robability Theory and Statistics Kasper K. erthelsen, Dept. For Mathematical Sciences kkb@math.aau.dk Literature: Walpole, Myers, Myers & Ye: robability and Statistics for Engineers and Scientists, rentice Hall, 8th ed. Slides and lecture overview: http://people.math.aau.dk/~kkb/undervisning/et610/ Lecture format: 2x45 min lecturing followed by exercises in group rooms 1 Lecture1
STTISTICS What is it good for? 90 80 70 60 2 Forecasting: Expectations for the future? How will the stock markets behave?? 50 40 30 20 10 0 1 2 3 4 nalysis of sales: How much do we sell, and when? Should we change or sales strategy? C Quality control: What is my rate of defective products? How can I best manage my production? What is the best way to sample? Lecture1
robability theory Sample space and events Consider an experiment Sample space S: S Event : S VENN DIGRM Example: S{1,2,,6} rolling a dice S{plat,krone} flipping a coin Example: {1,6} when rolling a dice Complementary event : S Example: {2,3,4,5} rolling a dice 3
robability theory Events Example: Rolling a dice S{1,2,3,4,5,6} {2,4,6} {1,2,3} S 5 4 6 2 1 3 Intersection: {2} Union: {1,2,3,4,6} Disjoint events: CD Ø C{1,3,5} and D{2,4,6} are disjoint S C D 4
robability theory Counting sample points Ways of placing your bets: Guess the results of 13 matches ossible outcomes: 1 X 2 Home win 3 possibilities 3 possibilities Draw way win 3 possibilities nswer: 3 3 3 3 3 13 The multiplication rule 5
robability theory Counting sample points Ordering n different objects Number of permutations??? There are n ways of selecting the first object n -1 ways of selecting second object 1 way of selecting the last object n factorial n n -1 1 n! ways The multiplication rule 3! 6 6
robability theory Counting sample points Multiplication rule: If k independent operations can be performed in n 1, n 2,, n k ways, respectively, then the k operations can be performed in n 1 n 2 n k ways Tree diagram: 7 T H T H T H T H T H T H T H Flipping a coin three times Head/Tail 2 3 8 possible outcomes
robability theory Counting sample points Number of possible ways of selecting r objects from a set of n destinct elements: Ordered Unordered n Without replacement r n! n r! n n! r r! n r! With replacement r n - 8
robability theory Counting sample points Example: nn, arry, Chris, and Dan should from a committee consisting of two persons, i.e. unordered without replacement. Number of possible combinations: 4 4! 2 2!2! Writing it out : C D C D CD 6 9
robability theory Counting sample points Example: Select 2 out of 4 different balls ordered and without replacement 4! Number of possible combinations: 4 2 4 2! Notice: Order matters! 12 10
robability theory robability Let be an event, then we denote the probability for S It always hold that 0 < < 1 Ø 0 S 1 Consider an experiment which has N equally likely outcomes, and let exactly n of these events correspond to the event. Then n N # successful outcomes # possible outcomes Example: Rolling a dice even number 3 6 1 2 11
12 robability theory robability Example: Quality control batch of 20 units contains 8 defective units. Select 6 units unordered and without replacement. Event : no defective units in our random sample. 20 Number of possible samples: N # possible 6 12 Number of samples without defective units: n 6 12 6 12!6!14! 77 0.024 20 6!6!20! 3230 6 # successful
robability theory robability Example: continued Event : exactly 2 defective units in our sample Number of samples with exactly 2 defective units: 12 8 n 4 2 12 8 4 2 12!8!6!14! 0.3576 20 4!8!2!6!20! 6 # successful 13
robability theory Rules for probabilities Intersection: + - + If and are disjoint: Union: + In particular: + 1 14
robability theory Conditional probability Conditional probability for given : where > 0 ayes Rule: Rewriting ayes rule: + 15
robability theory Conditional probability Example page 59: The distribution of employed/unemployed amongst men and women in a small town. Employed Unemployed Total Man 460 40 500 Woman 140 260 400 Total 600 300 900 man & employd 460/ 900 460 23 man employed 76.7% employd 600/ 900 600 30 man & unemployed 40/ 900 40 2 man unemployed 13. 3% unemployed 300/ 900 300 15 16
robability theory ayes rule Example: Lung disease & Smoking ccording to The merican Lung ssociation 7% of the population suffers from a lung disease, and 90% of these are smokers. mongst people without any lung disease 25.3% are smokers. Events: robabilities: : person has lung disease 0.07 : person is a smoker 0.90 0.253 What is the probability that at smoker suffers from a lung disease? 0.9 0.07 + 0.9 0.07 + 0.253 0.93 0.211 17
robability theory ayes rule extended version 1,, k are a partitioning of S 1 3 4 5 S 6 Law of total probability: k i 1 i i ayes formel udvidet: 2 18 r k i 1 r i r i
robability theory Independence Definition: Two events and are said to be independent if and only if or lternative Definition: Two events and are said to be independent if and only if Notice: Disjoint event mutually exclusive event are dependent! 19
robability theory Conditional probability Example: Employed Unemployed Total Man 460 40 500 Woman 140 260 400 Total 600 300 900 460/ 900 man employed 76. 7% 600/ 900 man 500/ 900 55. 6% Conclusion: the two events man and employed are dependent. 20
21 robability theory Rules for conditional probabilities C C C C robability of events and happening simultaneously robability of events, and C happening simultaneously roof: C C C C C C 1 3 2 2 1 2 1 k k k k k k General rule: