is a field of study concerned with collection, organization, summarization and analysis of, Summaries of Biostatistics

Download is a field of study concerned with collection, organization, summarization and analysis of and more Summaries Biostatistics in PDF only on Docsity! 7 Probability Objectives At the end of this lecture students will be familiar with some basic concept of probability. 2 Probability  Probabilities are written as:  Fractions from 0 to 1  Decimals from 0 to 1  Percent from 0% to 100% 5 Probability  If an event is certain to happen, then the probability of the event is 1 or 100%.  If an event will NEVER happen, then the probability of the event is 0 or 0%.  If an event is just as likely to happen as to not happen, then the probability of the event is ½, 0.5 or 50%. 6 Probability Impossible Unlikely Equal Chances Likely Certain 0 0.5 1 0% 50% 100% ½ 7 Probability  The probability of an event is written: P(event) = number of ways event can occur total number of outcomes 10 Definition  Experiment ==> is any process that can be repeated in which the results are uncertain. 1012 52 11  Sample space: collection of unique, non- overlapping possible outcomes of a random circumstance.  Simple event: one outcome in the sample space; a possible outcome of a random circumstance.  Event: a collection of one or more simple events in the sample space; often written as A, B, C, and so on Definition 12 2- Classical: It is well known that the probability of flipping a fair coin and getting a “tail” is 0.50. If a coin is flipped 10 times, is there a guarantee, that exactly 5 tails will be observed Methods for determining the probability cont. 15 3- Empirical: Assuming that an experiment can be repeated many times and assuming that there are one or more outcomes that can result from each repetition. Then, the probability of a given outcome is the number of times that outcome occurs divided by the total number of repetitions. Methods for determining the probability cont. 16 Probability theory is based on the three axioms stated by Kolmogorov:  1. The probability that each event will occur must be greater than or equal to 0 and less than or equal to 1.  2. The sum of the probabilities of all the mutually exclusive outcomes of the sample space is equal to 1.  3. The probability that either of two mutually exclusive events, A or B, will occur is the sum of the probabilities of their individual probabilities. 17 Conditional probabilities  It is the probability of an event on condition that certain criteria is satisfied  Example: If a subject was selected randomly and found to be female what is the probability that she has a blood group O. Here the total possible outcomes constitute a subset (females) of the total number of subjects.  This probability is termed probability of O given F P(O\F) = 20/50 = 0.40 20 Joint probability It is the probability of occurrence of two or more events together Example: Probability of being male & belong to blood group AB P(M and AB)= P(M∩AB) = 5/100 = 0.05 ∩ = intersection 21 Rules of probability 1- Multiplication rule Independence and multiplication rule P(A and B) = P(A) P(B) 22 1- Multiplication rule Dependence and the modified multiplication rule P(A and B) = P(A) P(B\A) Rules of probability 25 P(A) P(B) A and B are not independent P(B\A) ≠ P(B) P(B\A) 26 Problem 2 Ill Not Ill Total Ate Barbecue Did Not Eat Barbecue 90 20 30 60 120 80 Total 110 90 200 An outbreak of food poisoning occurs in a group of students who attended a party 27 P(A) A and B are mutually exclusive The occurrence of one event precludes the occurrence of the other P(B) P(A OR B) = P(A U B) = P(A) + P(B) Addition Rule 30 Example: The probability of being either blood type O or blood type A P(OUA) = P(O) + P(A) = (40/100)+(35/100) = 0.75 31 P(A) A and B are non mutually exclusive (Can occur together) Example: Male and smoker P(A OR B) = P(A U B) = P(A) + P(B) - P(A ∩ B) P(B) P(A ∩ B) 32 In a normally distributed population, the probability that a subject’s blood cholesterol level will be lower than 1 SD below the mean is 16% and the probability of being blood cholesterol level higher than 2 SD above the mean is 2.5%. What is the probability that a randomly selected subject will have a blood cholesterol level lower than 1 SD below the mean or higher than 2 SD above the mean. P(blood cholesterol level < 1 SD below the mean or 2 SD above the mean) = 16% + 2.5% = 18.5% Exercise 2: 35 In a study of the optimum dose of lignocaine required to reduce pain on injection of an intravenous agent used for induction of anesthesia, four dosing groups were considered (group A received no lignocaine, while groups B, C, and D received 0.1, 0.2, and 0.4 mg/kg, respectively). The following table shows the patients cross-classified by dose and pain score: Exercise 3: 36 Pain score Group Total A B C D 0 1 2 3 49 16 8 4 73 7 5 1 58 7 6 0 62 8 6 0 242 38 25 5 Total 77 86 71 76 310 Compute the following probabilities for a randomly selected patient: 1.being of group D and experiencing no pain 2.belonging to group B or having a pain score of 2 3.having a pain score of 3 given that he belongs to group A Exercise 3 cont. 37