APPLIED STATISTICS. Lecture 2 Introduction to Probability. Petr Nazarov Microarray Center

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1 Microarray Center LIED STTISTICS Lecture 2 Introduction to robability etr Nazarov petr.nazarov@crp-sante.lu Lecture 2. Introduction to probability

2 OUTLINE Lecture 2 Experiments, counting rules and assigning probabilities experiments counting rules, combinations, permutations assigning probabilities Events and ir probabilities event complement event intersection two events addition law Conditional probability conditional probability independent events multiplication law Bayes orem orem Bayes and it s application tabular approach Lecture 2. Introduction to probability 2

3 EXERIMENTS, COUNTING RULES ND ROBBILITIES Experiments Experiment Experiment process process that that generates generates well-defined well-defined outcomes outcomes Sample Sample point point n n element element sample sample space. space. sample sample point point represents represents an an experimental experimental outcome outcome Sample Sample space space The The set set all all experimental experimental outcomes. outcomes. Lecture 2. Introduction to probability 3

4 EXERIMENTS, COUNTING RULES ND ROBBILITIES robability robability numerical measure likelihood that that an an event will will occur. Lecture 2. Introduction to probability 4

5 EXERIMENTS, COUNTING RULES ND ROBBILITIES Counting Rules Multiple-step experiments Consider experiment tossing two coins. Let experimental outcomes be defined in terms pattern heads and tails appearing on upward faces two coins. How many experimental outcomes are possible for this experiment? The experiment tossing two coins can be thought as a two-step experiment in which step 1 is tossing first coin and step 2 is tossing second coin. Counting Counting rule rule for for a multi-step multi-step experiment experiment If If an an experiment experiment can can be be described described as as a sequence sequence kksteps steps with with n 1 possible 1 possible outcomes outcomes on on first first step, step, n 2 possible 2 possible outcomes outcomes on on second second step, step, and and so so on, on, n n total total number number experimental experimental outcomes outcomes is is given given by by n 1 n 1 n n k. k. How How many many outcomes outcomes can can appear appear after after 6 tossing tossing a coin? coin? Lecture 2. Introduction to probability 5

6 EXERIMENTS, COUNTING RULES ND ROBBILITIES Counting Rule 1 Tree Tree diagram diagram graphical graphical representation representation that that helps helps in in visualizing visualizing a multiple-step multiple-step experiment. experiment. Without Without tree tree diagram, diagram, one one might might think think only only three three experimental experimental outcomes outcomes are are possible possible for for two two tosses tosses a coin coin 0 heads, heads, 1 head, head, and and 2 heads heads Lecture 2. Introduction to probability 6

7 EXERIMENTS, COUNTING RULES ND ROBBILITIES To test state population, let s select 2 for a routine check-up Counting Rule 2 Mouse population: 5 mice How How many many variants variants such such a selection selection exits? exits? Counting Counting rule rule for for combinations combinations The The number number combinations combinations n objects objects taken taken from from N at at a time time is: is: N n C N n C n N N C n C N, n N! n! N n! So for mouse selection: C 2! 5! 5 2! Let s consider lottery 6 from 47 C 6! 47! ! Lecture 2. Introduction to probability images from 7

8 EXERIMENTS, COUNTING RULES ND ROBBILITIES To test state population, let s select 2 for 2 different check-up Counting Rule 3 Mouse population: 5 mice How How many many variants variants such such a selection selection exits? exits? Counting Counting rule rule for for permutations permutations The The number number permutation permutation n objects objects taken taken from from N at at a time time is: is: N n N! N n! So for mouse selection: 5 5! 2 20! 5 2 How many triplets we can build from 4 nucleotides without repetition? 4 4! ! Lecture 2. Introduction to probability images from 8

