CHAPTER 3 Probability Theory (Abridged Version)  B asic Definitions and Properties  C onditional Probability and Independence  B ayes’ Formula

2 POPULATION Random variable X SAMPLE of size n x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 …etc…. xnxn Data x i Relative Frequencies f (x i ) = f i /n x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ xkxk f (x k ) 1 Frequency Table Density Histogram X Total Area = 1 Probability TableProbability Histogram … at least if X is discrete. (Chapter 4)

3 Outcome Red Orange Yellow Green Blue Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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) Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways E

4 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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) Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways F Outcome Red Orange Yellow Green Blue

5 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways F Outcome Red Orange Yellow Green Blue #(F C ) = 2 ways

6 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways F Outcome Red Orange Yellow Green Blue #(F C ) = 2 ways

7 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways E F Intersection E ⋂ F = {Red, Yellow} “E and F” = Outcome Red Orange Yellow Green Blue #(F C ) = 2 ways

8 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways E F Intersection E ⋂ F = {Red, Yellow} “E and F” = #(E ⋂ F) = 2 Outcome Red Orange Yellow Green Blue #(F C ) = 2 ways

9 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways A B Intersection E ⋂ F = {Red, Yellow} “E and F” = #(E ⋂ F) = 2 Outcome Red Orange Yellow Green Blue Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  #(F C ) = 2 ways

10 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways Intersection E ⋂ F = {Red, Yellow} “E and F” = #(E ⋂ F) = 2 Outcome Red Orange Yellow Green Blue Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  #(F C ) = 2 ways “E or F” = E F

11 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways Intersection E ⋂ F = {Red, Yellow} “E and F” = #(E ⋂ F) = 2 Outcome Red Orange Yellow Green Blue Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  #(F C ) = 2 ways Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F” = E F #(E ⋃ F) = 4

12 AB A ⋂ BA ⋂ B A ⋂ BcA ⋂ Bc Ac ⋂ BAc ⋂ B “A only”“B only” Ac ⋂ BcAc ⋂ Bc “Neither A nor B ” “A and B” In general, for two events A and B…

13 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways Intersection E ⋂ F = {Red, Yellow} “E and F” = #(E ⋂ F) = 2 Outcome Red Orange Yellow Green Blue Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  #(F C ) = 2 ways Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F” = E F #(E ⋃ F) = 4 What about probability?

Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. 14 Sample Space The set of all possible outcomes of an experiment. DefinitionsPOPULATION (Pie Chart) 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 = #(S) = 5 #(E) = 3 ways #(F) = 3 ways Intersection E ⋂ F = {Red, Yellow} “E and F“ = #(E ⋂ F) = 2 Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  #(F C ) = 2 ways Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F” = #(E ⋃ F) = 4 OutcomeProbability Red0.20 Orange0.20 Yellow0.20 Green0.20 Blue Red Yellow Green Orange Blue E F “The probability of Red is equal to 0.20” P(Red) = 0.20 # trials … # Red # trials …… But what does it mean?? What happens to this “long run” relative frequency as # trials → ∞ ? All probs are > 0, and sum = 1.

15 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = #(S) = 5 #(E) = 3 ways #(F) = 3 ways Intersection E ⋂ F = {Red, Yellow} “E and F“ = #(E ⋂ F) = 2 Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  #(F C ) = 2 ways Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = #(E ⋃ F) = 4 OutcomeProbability Red0.20 Orange0.20 Yellow0.20 Green0.20 Blue Red Yellow Green Orange Blue E F What about probability of events? For any event E, P(E) =  P(Outcomes in E). BUT… General Fact: All probs are > 0, and sum = 1.

16 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = #(S) = 5 #(E) = 3 ways #(F) = 3 ways Intersection E ⋂ F = {Red, Yellow} “E and F“ = #(E ⋂ F) = 2 Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  #(F C ) = 2 ways Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = #(E ⋃ F) = 4 OutcomeProbability Red0.20 Orange0.20 Yellow0.20 Green0.20 Blue Red Yellow Green Orange Blue E F These outcomes are said to be “equally likely.” When this is the case, P(E) = #(E) / #(S), for any event E in the sample space S. All probs are > 0, and sum = 1.

