1. BIOSTATISTICS Topic: Probability 郭士逢 輔大生科系 2007 Note: These slides are made for teaching purpose only, with contents from the textbook, Biostatistics for the biological and health science, by M.M. Triola and M.F. Triola), and supplemental materials published by Pearson Education, Inc. in 2006

3. Definition Event –A collection of outcomes of a procedure Simple event –An outcome or an event that can not be further broken down Sample space –For a procedure consist of all possible simple events

4. Example - Gender of baby Procedure: 1 birth –Event: female –Sample space: {male, female} Procedure: 3 births –Event: 2 females and a male –Sample space: {fff, ffm, fmf, fmm, mff, mfm, mmf, mmm}

5. Notation of probabilities P denotes a probability A, B, C denote specific event P(A) denotes the probability of event A occuring

6. Defined the probability of an event Relative frequency approximation Classical approach for equally likely outcomes Subjective probabilities

7. Relative frequency approximation Conduct and observe a procedure and count the number of time that event A actually occurs Base on the result, P(A)s estimated as simulation

8. Classical approach For a given procedure has n different simple events, assuming each of those simple events has an equal chance of occurring

9. Subjective probabilities probability of event A s estimated by using knowledge of relevant circumstance

11. Law of large number As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability

12. Rules of probability The probability of an impossible event is 0 The probability of an event that is certain to occur is 1 For any event, the probability of A is between 0 and 1 inclusive

14. Complementary event Consist of all outcomes in which event A does not occur, and denoted by Example,105 out of 205 newborn babies are boys, then P(not boy) = P(girl) =100/205 = 0.488

15. Rounding off probabilities Express probability in fraction or decimal rounding off to 3 significant digits

16. Compound event Any event combining two or more simple events

19. Addition rule P(A or B) = P(in a single trial, event A occur s or event B occurs or they both occur) P(A+B) = P(A) + P(B) – P(A and B) Adding n such a way that every outcomes s counted only once

22. Definition Event A and B are disjoint or mutually exclusive, if they can not occur at the same time

24. Rules of complementary events

26. Multiplication rule P(A and B)=P(event A occurs in a first trial and event B in a second trial)

28. Conditional probability P(B|A) represents the probability of event B occurring after event has already occurred

29. Definition Event A and B are independent, if occurrence of one does not affect the probability of the occurrence of the other

30. Formal multiplication rule P(A and B) = P(A). P(B|A)

32. Probability of at least one “At least one” is equivalent to “one or more” The complement of getting at least one of a particular is get no item of that type

33. Conditional probability Conditional probability of an event is the probability with additional information that some other event has already occurred

34. Bayes’ theorem Dealing with sequential events Revise a probability value base on additional information that s later obtained

35. Definitions A prior probability is an initial probability obtained before any additional information A posterior probability is a probability that has been revised by using additional information that is later obtained

38. Definitions Absolute risk reduction = | P(event occurring in treatment group ) – P(event occurring in control group ) | From table 3-4 Absolute risk reduction =

39. Definitions Relative risk is the ratio P t / P c –Pt is the proportion of the characteristic in treatment group –Pc is the proportion in control group Using table 3-4 Absolute risk reduction = P t / P c =

40. Definitions Number needed to treat = 1 / absolute risk reduction Rounded up to next larger whole number

41. Definitions Actual odds against event A = –Expressed in the form of m:n Actual odds in favor of event A = –Expressed in the form of n:m

43. Definitions Odds ratio = Using table 3-4, odds ratio = ad / bc

44. Relative risk versus odds ratio Prospective study: relative risk, odds ratio Retrospective study: odds ratio only

47. Definitions Rate = –a = frequency count of the number of people that event occurred –b = total number of people exposed to the risk of the event occurring –k = multiplier number

48. Counting rule For a sequence of 2 events, first event can occur m ways and the second can occur n ways, the events together can occur a total of m*n ways

49. Factorial Denotes the product of decreasing positive whole number Example, 4! = 4*3*2*1 =24

50. Permutation rule Select r items from n available items If there is n items with some items are identical to others, number of permutation is

51. Combination rule Number of combination of r items selected from n different items