9 EXERIMENTS, COUNTING RULES ND ROBBILITIES Example: K&L Kentucky Kentucky ower ower & Light Light Company Company K&L K&L is is starting starting a a project project designed designed to to increase increase generating generating capacity capacity one one its its plants plants in in norrn norrn Kentucky. Kentucky. The The project project is is divided divided into into two two sequential sequential stages stages or or steps: steps: stage stage 1 1 design design and and stage stage 2 2 construction. construction. Even Even though though each each stage stage will will be be scheduled scheduled and and controlled controlled as as closely closely as as possible, possible, management management cannot cannot predict predict beforehand beforehand exact exact time time required required to to complete complete each each stage stage project. project. n n analysis analysis similar similar construction construction projects projects revealed revealed possible possible completion completion times times for for design design stage stage 2, 2, 3, 3, or or 4 4 months months and and possible possible completion completion times times for for construction construction stage stage 6, 6, 7, 7, or or 8 8 months. months. In In addition, addition, because because critical critical need need for for additional additional electrical electrical power, power, management management set set a a goal goal months months for for completion completion entire entire project. project. nderson et al Statistics for Business and Economics Lecture 2. Introduction to probability 9

10 EXERIMENTS, COUNTING RULES ND ROBBILITIES Example: K&L The goal is not reached. We need to assign probabilities to estimate risks! Lecture 2. Introduction to probability 10

11 EXERIMENTS, COUNTING RULES ND ROBBILITIES ssigning robabilities Basic Basic requirements requirements for for assigning assigning probabilities probabilities Two Two requirements requirements that that restrict restrict manner manner in in which which probability probability assignments assignments can can be be made: made: For For each each experimental experimental outcome outcome E i, i, probability probability must must be be 0 E E Considering Considering all all experimental experimental outcomes, outcomes, we we must must have have E E E E E E n n 1 But But how how to to assign assign it? it? 3 basic methods probability assigning: Classical Classical method method Frequency Frequency method method Subjective Subjective method method Theoretical Theoretical method method Lecture 2. Introduction to probability 11

12 EXERIMENTS, COUNTING RULES ND ROBBILITIES ssigning robabilities Classical Classical method method method method assigning assigning probabilities probabilities that that is is appropriate appropriate when when all all experimental experimental outcomes outcomes are are equally equally likely. likely. If n experimental outcomes are possible, a probability 1/n is assigned to each experimental outcome. When using this approach, two basic requirements for assigning probabilities are automatically satisfied. 11/6 21/6 31/6 41/6 51/6 61/6 H1/2 T1/2 2xH 2xT T+H 1/4 1/4 1/4 2xH1/4 2xT1/4 T+H1/2 1/4 Lecture 2. Introduction to probability 12

13 EXERIMENTS, COUNTING RULES ND ROBBILITIES ssigning robabilities Relative Relative frequency frequency method method method method assigning assigning probabilities probabilities that that is is appropriate appropriate when when data data are are available available to to estimate estimate proportion proportion time time experimental experimental outcome outcome occurs occurs if if experiment experiment is is repeated repeated a large large number number times. times. Example: clerk recorded number patients waiting for service in X-ray department for a local hospital at 9:00 a.m. on 20 successive days, and obtained following results. Number Number days outcome waiting occurred Total 20 Number waiting Relative frequency Total 1 Lecture 2. Introduction to probability 13

14 EXERIMENTS, COUNTING RULES ND ROBBILITIES ssigning robabilities Subjective Subjective method method method method assigning assigning probabilities probabilities on on basis basis judgment judgment What What is is a probability probability to to meet meet a dinosaur, dinosaur, going going out out from from this this building? building? Classical Classical method: method: 50% 50% Frequency Frequency method: method: no no data data Subjective Subjective method: method: no no way! way! Theoretical Theoretical method method method method assigning assigning probabilities probabilities on on basis basis knowledge knowledge probability probability distribution distribution to to be be discussed discussed later later Lecture 2. Introduction to probability 14

