A Simple Example. Sample Space and Event. Tree Diagram. Tree Diagram. Probability. Probability - 1. Probability and Counting Rules

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1 Probability and Counting Rules researcher claims that 10% of a large population have disease H. random sample of 100 people is taken from this population and examined. If 20 people in this random sample have the disease, what does it mean? How likely would this happen if the researcher is right? Simple Example What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing). So toss a coin twice. Do it! Did you get one head & one tail? What s it all mean? 1 2 Sample Space and Probability Random Experiment: (Probability Experiment) an experiment whose outcomes depend on chance. Sample Space (S): collection of all possible outcomes in random experiment. Event (E): a collection of outcomes 3 Sample Space and Event Sample Space: S {Head, Tail} S {Life span of a human} {x x 0, x R} S {ll individuals live in a community with or without having disease } Event: E {Head} E {Life span of a human is less than 3 years} E {The individuals who live in a community that has disease } 4 Sample Space {(H,1),(H,2),(H,3),(H,4),(T,1),(T,2),(T,3),(T,4)} Tree Diagram H T 2 x 4 1 (H,1) 2 (H,2) 3 (H,3) 4 (H,4) 1 (T,1) 2 (T,2) 3 (T,3) 4 (T,4) Tree Diagram What is the sample space for casting two dice experiment? Die 2 1 (1,1) Die x 6 36 outcomes (1,2) (1,3) (1,4) (1,) (1,6) 6 Probability - 1

2 Sample Space (Two Dice) Compound event Sample Space: S {(1,1),(1,2),(1,3),(1,4),(1,),(1,6), (2,1),(2,2),(2,3),(2,4),(2,),(2,6), (3,1),(3,2),(3,3),(3,4),(3,),(3,6), (4,1),(4,2),(4,3),(4,4),(4,),(4,6), (,1),(,2),(,3),(,4),(,),(,6), (6,1),(6,2),(6,3),(6,4),(6,),(6,6)} Simple event How many outcomes are Sum is 7? 7 Definition of Probability rough definition: (frequentist definition) Probability of a certain outcome to occur in a random experiment is the proportion of times that the this outcome would occur in a very long series of repetitions of the random experiment. Total Heads / Number of Tosses Number of Tosses Determining Probability How to determine probability? Empirical Probability Theoretical Probability (Subjective approach) Empirical Probability ssignment Empirical study: (Don t know if it is a balanced Coin?) Outcome Head Tail Total Empirical Probability ssignment Empirical probability assignment: Empirical Probability Distribution Empirical study: Number of times event E occurs E) Number of times experiment is repeated m n Probability of Head: Head) 12 Outcome Head % Empirical Probability Distribution Tail Total Probability Probability - 2

3 Theoretical Probability ssignment Make a reasonable assumption: What is the probability distribution in tossing a coin? ssumption: We have a balanced coi Theoretical Probability ssignment Theoretical probability assignment: Number of equally likely outcomes in event E E) Size of the sample space E) S) Probability of Head: 1 Head) 2. 0% Theoretical Probability Distribution Empirical study: Outcome Head Tail Total Probability Probability Distribution (Probability Model) If a balanced coin is tossed, Head and Tail are equally likely to occur, Head). 1/2 and Tail). 1/2 1/2 Probability ar Chart Total probability is 1. Empirical Probability Distribution 1 Head Tail 16 Relative and Probability Distributions Discrete Distribution Number of times visited a doctor from a random sample of 300 individuals from a community Class Relative ) ) ) ) ) ).01 Total Relative Distribution Probability - 3

4 Relative and Probability When selecting one individual at random from a population, the probability distribution and the relative frequency distribution are the same. 19 Probability for the Discrete Case If an individual is randomly selected from this group 300, what is the probability that this person visited doctor 3 times? 3 times) (42)/300 Class Relative Total or 14% 20 Discrete Distribution If an individual is randomly selected from this group 300, what is the probability that this person visited doctor 4 or times? Class Total or times) 4) + ) Relative It would be an empirical probability distribution, if the sample of 300 individuals is utilized for understanding a large population. 21 Properties of Probability Probability is always a value between 0 and 1. Total probability (all outcomes together) equals 1. Probability of either one of the disjoint events or to occur is the sum of their individual probabilities. or ) ) + ) 22 Probability Distribution (Probability Model) Complementation Rule If a balanced coin is tossed, Head (H) and Tail (T) are equally likely to occur, For any event E, all possible outcomes) H) + T) Probability Total probability is 1 ar Chart 1/2 H). and T). Head Tail 23 E does not occur) 1 E) Complement of E E * Some places use E c or E EE) E) E 24 Probability - 4

