Textbooks (required) course in probability
- 1. Textbooks (required): A First course in Probability (8th Ed), Sheldon Ross
- 2. Chapter(1) Combinatorial Analysis Note: This PowerPoint is only a summary and your main source should be the book.
- 3. 1.1 Introduction 1.2 The Basic Principle of Counting 1.3 Permutations 1.4 Combinations 1.5 Multinomial Coefficients Problems Theoretical Exercises Self-Test Problems and Exercises Contents
- 4. Suppose that two experiments are to be performed. Then if experiment 1 can result in any one of m possible outcomes and if for each outcome of experiment 1 there are n possible outcomes of experiment 2, then together there are mn possible outcomes of the two experiments. 1.2 The Basic Principle Of Counting
- 5. EXAMPLE 2a A small community consists of 10 women, each of whom has 3 children. If one woman and one of her children are to be chosen as mother and child of the year, how many different choices are possible? Solution. The choice of the woman as the outcome of the first experiment and the subsequent choice of one of her children as the outcome of the second experiment, we see from the basic principle that there are 10 * 3 = 30 possible choices.
- 6. The generalized basic principle of counting If r experiments that are to be performed are such that the first one may result in any of n1 possible outcomes; and if, for each of these n1 possible outcomes, there are n2 possible outcomes of the second experiment; and if, for each of the possible outcomes of the first two experiments, there are n3 possible outcomes of the third experiment; and if . . . , then there is a total of (n1 · n2 · · · nr) possible outcomes of the r experiments.
- 7. EXAMPLE 2b A college planning committee consists of 3 freshmen, 4 sophomores, 5 juniors, and 2 seniors. A subcommittee of 4, consisting of 1 person from each class, is to be chosen. How many different subcommittees are possible? Solution. We may regard the choice of a subcommittee as the combined outcome of the four separate experiments of choosing a single representative from each of the classes. It then follows from the generalized version of the basic principle that there are 3 * 4 * 5 * 2 = 120 possible subcommittees.
- 8. EXAMPLE 2c How many different 7-place license plates are possible if the first 3 places are to be occupied by letters and the final 4 by numbers? Solution. By the generalized version of the basic principle, the answer is 26 · 26 · 26 · 10 · 10 · 10 · 10 = 175,760,000
- 9. EXAMPLE 2e In Example 2c, how many license plates would be possible if repetition among letters or numbers were prohibited? Solution. In this case, there would be 26 · 25 · 24 · 10 · 9 · 8 · 7 = 78,624,000 possible license plates.
- 10. 1.3 Permutations There are 6 ordered arrangements of the three letters, a,b, and c. These are abc, acb, bac, bca, cab, cba. Each arrangement is called a permutation. Suppose now that we have n objects. Reasoning similar to that we have just used for the 3 letters then shows that there are n(n − 1)(n − 2) · · · 3 · 2 · 1 = n! different permutations of the n objects.
- 11. EXAMPLE 3a How many different batting orders are possible for a baseball team consisting of 9 players? Solution. There are 9! = 362,880 possible batting orders.
- 12. EXAMPLE 3b A class in probability theory consists of 6 men and 4 women. An examination is given, and the students are ranked according to their performance. Assume that no two students obtain the same score. (a) How many different rankings are possible? (b) If the men are ranked just among themselves and the women just among themselves, how many different rankings are possible? Solution. (a) Because each ranking corresponds to a particular ordered arrangement of the 10 people, the answer to this part is 10! = 3,628,800. (b) Since there are 6! possible rankings of the men among themselves and 4! possible rankings of the women among themselves, it follows from the basic principle that there are (6!)(4!) = (720)(24) = 17,280 possible rankings in this case.
- 13. EXAMPLE 3c Ms. Jones has 10 books that she is going to put on her bookshelf. Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book. Ms. Jones wants to arrange her books so that all the books dealing with the same subject are together on the shelf. How many different arrangements are possible? Solution. There are 4! 3! 2! 1! arrangements such that the mathematics books are first in line, then the chemistry books, then the history books, and then the language book. Similarly, for each possible ordering of the subjects, there are 4! 3! 2! 1! Possible arrangements. Hence, as there are 4! possible orderings of the subjects, the desired answer is 4! 4! 3! 2! 1! = 6912.
- 14. • Now, consider the case when some of the objects are indistinguishable from each other. In this case, there are different permutations of n objects, of which n1 are alike, n2 are alike, …,nr are alike. Example 3d: How many different letter arrangements can be found by using the letters PEPPER? Solution: =60
- 15. Example 3d: A chess tournament has 10 competitors of which 4 are Russian, 3 are from USA, 2 from UK, and 1 from Brazil. If the tournament result lists just nationalities of the players in the order in which they places, how many outcomes are possible? Solution: There are =12 600 possible outcomes.
- 16. 1.3 Combinations We are often interested in determining the number of different groups of r objects that could be formed from a total of n objects. In general, as n(n-1) …(n-r+1) represents the number of different ways that a group of r items could be selected from n items when the order of selection is relevant, and each group of r items will be counted r! times in this count, it follows that the number of different groups of r items that could be formed from a set of n items is given by the below formulation
- 17. EXAMPLE 4a A committee of 3 is to be formed from a group of 20 people. How many different committees are possible? Solution.
- 18. EXAMPLE 4b From a group of 5 women and 7 men, how many different committees consisting of 2 women and 3 men can be formed? What if 2 of the men are feuding and refuse to serve on the committee together?
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- 20. A useful combinatorial identity is
- 21. EXAMPLE 4d Expand (x + y) 3 . Solution.
- 23. 1.5 Multinomial Coefficients Here, we consider the following problem: A set of n distinct items is to be divided into r distinct groups of respective sizes n1,n2,…,nr, where How many different divisions are possible?
- 24. A police department in a small city consists of 10 officers. If the department policy is to have 5 of the officers patrolling the streets, 2 of the officers working full time at the station, and 3 of the officers on reserve at the station, how many different divisions of the 10 officers into the 3 groups are possible? EXAMPLE 5a Solution. There are 10! / (3! * 2! * 5)= 2520 possible divisions