Chapter 2 Probability  Sample Spaces and Events.2 - Axioms, Interpretations, and Properties of Probability.3 - Counting Techniques.4 - Conditional Probability.5 - Independence

2 POPULATION Random variable X SAMPLE of size n x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 …etc…. xnxn Data x i Relative Frequencies p(x i ) = f i /n x1x1 p(x1)p(x1) x2x2 p(x2)p(x2) x3x3 p(x3)p(x3) ⋮⋮ xkxk p(xk)p(xk) 1 Frequency Table Density Histogram X Total Area = 1 Probability TableProbability Histogram … at least if X is discrete. (Chapter 3)

3 Outcome Red Orange Yellow Green Blue 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) Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways E Consider the following experiment: Randomly select an individual from the population, and record its color. POPULATION

4 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) Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways F Outcome Red Orange Yellow Green Blue Consider the following experiment: Randomly select an individual from the population, and record its color. POPULATION

5 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 = Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways F Outcome Red Orange Yellow Green Blue #(F C ) = 2 ways Consider the following experiment: Randomly select an individual from the population, and record its color. POPULATION

6 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 = Red Yellow Green Orange Blue #(S) = 5 #(E) = 3 ways #(F) = 3 ways F Outcome Red Orange Yellow Green Blue #(F C ) = 2 ways Consider the following experiment: Randomly select an individual from the population, and record its color. POPULATION

7 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 = 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 Consider the following experiment: Randomly select an individual from the population, and record its color. POPULATION

8 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 = 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 Consider the following experiment: Randomly select an individual from the population, and record its color. POPULATION

9 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 = 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 Consider the following experiment: Randomly select an individual from the population, and record its color. POPULATION

10 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 = 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 Consider the following experiment: Randomly select an individual from the population, and record its color. POPULATION

11 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 = 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 Consider the following experiment: Randomly select an individual from the population, and record its color. POPULATION

12 AB A ⋂ BA ⋂ BA ⋂ BcA ⋂ Bc Ac ⋂ BAc ⋂ B “A only”“B only” Ac ⋂ BcAc ⋂ Bc “Neither A nor B” “A and B” In general, for any two events A and B, there are 4 disjoint intersections:  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

13 A B A ⋂ BA ⋂ BA ⋂ BcA ⋂ Bc Ac ⋂ BAc ⋂ B “A only”“B only”“A and B” In general, for any two events A and B, there are 4 disjoint intersections:  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 BcBc Ac ⋂ BcAc ⋂ Bc “Neither A nor B”

14 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 = 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 of outcomes? Consider the following experiment: Randomly select an individual from the population, and record its color. POPULATION

Consider the following experiment: Randomly select an individual from the population, and record its color. Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. POPULATION POPULATION (Pie Chart) 15 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 = #(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 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. Outcome Red Orange Yellow Green Blue OutcomeProbability Red0.20 Orange0.20 Yellow0.20 Green0.20 Blue

POPULATION (Pie Chart) Consider the following experiment: Randomly select an individual from the population, and record its color. Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. 16 Event Any subset of S (including the empty set , and S itself). E = “Primary Color”= {Red, Yellow, Blue} F = “Hot Color”= {Red, Orange, Yellow} “Cold Color” = {Green, Blue}“Not F” =Complement F C = #(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 “The probability of Red is equal to 0.20” P(Red) = 0.20 … …… But what does it mean?? All probs are > 0, and sum = 1. Outcome Red Orange Yellow Green Blue OutcomeProbability Red0.20 Orange0.20 Yellow0.20 Green0.20 Blue # R code for colors.r R = rep("Red", ) O = rep("Orange", ) Y = rep("Yellow", ) G = rep("Green", ) B = rep("Blue", ) pop = c(R, O, Y, G, B) plot.new() plot(0, 0, type = "n", axes = F, xlim = c(0, 300), ylim = c(0, 1), xlab = "n = # Trials", ylab = "#(Red) / n") axis(1) axis(2) box() lines(c(0, 300), c(.2,.2), lty = 2) i = 0 for (n in 1:300) { color = sample(pop, 1, replace = TRUE) if (color == "Red") i = i+1 relfreq = i/n points(n, relfreq, pch=19, cex =.5, col = "red") }

POPULATION (Pie Chart) Consider the following experiment: Randomly select an individual from the population, and record its color. Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. 17 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 = #(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 Red Yellow Green Orange Blue E F “The probability of Red is equal to 0.20” P(Red) = 0.20 … …… But what does it mean?? All probs are > 0, and sum = 1. Outcome Red Orange Yellow Green Blue OutcomeProbability Red0.20 Orange0.20 Yellow0.20 Green0.20 Blue

18 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 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. Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color.

P(  ) = 0 These outcomes are said to be “equally likely.” 19 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 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. Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color.

P(  ) = 0 These outcomes are said to be “equally likely.” 20 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 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. Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 21 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 = 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 All probs are > 0, and sum = 1. P(E) = 0.60

Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 22 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 = 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. P(F) = 0.45

Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 23 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 = 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.

Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 24 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 = 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.

Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color. OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 25 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 = 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 26 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 = 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 P(E ⋃ F) = Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color.

All probs are > 0, and sum = 1. P(E ⋃ F) = OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 27 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 = 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 Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color.

All probs are > 0, and sum = 1. P(E ⋃ F) = P(E) OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue P(  ) = 0 28 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 = 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) Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color.

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. 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 = 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. Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color.

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. 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 = 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. Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color.

BBCBC A P(A ⋂ B ) P(A ⋂ B C ) P(A)P(A) ACAC P(A C ⋂ B ) P(A C ⋂ B C ) P(AC)P(AC) P(B)P(B) P(BC)P(BC) 1.0 P(E ⋃ F) = P(E) + P(F) P(E ⋃ F) = P(E) + P(F) – P(E ⋂ F) F E If events E and F are disjoint, then P(E ⋂ F) = 0. So… In general, for any two events A and B, there are 4 disjoint intersections: AB A ⋂ BA ⋂ BA ⋂ BcA ⋂ Bc Ac ⋂ BAc ⋂ B “A only”“B only” Ac ⋂ BcAc ⋂ Bc “Neither A nor B” “A and B” Probability Table

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 OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue 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) P(E) = 0.60 P(F) = 0.45 Red Yellow Green Orange Blue E F Probability Table All probs are > 0, and sum = 1. Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue 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) 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. Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color.

OutcomeProbability Red0.10 Orange0.15 Yellow0.20 Green0.25 Blue 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}. 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. Perform repeated trials of the following experiment: Randomly select an individual from the population, and record its color.

~ 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: 36 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 37 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

38 AB C In general, for three events A, B, and C… A ⋂ B A ⋂ C B ⋂ C

39 AB C In general, for three events A, B, and C… A ⋂ B

40 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…

41 AB C A ⋂ B ⋂ C A ⋂ B ⋂ C c A 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… “All three events occur” A ⋂ Bc ⋂ CA ⋂ Bc ⋂ C

42 AB C A ⋂ B ⋂ C A ⋂ B ⋂ C c A 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… “Exactly two events occur” A ⋂ Bc ⋂ CA ⋂ Bc ⋂ C

43 AB C A ⋂ B ⋂ C A ⋂ B ⋂ C c A 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… “At least two events occur” A ⋂ Bc ⋂ CA ⋂ Bc ⋂ C

44 AB C A ⋂ B ⋂ C A ⋂ B ⋂ C c A 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… “Exactly one event occurs” A ⋂ Bc ⋂ CA ⋂ Bc ⋂ C