P(  ) = 0 These outcomes are said to be “equally likely.” 17 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = #(S) = 5 #(E) = 3 ways #(F) = 3 ways Intersection E ⋂ F = {Red, Yellow} “E and F” = #(E ⋂ F) = 2 Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  #(F C ) = 2 ways Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = #(E ⋃ F) = 4 OutcomeProbability Red0.20 Orange0.20 Yellow0.20 Green0.20 Blue Red Yellow Green Orange Blue E F P(E) = 3/5 = 0.6 P(F) = 3/5 = 0.6 P(F C ) = 2/5 = 0.4 P(E ⋂ F) = 0.4 P(E ⋃ F) = 4/5 = 0.8 All probs are > 0, and sum = 1.

P(  ) = 0 These outcomes are said to be “equally likely.” 18 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = #(S) = 5 Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = OutcomeProbability Red0.20 Orange0.20 Yellow0.20 Green0.20 Blue Red Yellow Green Orange Blue E F P(E) = 3/5 = 0.6 P(F) = 3/5 = 0.6 P(F C ) = 2/5 = 0.4 P(E ⋂ F) = 0.4 P(E ⋃ F) = 4/5 = 0.8 OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue These outcomes are NOT “equally likely.” All probs are > 0, and sum = 1.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 19 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = Red Yellow Green Orange Blue E F P(E) = 0.60 All probs are > 0, and sum = 1.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 20 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = Red Yellow Green Orange Blue E F P(E) = 0.60 P(F) = 0.45 All probs are > 0, and sum = 1.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 21 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = Red Yellow Green Orange Blue E F P(E) = 0.60 P(F) = 0.45 P(F C ) = 1 – P(F) = 0.55 All probs are > 0, and sum = 1.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 22 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = Red Yellow Green Orange Blue E F P(E) = 0.60 P(F) = 0.45 P(E ⋂ F) = 0.3 P(F C ) = 1 – P(F) = 0.55 All probs are > 0, and sum = 1.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 23 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = Red Yellow Green Orange Blue E F P(E) = 0.60 P(F) = 0.45 P(E ⋂ F) = 0.3 P(E ⋃ F) = 0.75 P(F C ) = 1 – P(F) = 0.55 All probs are > 0, and sum = 1.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 24 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = Red Yellow Green Orange Blue E F P(E) = 0.60 P(F) = 0.45 P(E ⋂ F) = 0.3 P(E ⋃ F) = 0.75P(E ⋃ F) = P(F C ) = 1 – P(F) = 0.55 All probs are > 0, and sum = 1.

P(E ⋃ F) = OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 25 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = Red Yellow Green Orange Blue E F P(E) = 0.60 P(F) = 0.45 P(E ⋂ F) = 0.3 P(E ⋃ F) = 0.75P(E ⋃ F) = P(E) P(F C ) = 1 – P(F) = 0.55 All probs are > 0, and sum = 1.

P(E ⋃ F) = P(E) OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 26 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = {Red, Orange, Yellow, Blue}“E or F“ = P(E) = 0.60 P(F) = 0.45 P(E ⋂ F) = 0.3 P(E ⋃ F) = 0.75 P(F C ) = 1 – P(F) = 0.55 Red Yellow Green Orange Blue E F P(E ⋃ F) = P(E) + P(F) All probs are > 0, and sum = 1.

P(E ⋃ F) = P(E) + P(F) OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(E ⋃ F) = 0.75 P(  ) = 0 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = “E or F“ = P(E) = 0.60 P(F) = 0.45 P(E ⋂ F) = 0.3 P(F C ) = 1 – P(F) = 0.55 Red Yellow Green Orange Blue E F P(E ⋃ F) = P(E) + P(F) – P(E ⋂ F) {Red, Orange, Yellow, Blue} All probs are > 0, and sum = 1.

