Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42CHAPTER 3Probability Theory•3.1 - Basic Definitions and Properties•3.2 - Conditional Probability and Independence•3.3 - Bayes’ Formula•3.4 - Applications (biomedical)2POPULATIONRandom variable XSAMPLE of size nx1x2x3 x4 x5 x6 …etc….xn Data xiRelative Frequenciesf (xi ) = fi /nx1f (x1) x2f (x2) x3f (x3) ⋮ ⋮xkf (xk)1Frequency TableDensity HistogramXTotal Area = 1)()()(xfxxsxfxxnn221???Probability Table Probability Histogram… at least if X is discrete.(Chapter 4)3OutcomeRedOrangeYellowGreenBlue• Sample Space The set of all possible outcomes of an experiment.Definitions• An outcome is the result of an experiment on a population.S = {Red, Orange, Yellow, Green, Blue}Venn Diagram• Event Any subset of S (including the empty set , and S itself).E = “Primary Color” = {Red, Yellow, Blue}(using basic Set Theory)RedYellowGreenOrangeBlue#(S) = 5#(E) = 3 waysEConsider the following experiment: Randomly select an individual from the population, and record its color.POPULATION4• Sample Space The set of all possible outcomes of an experiment.Definitions• An outcome is the result of an experiment on a population.S = {Red, Orange, Yellow, Green, Blue}.Venn Diagram• Event Any subset of S (including the empty set , and S itself).E = “Primary Color” = {Red, Yellow, Blue}F = “Hot Color” = {Red, Orange, Yellow}(using basic Set Theory)RedYellowGreenOrangeBlue#(S) = 5#(E) = 3 ways#(F) = 3 waysFOutcomeRedOrangeYellowGreenBlueConsider the following experiment: Randomly select an individual from the population, and record its color.POPULATION5• Sample Space The set of all possible outcomes of an experiment.Definitions• An outcome is the result of an experiment on a population.S = {Red, Orange, Yellow, Green, Blue}.Venn Diagram• Event Any subset of S (including the empty set , and S itself).E = “Primary Color” = {Red, Yellow, Blue}F = “Hot Color” = {Red, Orange, Yellow}(using basic Set Theory)“Cold Color”= {Green, Blue}“Not F” =Complement F C =RedYellowGreenOrangeBlue#(S) = 5#(E) = 3 ways#(F) = 3 waysFOutcomeRedOrangeYellowGreenBlue#(FC) = 2 waysConsider the following experiment: Randomly select an individual from the population, and record its color.POPULATION6• Sample Space The set of all possible outcomes of an experiment.Definitions• An outcome is the result of an experiment on a population.S = {Red, Orange, Yellow, Green, Blue}.Venn Diagram• Event Any subset of S (including the empty set , and S itself).E = “Primary Color” = {Red, Yellow, Blue}F = “Hot Color” = {Red, Orange, Yellow}(using basic Set Theory)“Cold Color”= {Green, Blue}“Not F” =Complement F C =RedYellowGreenOrangeBlue#(S) = 5#(E) = 3 ways#(F) = 3 waysFOutcomeRedOrangeYellowGreenBlue#(FC) = 2 waysConsider the following experiment: Randomly select an individual from the population, and record its color.POPULATION7• Sample Space The set of all possible outcomes of an experiment.Definitions• An outcome is the result of an experiment on a population.S = {Red, Orange, Yellow, Green, Blue}.Venn Diagram• Event Any subset of S (including the empty set , and S itself).E = “Primary Color” = {Red, Yellow, Blue}F = “Hot Color” = {Red, Orange, Yellow}(using basic Set Theory)“Cold Color”= {Green, Blue}“Not F” =Complement F C =RedYellowGreenOrangeBlue#(S) = 5#(E) = 3 ways#(F) = 3 waysEFIntersection E ⋂ F = {Red, Yellow}“E and F” =OutcomeRedOrangeYellowGreenBlue#(FC) = 2 waysConsider the following experiment: Randomly select an individual from the population, and record its color.POPULATION8• Sample Space The set of all possible outcomes of an experiment.Definitions• An outcome is the result of an experiment on a population.S = {Red, Orange, Yellow, Green, Blue}.Venn Diagram• Event Any subset of S (including the empty set , and S itself).E = “Primary Color” = {Red, Yellow, Blue}F = “Hot Color” = {Red, Orange, Yellow}(using basic Set Theory)“Cold Color”= {Green, Blue}“Not F” =Complement F C =RedYellowGreenOrangeBlue#(S) = 5#(E) = 3 ways#(F) = 3 waysEFIntersection E ⋂ F = {Red, Yellow}“E and F” =#(E ⋂ F) = 2OutcomeRedOrangeYellowGreenBlue#(FC) = 2 waysConsider the following experiment: Randomly select an individual from the population, and record its color.POPULATION9• Sample Space The set of all possible outcomes of an experiment.Definitions• An outcome is the result of an experiment on a population.S = {Red, Orange, Yellow, Green, Blue}.Venn Diagram• Event Any subset of S (including the empty set , and S itself).E = “Primary Color” = {Red, Yellow, Blue}F = “Hot Color” = {Red, Orange, Yellow}(using basic Set Theory)“Cold Color”= {Green, Blue}“Not F” =Complement F C =RedYellowGreenOrangeBlue#(S) = 5#(E) = 3 ways#(F) = 3 waysABIntersection E ⋂ F = {Red, Yellow}“E and F” =#(E ⋂ F) = 2OutcomeRedOrangeYellowGreenBlueNote: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  #(FC) = 2 waysConsider the following experiment: Randomly select an individual from the population, and record its color.POPULATION10• Sample Space The set of all possible outcomes of an experiment.Definitions• An outcome is the result of an experiment on a population.S = {Red, Orange, Yellow, Green, Blue}.Venn Diagram• Event Any subset of S (including the empty set , and S itself).E = “Primary Color” = {Red, Yellow, Blue}F = “Hot Color” = {Red, Orange, Yellow}(using basic Set Theory)“Cold Color”= {Green, Blue}“Not F” =Complement F C =RedYellowGreenOrangeBlue#(S) = 5#(E) = 3 ways#(F) = 3 waysIntersection E ⋂ F = {Red, Yellow}“E and F” =#(E ⋂ F) = 2OutcomeRedOrangeYellowGreenBlueNote: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  #(FC) = 2 ways“E or F” =EFConsider the following experiment: Randomly select an individual from the population, and record its color.POPULATION11• Sample Space The set of all possible outcomes of an