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Chap 2 Combinatorial Methods Ghahramani 3rd edition

Chap 2 Combinatorial Methods Ghahramani 3rd edition. Outline. 2.1 Introduction 2.2 Counting principle 2.3 Permutations 2.4 Combinations 2.5 Stirling ’ s formula. 2.1 Introduction. If the sample space is finite and furthermore sample points are all equally likely, then

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Chap 2 Combinatorial Methods Ghahramani 3rd edition

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  1. Chap 2 Combinatorial MethodsGhahramani 3rd edition

  2. Outline 2.1 Introduction 2.2 Counting principle 2.3 Permutations 2.4 Combinations 2.5 Stirling’s formula

  3. 2.1 Introduction • If the sample space is finite and furthermore sample points are all equally likely, then P(A)=N(A)/N(S) So we study combinatorial analysis here, which deals with methods of counting.

  4. 2.2 Counting principle • Ex 2.1 How many outcomes are there if we throw 5 dice? • Ex 2.2 In tossing 4 fair dice, P(at least one 3 among these 4 dice)=? • Ex 2.3 Virginia wants to give her son, Brian, 14 different baseball cards within a 7-day period. If Virginia gives Brian cards no more than once a day, in how many way can this be done? • Ex 2.6(Standard Birthday Problem) P(at least two among n people have the same Bday)=?

  5. Counting principle • Thm 2.3 A set with n elements has 2n subsets. • Ex 2.9 Mark has $4. He decides to bet $1 on the flip of a fair coin 4 times. What is the probability that (a) he breaks even; (b) he wins money?(use tree diagram)

  6. 2.3 Permutations • Ex 2.10 3 people, Brown, Smith, and Jones, must be scheduled for job interviews. In how many different orders can this be done? • Ex 2.11 2 anthropology, 4 computer science, 3 statistics, 3 biology, and 5 music books are put on a bookshelf with a random arrangement. What is the probability that the books of the same subject are together?

  7. Permutations • Ex 2.12 If 5 boys and 5 girls sit in a row in a random order, P(no two children of the same sex sit together)=? • Thm 2.4 The number of distinguishable permutations of n objects of k different types, where n1 are alike, n2 are alike, …, nk are alike and n=n1+n2+…+nk is

  8. Permutations • Ex 2.13 How many different 10-letter codes can be made using 3 a’s, 4 b’s, and 3 c’s? • Ex 2.14 In how many ways can we paint 11 offices so that 4 of them will be painted green, 3 yellow, 2 white, and the remaining 2 pink? • Ex 2.15 A fair coin is flipped 10 times. P(exactly 3 heads)=?

  9. 2.4 Combinations • Ex 2.16 In how many ways can 2 math and 3 biology books be selected from 8 math and 6 biology books? • Ex 2.17 45 instructors were selected randomly to ask whether they are happy with their teaching loads. The response of 32 were negative. If Drs. Smith, Brown, and Jones were among those questioned. P(all 3 gave negative responses)=?

  10. Combinations • Ex 2.18 In a small town, 11 of the 25 schoolteachers are against abortion, 8 are for abortion, and the rest are indifferent. A random sample of 5 schoolteachers is selected for an interview. (a)P(all 5 are for abortion)=? (b)P(all 5 have the same opinion)=? • Ex 2.19 In Maryland’s lottery, player pick 6 integers between 1 and 49, order of selection being irrelevant. P(grand prize)=? P(2nd prize)=? P(3rd prize)=?

  11. Combinations • Ex 2.20 7 cards are drawn from 52 without replacement. P(at least one of the cards is a king)=? • Ex 2.21 5 cards are drawn from 52. P(full house)=? • Ex 2.24 A professor wrote n letters and sealed them in envelopes. P(at least one letter was addressed correctly)=? Hint: Let Ei be the event that ith letter is addressed correctly. Compute P(E1U…UEn) by inclusion-exclusion principle.

  12. Combinations • Thm 2.5 (Binomial expansion) • Ex 2.25 What is the coefficient of x2y3 in the expansion of (2x+3y)5? • Ex 2.26 Evaluate the sum

  13. Combinations • Ex 2.27 Evaluate the sum • Ex 2.28 Prove that

  14. Combinations • Ex 2.29 Prove the inclusion-exclusion principle. • Ex 2.30 Distribute n distinguishable balls into k distinguishable cells so that n1 balls are distributed into the first cell, n2 balls into the second cell, …, nk balls into the kth cell, where n1+n2+…+nk=n. How many possible ways? Sol:

  15. Combinations • Thm 2.6 (Multinomial expansion).

  16. 2.5 Stirling’s formula

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