P(E ⋃ F) = P(E) + P(F) – P(E ⋂ F) OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(E ⋃ F) = 0.75 P(  ) = 0 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = “E or F“ = P(E) = 0.60 P(F) = 0.45 P(E ⋂ F) = 0.3 P(F C ) = 1 – P(F) = 0.55 Red Yellow Green Orange Blue E F P(E ⋃ F) = P(E) + P(F) – P(E ⋂ F) = – 0.30 {Red, Orange, Yellow, Blue} All probs are > 0, and sum = 1.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(E ⋃ F) = 0.75 P(  ) = 0 Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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 = Intersection E ⋂ F = {Red, Yellow} “E and F” = Note: A = {Red, Green} ⋂ B = {Orange, Blue} = A and B are disjoint, or mutually exclusive events  Union E ⋃ F = “E or F“ = P(E) = 0.60 P(F) = 0.45 P(E ⋂ F) = 0.3 P(F C ) = 1 – P(F) = 0.55 Red Yellow Green Orange Blue E F P(E ⋃ F) = P(E) + P(F) – P(E ⋂ F) = – 0.30 {Red, Orange, Yellow, Blue} All probs are > 0, and sum = 1.

EECEC F P(F)P(F) FCFC P(FC)P(FC) P(E)P(E) P(EC)P(EC) 1.0 OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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) P(E) = 0.60 P(F) = 0.45 Red Yellow Green Orange Blue E F EECEC F P(E ⋂ F ) P(E C ⋂ F ) P(F)P(F) FCFC P(E ⋂ F C ) P(E C ⋂ F C ) P(FC)P(FC) P(E)P(E) P(EC)P(EC) 1.0 Probability Table All probs are > 0, and sum = 1.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) 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) P(E) = 0.60 P(F) = 0.45 Red Yellow Green Orange Blue E F EECEC F FCFC Probability Table All probs are > 0, and sum = 1.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue Sample Space The set of all possible outcomes of an experiment. Definitions Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION (Pie Chart) An outcome is the result of an experiment on a population. S = {Red, Orange, Yellow, Green, Blue}. 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) P(E) = 0.60 P(F) = E F EECEC F FCFC Probability Table Venn Diagram All probs are > 0, and sum = 1.

~ Summary of Basic Properties of Probability ~ Population Hypothesis  Experiment  Sample space of possible outcomes  Event E ⊆  Probability P(E) = ? Def: P(E) = “limiting value” of as experiment is repeated indefinitely. P(E) =  P(outcomes) = always a number between 0 and 1. (That is, 0 ≤ P(E) ≤ 1.) If AND ONLY IF all outcomes in are equally likely, then P(E) = If E and F are any two events, then so are the following: 33 EventDescriptionNotationTerminologyProbab Not E“E does not occur.” complement of E 1 – P(E) E and F “Both E and F occur simultaneously.” E ⋂ F intersection of E and F - E or F “Either E occurs, or F occurs (or both).” E ⋃ F union of E and F P(E) + P(F) – P(E ⋂ F ) ECEC ECEC E E F F E “If E occurs, then F occurs.” E ⊆ F E is a subset of F P(E ⋂ F )

What percentage receives T 1 only? Example: Two treatments exist for a certain disease, which can either be taken separately or in combination. Suppose:  70% of patient population receives T 1  50% of patient population receives T 2  30% of patient population receives both T 1 and T 2 34 T 1 T 2 T1c ⋂ T2T1c ⋂ T2 T1 ⋂ T2cT1 ⋂ T2c T1 ⋂ T2T1 ⋂ T2 (w/ or w/o T 2 ) (w/ or w/o T 1 ) (w/o T 2 ) P(T 2 ) = 0.5P(T 1 ⋂ T 2 ) = 0.3 P(T 1 ⋂ T 2 c ) = 0.7 – 0.3 = 0.4…. i.e., 40% What percentage receives T 2 only? (w/o T 1 ) P(T 1 c ⋂ T 2 ) = 0.5 – 0.3 = 0.2…. i.e., 20% What percentage receives neither T 1 nor T 2 ? P(T 1 c ⋂ T 2 c ) = 1 – ( ) = 0.1…. i.e., 10% T1c ⋂ T2cT1c ⋂ T2c P(T 1 ) = 0.7 T1T1 T1cT1c T2T2 T2cT2c Column marginal sums Row marginal sums

