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02. Combination & Permutation

C P H 郭俊利 2009/03/08. 02. Combination & Permutation. 1.5 ~ 2.2. Outline. Review Sets and probability axioms Combination Permutation Binomial formula Random variable Homework 1. Example 1 (1/2). Let E, F, and G be three events (E − EF) ∪ F = E ∪ F ?

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02. Combination & Permutation

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  1. C P H 郭俊利 2009/03/08 02. Combination & Permutation

  2. 1.5 ~ 2.2 Outline • Review • Sets and probability axioms • Combination • Permutation • Binomial formula • Random variable • Homework 1

  3. Example 1 (1/2) • Let E, F, and G be three events • (E − EF) ∪ F = E ∪ F ? • FcG ∪ EcG = G(F ∪ E)c? • EF ∪ EG ∪ FG ⊂ E ∪ F ∪ G ?

  4. Example 1 (2/2) • (E − EF) ∪ F • FcG ∪ EcG • EF ∪ EG ∪ FG E F G

  5. n! k! (n - k)! Combination • Cnk = (nk) = • n choose c • The number of n chosen c • Example: C42

  6. Example 2 • Full house • Two pairs • Flush • P(A∩B∩C) = P(A) P(B|A) P(C|A∩B) (Last week)

  7. (n – 1) ! m! (n – 1 – m) ! (n – 1) ! (m – 1) ! (n – m) ! = + Pascal Law • Cnm = Cn-1m + Cn-1m-1 . . . • Cnm Cn-1m-1 × 2 • C63 C52 × 2 = 20 ≒ ≒

  8. 7! 2! 3! 2! Permutation • Pnm = Cnm m! • Example: P42 • Example: Different

  9. Partition • 10 students are classified into 3 groups by the rule (3, 3, 4) A B C 10 3, 3, 4 10 ! 3! 3! 4! C103 C73 C44 ( ) =

  10. Example 3 • 10 students are classified into 3 groups by the rule (3, 3, 4) A3 A3 B4 1 2 ! 10 3, 3, 4 ( )

  11. H • Hnm = Cn+m-1m • Example: H34

  12. Example 4 • From the set of integers {1, 2, 3,. . . , 100000} a number is selected at random. What is the probability that the sum of its digits is 8? Hint: Establish a one-to-one correspondence between the set of integers from {1, 2,. . . ,100000} the sum of whose digits is 8, and the set of possible ways 8 identical objects can be placed into 5 distinguishable cells. • Advanced thinking: If the sum is 11, what is the probability?

  13. Binomial Formula (1/2) • To roll a dice five times • The probability: Number 6 is shown twice • The probability: Even is shown twice p(k) = Cnk pk (1 – p)k p is small letter, not Permutation C52 (1/6)2 (5/6)3

  14. Binomial Formula (2/2) • (x + 1)2 = x2 + 2x + 1 • (x + 1)3 = x3 + 3x2 + 3x +1 • (x + 1)n = Cnnxn + … + Cn1x1 + Cn0x0 If x = 1, … Example: ΣC10k = ? 10 K = 1

  15. 36 Random Variable • Binomial Random Variable • The random variable is a real-valued function of the outcome of the experiment. p(x) = Cnx px (1 – p)x p(x) 6 5 4 3 2 1 The sum of two dices is x, what is p(x) ? x 1 2 3 4 … 7 … 11 12

  16. Expected Value • Example: • The expectation of throwing a dice is 3.5 • The answer to a question is 80% correctly, the grade may be 80. Expectation E[X] = Σ x p(x) p(x) > 0

  17. Example 5 • The shooting average of A is 2/3 The shooting average of B is 3/4 The shooting average of C is 4/5 (1) P (at least one hit) = (2) P (one hit) = P (two hits) = P (three hits) = (3) (A hit | one hit) = (4) E (how many hits) =

  18. Example 6 • x is the x-th getting a white ball. p(x) = ? • p(1) = P (1st = W) = 5/8 • p(2) = P (2nd = W) = • p(3) = P (3rd = W) =

  19. Problems • Chapter 1 • 1 ☆ (simple to trivial) • 9 ☆☆ • 10 ☆☆ (disjoint characteristic) • 14 ☆☆ (last week) • 22 ☆☆☆☆ (like as example 6) • 26 ☆☆☆☆☆ • 31 ☆☆☆

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