5 Complementation Rule Complementation Rule If an unbalanced coin has a probability of 0.7 to turn up Head each time tossing this coin. What is the probability of not getting a Head for a random toss? not getting Head) If the chance of a randomly selected individual living in community to have disease H is.001, what is the probability that this person does not have disease H? having disease H).001 not having disease H) 1 having disease H) Topics Rules of Probability Notations and Venn Diagram ddition Rules Conditional Probability Multiplication Rules Odds Ratio & Relative Risk 27 S {1, 2, 3, 4,, 6} {1, 2, 3} {3, 6} Intersection of events: <> and Example: {3} Union of events: <> or Example: {1, 2, 3, 6} S Venn Diagram (with elements listed) 28 ddition Rules for Probability Mutually Exclusive Events and are mutually exclusive (disjointed) events ddition Rule 1 (Special ddition Rule) In an experiment of casting an unbalanced die, the event and are defined as the following: an odds number 6 and given ).61, ) 0.1. What is the probability of getting an odds number or a six? and are mutually exclusive dditive Rule: ) ) + ) 29 ) ) + ) Probability -

6 ddition Rule 1 - generalized (for multiple events) In 1, 2,,, k are defined k mutually exclusive (m.e.) then 1 2 k ) 1 )+ 2 )+ + k ) Example: When casting a balanced die, the probability of getting 1 or 2 or 3 is 1 2 3) 1) + 2) + 3) 1/6 + 1/6 + 1/6 1/2 31 Example: When casting a unbalanced die, such that 1).2, 2).1, 3).3, the probability of getting 1 or 2 or 3 is 1 2 3) 1) + 2) + 3) General ddition Rule Not M.E.! Venn Diagram (with counts) Given total of 100 subjects )? Sample Space? 4 ) ) + ) ) 33 Smokers, ) 0 Lung Cancer, ) 2 ) 20 Joint Event 34 Venn Diagram (with relative frequencies) Given a sample space ).?.4 Probability of Joint Events In a study, an individual is randomly selected from a population, and the event and are defined as the followings: the individual is a smoker. the individual has lung cancer ).0, ).2 and ) ) smoker and has lung cancer) Smokers, ).0 Lung Cancer, ).2 ).20 Joint Event 3 What is the probability or ) )? 36 Probability - 6

7 ddition Rule 2 (General ddition Rule) What is the probability that this randomly selected person is a smoker or has lung cancer? or )? or ) ) + ) - and ) Contingency Table Cancer, No Cancer, c Total Smoke, Not Smoke, c Venn Diagram Conditional Probability The conditional probability of event to occur given event has occurred (or given the condition ) is denoted as ) and is, if ) is not zero, E) # of equally likely outcomes in E, ) ) ) or ) ) ) Conditional Probability Smoke S Not Smoke S Cancer C 20 2 No Cancer C ) ) ) Total C S) 20/0.4 C S ' ) / Conditional Probability Smoke S Not Smoke S Cancer C 20 (.2) (.0) 2 C) (.2) C S).2/..4 C S ' ).0/..1 No Cancer C 30 (.3) 4 (.4) 7 C)(.7) ) ) ) Total 0 S) (.) 0 S ) (.) 100 (1.0) C S) What is C S )? 4 (Relative Risk ) 41 Independent Events Events and are independent if ) ) or ) ) or and ) ) ) 42 Probability - 7

8 Multiplication Rule 1 (Special Multiplication Rule) If event and are independent, and ) ) ) Example If a balanced die is rolled twice, what is the probability of having two 6 s? 6 1 the event of getting a 6 on the 1st trial 6 2 the event of getting a 6 on the 2nd trial 6 1 ) 1/6, 6 2 ) 1/6, 6 1 and 6 2 are independent events 6 1 and 6 2 ) 6 1 ) 6 2 ) (1/6)(1/6) 1/ Independent Events 10% of the people in a large population has disease H. If a random sample of two subjects was selected from this population, what is the probability that both subjects have disease H? H i : Event that the i-th randomly selected subject has disease H. H 2 H 1 ) H 2 ) [Events are almost independent] H 1 H 2 )? H 1 ) H 2 ).1 x Independent Events If events 1, 2,, k are independent, then 1 and 2 and and k ) 1 ) 2 ) k ) What is the probability of getting all heads in tossing a balanced coin four times experiment? H 1 ) H 2 ) H 3 ) H 4 ) (.) Independent Events 10% of the people in a large population has disease H. If a random sample of 3 subjects was selected from this population, what is the probability that all subjects have disease H? H i : Event that the i-th randomly selected subject has disease H. [Events are almost independent] H 1 H 2 H 3 )?H 1 ) H 2 ) H 3 ).1 x.1 x Multiplication Rule 2 (General Multiplication Rule) For any two events and, and ) ) ) ) ) ) ) and ) ) and ) ) 48 Probability - 8