35 AB C A ⋂ B ⋂ C A ⋂ B ⋂ C c A ⋂ Bc⋂ CA ⋂ Bc⋂ CA c ⋂ B ⋂ C A ⋂ Bc ⋂ CcA ⋂ Bc ⋂ Cc A c ⋂ B ⋂ C c Ac ⋂ Bc ⋂ CAc ⋂ Bc ⋂ C “A only” “C only” “B only” Ac ⋂ Bc ⋂ CcAc ⋂ Bc ⋂ Cc “Neither A nor B nor C” In general, for three events A, B, and C…

CHAPTER 3 Probability Theory (Abridged Version)  B asic Definitions and Properties  C onditional Probability and Independence  B ayes’ Formula

Probability of “Primary Color,” given “Hot Color” = ? EECEC F FCFC OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue POPULATION E = “Primary Color”= {Red, Yellow, Blue} F = “Hot Color”= {Red, Orange, Yellow} P(E) = 0.60 P(F) = 0.45 Probability Table Venn Diagram E F Blue Green Orange Red Yellow

E F Blue Green Orange Red Yellow Probability of “Primary Color,” given “Hot Color” = ? EECEC F FCFC E = “Primary Color”= {Red, Yellow, Blue} F = “Hot Color”= {Red, Orange, Yellow} P(E) = 0.60 P(F) = 0.45 Probability Table Venn Diagram E F Blue Green Orange Red Yellow OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue POPULATION P(E | F) Conditional Probability = P(F | E)

EECEC F FCFC E = “Primary Color”= {Red, Yellow, Blue} F = “Hot Color”= {Red, Orange, Yellow} P(E) = 0.60 P(F) = 0.45 Probability Table Venn Diagram E F Blue Green Orange Red Yellow Probability of “Primary Color,” given “Hot Color” = ? P(E | F) OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue POPULATION Conditional Probability P(F | E) 0.5 =

EECEC F FCFC E = “Primary Color”= {Red, Yellow, Blue} F = “Hot Color”= {Red, Orange, Yellow} P(E) = 0.60 P(F) = 0.45 Probability Table Venn Diagram E F Blue Green Orange Red Yellow Probability of “Primary Color,” given “Hot Color” = ? P(E | F)P(E C | F) = 1 – = OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue POPULATION Conditional Probability P(F | E) 0.5

EECEC F FCFC E = “Primary Color”= {Red, Yellow, Blue} F = “Hot Color”= {Red, Orange, Yellow} P(E) = 0.60 P(F) = 0.45 Probability Table Venn Diagram E F Blue Green Orange Red Yellow Probability of “Primary Color,” given “Hot Color” = ? P(E | F)P(E C | F) P(E | F C ) = 1 – = OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue POPULATION Conditional Probability P(F | E) Red Yellow

42 Def: The conditional probability of event A, given event B, is denoted by P(A|B), and calculated via the formula Thus, for any two events A and B, it follows that P(A ⋂ B) = P(A | B) × P(B). B occurs with prob P(B) Given that B occurs, A occurs with prob P(A | B) Both A and B occur, with prob P(A ⋂ B) Example: P(Live to 75) × P(Live to 80 | Live to 75) = P(Live to 80) Tree Diagrams P(B)P(B) P(Bc)P(Bc) P(A | B) P(A c | B) P(A | B c ) P(A c | B c ) P(A ⋂ B)P(A ⋂ B) P(Ac ⋂ B)P(Ac ⋂ B) P(A ⋂ Bc)P(A ⋂ Bc) P(Ac ⋂ Bc)P(Ac ⋂ Bc) EventAAcAc B P(A ⋂ B)P(A ⋂ B)P(Ac ⋂ B)P(Ac ⋂ B) BcBc P(A ⋂ Bc)P(A ⋂ Bc)P(Ac ⋂ Bc)P(Ac ⋂ Bc) A B A ⋂ BA ⋂ B A ⋂ BcA ⋂ Bc Ac ⋂ BAc ⋂ B Ac ⋂ BcAc ⋂ Bc Multiply together “branch probabilities” to obtain “intersection probabilities” AB

