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stat1801_1 - Probability deductive reasoning population P H...

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Probability population sample deductive reasoning ( 29 ( 29 5 . 0 = = T P H P ten times ( 29 ( 29 10 5 . 0 heads all = P
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Statistical Inference population sample inductive reasoning ( 29 ( 29 ? = = T P H P ten times All heads. Is it a fair coin?
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Classical Probability N equally probable possible outcomes …… f satisfy event E ( 29 N f E P = ( 29 25 . 0 52 13 heart = = P ( 29 6 1 36 6 dice rolled on two 7 of total = = P ( 29 ( 29 ( 29 n n P n P 365 365 365 365 1 365 364 365 birthdays different 365 = × × × + - × × × =
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Counting Procedures Example: Possible outcomes of tossing three coins H T HH HT TH TT HHH HHT HTH HTT THH THT TTH TTT 8 2 2 2 = × × = N step 1 in ways st 1 n step 2 in ways nd 2 n step k in ways th k n steps in ways 2 1 k n n n k
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Permutation A specific arrangement of objects in a definite order. Example: possible arrangements of three persons in a row A B C AB AC BA BC CA CB ABC ACB BAC BCA CAB CBA 6 1 2 3 = × × = N ( 29 ( 29 ( 29 ! 1 2 1 objects distinct of ns permutatio of No. n n n n = - =
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Permutation 6 objects: A B C D E F a row round table bracelet 6! = 720 A B C D E F A B D C F E (6-1)! = 120 A B D C F E (6-1)!/2 = 60 A F D E B C
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Permutation n distinct objects pick r objects and arrange ( 29 ( 29 ( 29 ! ! 1 1 : subset ordered of No. r n n r n n n P r n - = + - - = Example: Tierce outcomes in horse race with 14 horses 5 6 14 7 1 2 1 2 7 2184 ! 11 ! 14 3 14 = = = P N
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Permutation Objects not all distinct L E T T E R Treat as distinct L E 1 T 1 T 2 E 2 R 7! permutations Duplicated permutations L E 2 T 2 T 1 E 1 R 2! × 2! 1260 ! 2 ! 2 ! 7 ns permutatio of no. = = ! ! ! ! ... 2 1 2 1 r r n n n n n n n n = In general for n 1 type 1 objects, … n r type r objects, with n = n 1 + n 2 +…+ n r , the number of permutations is
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Combination A subset of r objects from n distinct objects, without regard to order. Example: Choose three letters from {A, B, C, D} Combination Permutations {A, B, C} (A,B,C),(A,C,B),(B,A,C),(B,C,A),(C,A,B),(C,B,A) {A, B, D} (A,B,D),(A,D,B),(D,A,B),(D,B,A),(B,A,D),(B,D,A) {A, C, D} (A,C,D),(A,D,C),(C,A,D),(C,D,A),(D,A,C),(D,C,A) {B, C, D} (B,C,D),(B,D,C),(C,B,D),(C,D,B),(D,B,C),(D,C,B) ( 29 24 1 24 ! 3 4 ! 4 3 4 = = - = P 4 6 24 ! 3 3 4 3 4 = = = P C ( 29 ! ! ! ! r n r n r P C r n r n - = =
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Combination Example: 6 numbers chosen from {1, 2, ….., 49} 13983816 ! 43 ! 6 ! 49 6 49 = = = C N Example: draw five poker cards, P (full house) = ? Full house: 2598960 : outcomes possible of no. 5 52 = = C N 3744 2 : houses full possible of no. 2 4 3 4 2 13 = × × × = C C C f ( 29 00144 . 0 2598960 3744 house full = = P
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Combination Example: Second law of thermodynamics open stopcock Total no. of possible outcomes 30 100 10 267 . 1 2 × = No. of possible outcomes with 30 molecules on left 25 30 100 10 937 . 2 × = C 100 molecules P (30 molecules) = ?
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