9 Multiplication Rule 2 Multiplication Rule 2 If in the population, 0% of the people smoked, and 40% of the smokers have lung cancer, what percentage of the population that are smoker and have lung cancer? S) 0% of the subjects smoked C S) 40% of the smokers have cancer C and S) C S) S).4 x ox 1 contains 2 red balls and 1 blue ball ox 2 contains 1 red ball and 3 blue balls coin is tossed. If Head turns up a ball is drawn from ox 1, and if Tail turns up then a ball is drawn from ox 2. Find the probability of selecting a red ball. H and Red) + T and Red) H)Red H) + T)Red T) (1/2) x (2/3) + (1/2) x (1/4) 1/3 + 1/8 11/24 0 Counting Rules Fundamental Principle of Counting Counting Rule: (use of the notation k n ) In a sequence of n events in which the first one has k 1 possibilities and the second event has k 2 and the third has k 3, and so forth, the total possibilities of the sequence will be k 1 k 2 k 3 k n 2 x 4 8 possible outcomes (coin & wheel) 6 X 6 36 possible outcomes (2 dice) k 1 k 2 Example: There are three shirts of different colors, two jackets of different styles and five pairs of pants in the closet. How many ways can you dress yourself with one shirt, one jacket and one pair of pants selected from the closet? Sol: 3 x 2 x 30 ways 1 K 1 x k 2 x k 3 2 Permutations How many different four-letter code words can be formed by using the four letters in the word SE without repeating use of the same letter? k 1 4 k 2 3 k 3 2 k S E Factorial Formula For any counting number n 1 x 2 x x (n 1) x n 0! 1 Example: 3! 1 x 2 x Probability - 9

10 How many different four-letter code words can be formed by using the four letters in the word SE and the same letter can be used repeatedly? k 1 4 k 2 4 k 3 4 k S E How many different four-letter code words can be formed by using the four letters selected from letters through Z and the same letter can not be used repeatedly? k 1 26 k 2 2 k 3 24 k ,800 S E 6 How many different four-letter code words can be formed by using the four letters selected from letters through Z and the same letter can be used repeatedly? k 1 26 k 2 26 k 3 26 k ,976 S E Permutation Rule Permutation Rule: The number of possible permutations of r objects from a collection of n distinct objects is P r n ( n r)! Order does count! 7 8 Permutations P r n ( n r)! Combination Rule How many ways can a four-digit code be formed by selecting 4 distinct digits from nine digits, 1 through 9, without repeating use of the same digit? 9 P 4 9! 9! ! (9-4)!!! Combination Rule: The number of possible combinations of r objects from a collection of n distinct objects is C r n Order does not count! r!( n r)! 60 Probability - 10

11 Combinations C r n r!( n r)! How many ways can a committee be formed by selecting 3 people from a group 10 candidates? n 10, r 3 10! 10! ! 10C 3 3! (10-3)! 3! 7! 3! 7! ! Combinations How many ways can a combination of 4 distinct digits be selected from nine digits, 1 through 9? 9 C 4 9! 4! (9-4)! ! 4!! C r n r!( n r)! ! 4!! C, C, C, C, C, C are the same combination Combinations How many ways can 6 distinct numbers be selected from a set of 47 distinct numbers? n 47, r 6 47! ! 47C 6 6! 41! 6! 41! ! 10,737,73 C r n r!( n r)! 63 Probability Using Counting (Theoretical pproach) In a lottery game, one selects 6 distinct numbers from a set of 47 distinct numbers, what is the probability of winning the jackpot? n 47, r 6 47C 6 47!/(6! 41!) 10,737,73 Jackpot) 1 10,737,73 (Theoretical Prob.) 64 irthday Problem In a group of randomly select 23 people, what is the probability that at least two people have the same birth date? (ssume there are 36 days in a year.) at least two people have the same birth date) Too hard!!! 1 everybody has different birth date) 1 [36x364x x( )] / inomial Probability What is the probability of getting two 6 s in casting a balanced die times experiment? S S S S S ) (1/6) 2 x (/6) S S S S S ) (1/6) 2 x (/6) 3 S S S S S ) (1/6) 2 x (/6) 3! How many of them? !3! Probability (two 6 s) x Probability - 11