43 Example: Bob must take two trains to his home in Manhattan after work: the A and the B, in either order. At 5:00 PM… The A train arrives first with probability 0.65, and takes 30 mins to reach its last stop at Times Square. The B train arrives first with probability 0.35, and takes 30 mins to reach its last stop at Grand Central Station. At Times Square, Bob exits, and catches the second train. The A arrives first with probability 0.4, then travels to Brooklyn. The B train arrives first with probability 0.6, and takes 30 minutes to reach a station near his home. At Grand Central Station, the A train arrives first with probability 0.8, and takes 30 minutes to reach a station near his home. The B train arrives first with probability 0.2, then travels to Queens. With what probability will Bob be exiting the subway at 6:00 PM?

44 Example: 5:00 5:30 6: MULTIPLY: ADD: 0.67 Bob must take two trains to his home in Manhattan after work: the A and the B, in either order. At 5:00 PM… The A train arrives first with probability 0.65, and takes 30 mins to reach its last stop at Times Square. The B train arrives first with probability 0.35, and takes 30 mins to reach its last stop at Grand Central Station. At Times Square, Bob exits, and catches the second train. The A arrives first with probability 0.4, then travels to Brooklyn. The B train arrives first with probability 0.6, and takes 30 minutes to reach a station near his home. At Grand Central Station, the A train arrives first with probability 0.8, and takes 30 minutes to reach a station near his home. The B train arrives first with probability 0.2, then travels to Queens. With what probability will Bob be exiting the subway at 6:00 PM?

45 POPULATION

OutcomeProbability Red0.10 Orange0.18 Yellow0.17 Green0.22 Blue POPULATION EECEC F FCFC Probability Table Venn Diagram E F Blue Green Orange Red Yellow

EECEC F FCFC E = “Primary Color”= {Red, Yellow, Blue} F = “Hot Color”= {Red, Orange, Yellow} P(E) = 0.60 Probability Table Venn Diagram E F Blue Green Orange Red Yellow OutcomeProbability Red0.10 Orange0.18 Yellow0.17 Green0.22 Blue POPULATION Conditional Probability P(E | F) P(F | E) 0.60 = P(E) 0.45 = P(F) P(F) = 0.45

EECEC F FCFC E = “Primary Color”= {Red, Yellow, Blue} F = “Hot Color”= {Red, Orange, Yellow} P(E) = 0.60 Probability Table Venn Diagram E F Blue Green Orange Red Yellow OutcomeProbability Red0.10 Orange0.18 Yellow0.17 Green0.22 Blue POPULATION Conditional Probability P(E | F) P(F | E) = P(E) = P(F) P(F) = 0.45 Events E and F are “statistically independent”

49 Example: According to the American Red Cross, US pop is distributed as shown. Rh Factor Blood Type +– Row marginals: O A B AB Column marginals: Def: Two events A and B are said to be statistically independent if P(A | B) = P(A), Example: Are events A = “Ace” and B = “Black” statistically independent? P(A) = 4/52 = 1/13, P(B) = 26/52 = 1/2, P(A ⋂ B) = 2/52 = 1/26 YES! Neither event provides any information about the other. Are “Type O” and “Rh+” statistically independent? = P(O) = P(Rh+) Is.384 =.461 ×.833? P(O ⋂ Rh+) =.384 YES! which is equivalent to P(A ⋂ B) = P(A | B) × P(B). If either of these two conditions fails, then A and B are statistically dependent. P(A)P(A)