15 EXERIMENTS, COUNTING RULES ND ROBBILITIES Example: K&L Management Management decided decided to to conduct conduct a study study completion completion times times for for similar similar projects projects undertaken undertaken by by K&L K&L over over past past three three years. years. The The results results a study study similar similar projects projects are are summarized summarized in in Table. Table. fter fter reviewing reviewing results results study, study, management management decided decided to to employ employ relative relative frequency frequency method method assigning assigning probabilities. probabilities. Management Management could could have have provided provided subjective subjective probability probability estimates, estimates, but but felt felt that that current current project project was was quite quite similar similar to to previous previous projects. projects. Thus, Thus, relative relative frequency frequency method method was was judged judged best. best. Lecture 2. Introduction to probability 15

16 EVENTS ND THEIR ROBBILITIES Event Event Event collection collection sample sample points. points. Let us denote event successful completion as C, so C{2, 6, 2, 7, 2, 8, 3, 6, 3, 7, 4, 6} robability robability an an event event The The probability probability any any event event is is equal equal to to sum sum probabilities probabilities sample sample points points in in event. event. C 0.7 Lecture 2. Introduction to probability 16

17 EVENTS ND THEIR ROBBILITIES Complement Event Complement Complement event event The The event event consisting consisting all all sample sample points points that that are are not not in in.. C + 1 Venn Venn diagram diagram graphical graphical representation representation for for showing showing symbolically symbolically sample sample space space and and operations operations involving involving events. events. Usually Usually sample sample space space is is represented represented by by a rectangle rectangle and and events events are are represented represented as as circles circles within within sample sample space. space. if C 0.7 C C 0.3 Lecture 2. Introduction to probability 17

18 EVENTS ND THEIR ROBBILITIES Union, Intersection and ddition Law for Two Events Union Union and and В The The event event containing containing all all sample sample points points belonging belonging to to or or В or or both. both. The The union union is is denoted denoted B. B. Intersection Intersection and and В The The event event containing containing sample sample points points belonging belonging to to both both and and B. B. The The intersection intersection is is denoted denoted А B. B. ddition ddition Law Law The The event event containing containing sample sample points points belonging belonging to to both both and and B. B. The The intersection intersection is is denoted denoted А B. B. B + B B Lecture 2. Introduction to probability 18

19 EVENTS ND THEIR ROBBILITIES ractical Example Consider a study study conducted by by personnel manager a big big computer stware company. The The study study showed that that 30% 30% employees who who left left firm firm within within two two years years did did so so primarily because y y were were dissatisfied with with ir ir salary, salary, 20% 20% left left because y y were were dissatisfied with with ir ir work work assignments, and and 12% 12% former former employees indicated dissatisfaction with with both both ir ir salary salary and and ir ir work work assignments. What What is is probability that that an an employee who who leaves leaves within within two two years years does does so so because dissatisfaction with with salary, salary, dissatisfaction with with work work assignment, or or both? both? Let Let S event event that that employee employee leaves leaves because because salary salary W event event that that employee employee leaves leaves because because work work assignment assignment We We have have S S 0.30, 0.30, W W 0.20, 0.20, and and S S W W Using Using equation equation for for addition addition law, law, we we have have S S W W S S + W W S S W W nderson et al Statistics for Business and Economics Lecture 2. Introduction to probability 19

20 EVENTS ND THEIR ROBBILITIES Mutual Exclusive Events Mutually Mutually exclusive exclusive events events Events Events that that have have no no sample sample points points in in common; common; that that is, is, А Вis Вis empty empty and and B B ddition ddition Law Law for for mutually mutually exclusive exclusive events events B + B What What is is probability that that a die die shows shows 5 or or 6? Lecture 2. Introduction to probability 20

21 CONDITIONL ROBBILITY Conditional robability Conditional Conditional probability probability The The probability probability an an event event given given that that anor anor event event already already oc-curred. oc-curred. The The conditional conditional probability probability given given В is is denoted denoted B B B B B B B Example: Example: Let's Let's consider consider situation situation promotion promotion status status male male and andfemale ficers ficers a major major metropolitan metropolitan police police force force in in eastern eastern United United States. States. The The police police force force consists consists ficers, ficers, men men and and women. women. Over Over past past two two years, years, ficers ficers on on police police force force received received promotions. promotions. The The specific specific breakdown breakdown promotions promotions for for male male and and female female ficers ficers is is shown shown in in Table. Table. Female Female association association raised raised a discrimination discrimination case case!! Lecture 2. Introduction to probability 21

