n-0 + C(n,1*p^(1*(1-p^(n C(n,c*p^(c*(1-p^(n-c Partitioning A combination is partition of a set in two (one part containing k elements and the other part containing n-k elements Given a set of n elements we would like to partition it to r sets consisting of n 1, n 2,n r elements where n 1 + n 2 +n r =n. The total number of such choices is: n n n1 n n1n 2 n n1 nr 1 n1 n2 n3 nr 11

12 Partitioning and the multinomial coefficient n n n1 n n1n 2 n n1 nr 1 n! n1 n2 n3 nr n1! n2!... nr! This is called the multinomial coefficient and is written as: n n1, n2, nr Splitting into groups A class consisting of 4 graduate and 12 undergraduate students is randomly divided into 4 groups of 4. What is the P that each group includes a graduate student. How many groupings are there? (16 4,4,4,4 How may are relevant? 4! (12 3,3,3,3 So the probability is 4! (12 3,3,3,3 / (16 4,4,4,4 12

13 Anagrams How many different letter sequences can be obtained by arranging the letters in the word TATTOO? partition the letter positions. 6!/(3!2!1! = 60 first consider each letter distinct, then discount the rearrangements 6! / (3!2!1! Summary of Chapter 1 Three methods for calculating probabilities: a The counting method (all outcomes are equally likely b The sequential method (trees, and conditional probabilities on the branches of trees c The divide-and-conquer method (TPT (P(B is found by conditioning on different events A i, where {A i } form a partition of the sample space 13

14 Example From a group of 5 women and 7 men, how many different committees consisting of 2 women and 3 men can be formed? c(5,2*c(7,3 What if two of the men refuse to serve on the committee together? c(5,2*( c(7,3 c(5,3 - c(5,2 - c(5,2 Example A production lot of size 100 is known to have 5 defective items. A random sample of 10 items is selected without replacement. (i What is the probability that the sample has no defectives? (ii What is the probability that the sample has all the defectives? 14

15 Example A president, treasurer, and secretary, all different, are to be chosen from a club consisting of 10 people. How many different choices of officers are possible if: There are no restrictions; A and B will not serve together; C and D will serve together or not at all; E must be an officer; F will serve only if he is the president. Example Consider 3 classes, each consisting of n students. From this group of 3n students, a group of 3 students is to be chosen. How many choices are possible? How many choices are there in which all 3 students are in the same class? How many choices are there in which 2 of 3 students are in the same class, and 1 from another? How many choices are there in which all 3 students are in different classes? 15

16 Example 15 telephones (5 cellular, 5 cordless, 5 corded have just been received at an authorized service center. What is the probability that all the cordless phones are among the first 10 to be serviced? What is the probability that after servicing 10 of these phones, phones of only 2 of the 3 types remain to be serviced? What is the probability that two phones of each type are among the first 6 serviced? Example 10 children are to be divided into an A team and a B team of 5 each. The A team will play in one league and the B team in another. How many different divisions are possible? If the two teams are going to play against each other,...? 16

17 A and B, A or B If A and B are any two events: P(A and B = unspecified P(A or B = P(A + P(B P(A and B If A and B are disjoint events: P(A and B = 0 P(A or B = P(A + P(B If A and B are independent events: P(A and B = P(A P(B P(A or B = P(A + P(B P(AP(B = 1 P(A c P(B c 17