 A and B are statistically independent if: P(A | B) = P(A) IMPORTANT FORMULAS  P(A c ) = 1 – P(A)  P(A ⋃ B) = P(A) + P(B) – P(A ⋂ B) 50 = 0 if A and B are disjoint  P(A ⋂ B) = P(A | B) P(B) P(A ⋂ B) = P(A) P(B)  DeMorgan’s Laws (A ⋃ B) c = A c ⋂ B c “Not (A or B)” = “Not A” and “Not B” = “Neither A nor B” (A ⋂ B) c = A c ⋃ B c “Not (A and B)” = “Not A” or “Not B” A B AB

Example: In a population of individuals:  60% of adults are male P(B | A) = 0.6  40% of males are adults P(A | B) = 0.4  30% are men P(A ⋂ B) = 0.3 What percentage are adults? 51 A = Adult B = Male What percentage are males? Are “adult” and “male” statistically independent in this population? 0.3 Men BoysWomen Girls

Example: In a population of individuals:  60% of adults are male P(B | A) = 0.6  40% of males are adults P(A | B) = 0.4  30% are men P(A ⋂ B) = 0.3 ⟹ P(B ⋂ A) = 0.6 P(A) 0.3 P(A) = 0.3 / 0.6 What percentage are adults? 52 A = Adult B = Male What percentage are males? Are “adult” and “male” statistically independent in this population? 0.3 ⟹ P(A ⋂ B) = 0.4 P(B) 0.3 P(B) = 0.3 / AdultChild Male Female P(A | B) = P(A)? OR P(B | A) = P(B)? OR P(A ⋂ B) = P(A) P(B)? NO 0.4 ≠ ≠ 0.75 P(A) = 0.3 / 0.6 = 0.5, or 50% 0.5 – 0.3 = … P(B) = 0.3 / 0.4 = 0.75, or 75% 0.75 – 0.3 = … 0.3 ≠ (0.5)(0.75) Men BoysWomen Girls

CHAPTER 3 Probability Theory  B asic Definitions and Properties  C onditional Probability and Independence  B ayes’ Formula

P(B)P(B)P(B ) Bayes’ Formula Exactly how does one event A affect the probability of another event B? 54 AP(B)P(B) prior probability posterior probability P(B  A)P(A)P(B  A)P(A) But what if the numerator and denominator are not explicitly given?

Example: Vitamin B-complex deficiency among general population B1B1 Thiamine B2B2 Riboflavin B3B3 Niacin B4B4 No B deficiency Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. “10% of pop is B 1 -deficient (only), 20% is B 2 -deficient (only), and 30% is B 3 -deficient (only). The remaining 40% is not B-deficient.” Given:

Example: Vitamin B-complex deficiency among general population B1B1 Thiamine B2B2 Riboflavin B3B3 Niacin B4B4 No B deficiency Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. P(B 2 ) =.20P(B 3 ) =.30P(B 1 ) =.10 P(B 4 ) =.40 A = Alcoholic A c = Not Alcoholic Given: P(A ∩ B 1 ) To find these intersection probabilities, we need more information! Prior probs 1.00 P(A ∩ B 2 )P(A ∩ B 3 ) P(A ∩ B 4 ) P(A c ∩ B 1 ) P(A c ∩ B 2 )P(A c ∩ B 3 )P(A c ∩ B 4 )

Example: Vitamin B-complex deficiency among general population B1B1 Thiamine B2B2 Riboflavin B3B3 Niacin B4B4 No B deficiency Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. P(B 2 ) =.20P(B 3 ) =.30P(B 1 ) =.10 P(B 4 ) =.40 Also given… “Alcoholics comprise 35%, 30%, 25%, and 20% of the B 1, B 2, B 3, B 4 groups, respectively.” Given: A = Alcoholic A c = Not Alcoholic Prior probs 1.00 P(A ∩ B 1 )P(A ∩ B 2 )P(A ∩ B 3 ) P(A ∩ B 4 ) P(A c ∩ B 1 ) P(A c ∩ B 2 )P(A c ∩ B 3 )P(A c ∩ B 4 )