22 CONDITIONL ROBBILITY Example: olice Joint probability table Lecture 2. Introduction to probability 22

23 CONDITIONL ROBBILITY Example: olice What What conclusion conclusion can can you you make, make, seeing seeing this this result? result? Only Only that that a randomly randomly taken taken man man will will be be promoted promoted with with x2 x2 higher higher probability probability than than a randomly randomly selected selected woman. woman. NO NO CONCLUSION CONCLUSION CN CN BE BE MDE MDE CONCERNING CONCERNING THE THE DISCRIMINTION DISCRIMINTION!!!!!! Lecture 2. Introduction to probability 23

24 CONDITIONL ROBBILITY Independent Events B B B Multiplication Multiplication law law probability probability law law used used to to compute compute probability probability intersection intersection two two events. events. B B Independent Independent events events Two Two events events and and В that that have have no no influence influence on on each each or. or. B B B B B fail 0.2 What What is is probability probability each each outcome? outcome? success 0.8 Lecture 2. Introduction to probability 24

25 Lecture 2. Introduction to probability 25 BYES THEOREM The Theorem Bayes Bayes orem method used to compute posterior probabilities. Bayes orem method used to compute posterior probabilities. rior probabilities Initial estimates probabilities events. rior probabilities Initial estimates probabilities events. osterior probabilities Revised probabilities events based on additional information osterior probabilities Revised probabilities events based on additional information n n i i i B B B B B L B B B B B 2 1 B B B B B n i i L

26 BYES THEOREM Example: Bayes Theorem Medical Medical researchers researchers know know that that probability probability getting getting lung lung cancer cancer if if a person person smokes smokes is is The The probability probability that that a non-smoker non-smoker will will get get lung lung cancer cancer is is It It is is also also known known that that 11% 11% population population smokes. smokes. What What is is probability probability that that a person personwith with lung lung cancer cancer will will have have been been a smoker? smoker? Non-Smokers 89% OULTION Smokers 11% No cancer 97% Cancer 3% Cancer 34% No cancer 66% Lecture 2. Introduction to probability 26

27 CONDITIONL ROBBILITY Example: Bayes Theorem Medical Medical researchers researchers know know that that probability probability getting getting lung lung cancer cancer if if a a person person smokes smokes is is The The probability probability that that a a non-smoker non-smoker will will get get lung lung cancer cancer is is It It is is also also known known that that 11% 11% population population smokes. smokes. What What is is probability probability that that a a person person with with lung lung cancer cancer will will have have been been a a smoker? smoker? Let s define events: s smoker n non-smoker C get a lung cancer N no cancer B B B s C s 0.11 C s 0.34 C n 0.03 s C? s C C B B C C i i i B + B C C + B C s s n 0.89 n C n N opulation 100 % s 0.11 s C s N C n C + s C s C Lecture 2. Introduction to probability 27

28 CONDITIONL ROBBILITY Example: Bayes Theorem and Tabular pproach Medical Medical researchers researchers know know that that probability probability getting getting lung lung cancer cancer if if a person person smokes smokes is is The The probability probability that that a non-smoker non-smoker will will get get lung lung cancer cancer is is It It is is also also known known that that 11% 11% population population smokes. smokes. What What is is probability probability that that a person personwith with lung lung cancer cancer will will have have been been a smoker? smoker? Let s define events: s smoker n non-smoker C get a lung cancer s 0.11 C s 0.34 C n 0.03 s C? B B B B B Lecture 2. Introduction to probability 28

29 QUESTIONS? Thank you for your attention to be continued Lecture 2. Introduction to probability 29