Example: Vitamin B-complex deficiency among general population B1B1 Thiamine B2B2 Riboflavin B3B3 Niacin B4B4 No B deficiency Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. P(B 2 ) =.20P(B 3 ) =.30P(B 1 ) =.10 P(B 4 ) =.40 Also given… P(A | B 1 ) =.35P(A | B 2 ) =.30P(A | B 3 ) =.25P(A | B 4 ) =.20 Prior probs 1.00 Given: A = Alcoholic A c = Not Alcoholic P(A ∩ B 1 )P(A ∩ B 2 )P(A ∩ B 3 ) P(A ∩ B 4 ) P(A c ∩ B 1 ) P(A c ∩ B 2 )P(A c ∩ B 3 )P(A c ∩ B 4 )

Example: Vitamin B-complex deficiency among general population B1B1 Thiamine B2B2 Riboflavin B3B3 Niacin B4B4 No B deficiency Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. Also given… Prior probs 1.00 Given: A = Alcoholic A c = Not Alcoholic P(A ∩ B 1 )P(A ∩ B 2 )P(A ∩ B 3 ) P(A ∩ B 4 ) P(A c ∩ B 1 ) P(A c ∩ B 2 )P(A c ∩ B 3 )P(A c ∩ B 4 ) P(A | B 1 ) =.35P(A | B 2 ) =.30P(A | B 3 ) =.25P(A | B 4 ) =.20 P(B 2 ) =.20P(B 3 ) =.30P(B 1 ) =.10 P(B 4 ) =.40 P(A ∩ B) = P(A | B) P(B) Recall:

Example: Vitamin B-complex deficiency among general population B1B1 Thiamine B2B2 Riboflavin B3B3 Niacin B4B4 No B deficiency Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. Also given… Prior probs 1.00 Given: A = Alcoholic A c = Not Alcoholic P(A c ∩ B 1 ) P(A c ∩ B 2 )P(A c ∩ B 3 )P(A c ∩ B 4 ) P(A ∩ B) = P(A | B) P(B) Recall: P(A | B 1 ) =.35P(A | B 2 ) =.30P(A | B 3 ) =.25P(A | B 4 ) =     P(B 2 ) =.20P(B 3 ) =.30P(B 1 ) =.10 P(B 4 ) =.40

Example: Vitamin B-complex deficiency among general population B1B1 Thiamine B2B2 Riboflavin B3B3 Niacin B4B4 No B deficiency Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. Prior probs 1.00 Given: A = Alcoholic A c = Not Alcoholic.10     P(B 2 ) =.20P(B 3 ) =.30P(B 1 ) =.10 P(B 4 ) =.40 P(B 1 | A) = ?P(B 2 | A) = ?P(B 3 | A) = ? P(B 4 | A) = ? Posterior probabilities

B1B1 Thiamine B2B2 Riboflavin B3B3 Niacin B4B4 No B deficiency Example: Vitamin B-complex deficiency among general population Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. P(B 2 ) =.20P(B 3 ) =.30P(B 1 ) =.10 P(B 4 ) =.40 P(B 1 | A) = ?P(B 2 | A) = ?P(B 3 | A) = ? P(B 4 | A) = ? P(A) =.25 P(A c ) = P(B 1 ∩ A) P(A) Prior probs Given: A = Alcoholic A c = Not Alcoholic Posterior probabilities

B1B1 Thiamine B2B2 Riboflavin B3B3 Niacin B4B4 No B deficiency Example: Vitamin B-complex deficiency among general population Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. P(B 1 | A) =P(B 2 | A) =P(B 3 | A) = P(B 4 | A) = P(A) =.25 P(A c ) = Prior probs Given: A = Alcoholic A c = Not Alcoholic Posterior probabilities 1.00 P(B 2 ) =.20P(B 3 ) =.30P(B 1 ) =.10 P(B 4 ) =.40

B1B1 Thiamine B2B2 Riboflavin B3B3 Niacin B4B4 No B deficiency Example: Vitamin B-complex deficiency among general population Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. P(B 1 | A) =.14P(B 2 | A) =.24P(B 3 | A) =.30 P(B 4 | A) = P(A) =.25 P(A c ) = INCREASE DECREASE NO CHANGE; A and B 3 are independent! Prior probs Given: A = Alcoholic A c = Not Alcoholic Posterior probabilities P(B 2 ) =.20P(B 3 ) =.30P(B 1 ) =.10 P(B 4 ) =.40

B1B1 Thiamine B2B2 Riboflavin B3B3 Niacin B4B4 No B deficiency Example: Vitamin B-complex deficiency among general population Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. P(B 1 | A c ) = ??P(B 2 | A c ) = ??P(B 3 | A c ) = ?? P(B 4 | A c ) = ?? P(A) =.25 P(A c ) = Exercise: Prior probs Given: A = Alcoholic A c = Not Alcoholic Posterior probabilities P(B 2 ) =.20P(B 3 ) =.30P(B 1 ) =.10 P(B 4 ) =.40

Example: Vitamin B-complex deficiency among general population Assume B 1, B 2, B 3, B 4 “partition” the population, i.e., they are disjoint and exhaustive. A (Yes) A c (No) Alcoholic etc. Non- deficient Thiamine- deficient Riboflavin- deficient Niacin- deficient C1C1 C2C2 C5C5 C6C6 C4C4 C3C3 C7C7 C8C8 C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8

Prior probabilities: BAYES’ FORMULA Assume B 1, B 2, …, B n “partition” the population, i.e., they are disjoint and exhaustive. AAcAAc B 1 B 2 B 3 ……etc……. B n Given… P(B 1 ) P(B 2 ) P(B 3 ) ……etc……. P(B n ) Conditional probabilities: P(A|B 1 ) P(A|B 2 ) P(A|B 3 ) ……etc……. P(A|B n ) 1 Then… Posterior probabilities: P(B 1 |A) P(B 2 |A) P(B 3 |A) ……etc……. P(B n |A) are computed via P(B i | A) = P(B i ∩ A) P(A) “LAW OF TOTAL PROBABILITY” P(A | B i ) P(B i ) P(A | B 1 ) P(B 1 ) + P(A | B 2 ) P(B 2 ) + …+ P(A | B n ) P(B n ) = P(A) = P(A | B 1 ) P(B 1 ) + P(A | B 2 ) P(B 2 ) + …+ P(A | B n ) P(B n ) for i = 1, 2, 3,…, n P(A ∩ B 1 )P(A ∩ B 2 )P(A ∩ B 3 )P(A ∩ B n ) ……etc……. P(A)P(A) P(A c ∩ B 1 ) P(Ac ∩B2)P(Ac ∩B2)P(A c ∩ B 3 ) ……etc……. P(A c ∩ B n ) P(Ac)P(Ac)

Prior probabilities: BAYES’ FORMULA Assume B 1, B 2, …, B n “partition” the population, i.e., they are disjoint and exhaustive. AAcAAc B 1 B 2 B 3 ……etc……. B n Given… P(B 1 ) P(B 2 ) P(B 3 ) ……etc……. P(B n ) 1 Then… Posterior probabilities: P(B 1 |A) P(B 2 |A) P(B 3 |A) ……etc……. P(B n |A) are computed via P(B i | A) = P(B i ∩ A) P(A) P(A | B i ) P(B i ) P(A | B 1 ) P(B 1 ) + P(A | B 2 ) P(B 2 ) + …+ P(A | B n ) P(B n ) = for i = 1, 2, 3,…, n P(A ∩ B 1 )P(A ∩ B 2 )P(A ∩ B 3 )P(A ∩ B n ) ……etc……. P(A)P(A) P(A c ∩ B 1 ) P(Ac ∩B2)P(Ac ∩B2)P(A c ∩ B 3 ) ……etc……. P(A c ∩ B n ) P(Ac)P(Ac) ……etc……