Teaching Basics of the Knowledge Space Theory

Development Team:

Nadine Jacobsen

Franziska Klein

Ana Lapa

Alexandra Sarafoglou

Alessandro Silvestri

Theoretical Background: What are knowledge spaces?

So you may wonder what this knowledge space you just saw might be?

Knowledge spaces are a formal (mathematical) theory started in the eighties for the assessment of knowledge in human beings. The initial objective was to construct "an efficient machine for the assessment of knowledge". So what distinguishes the style from the Classical test theory and others? In KST no attempt is made to compute a numerical score for representing how much a student has learned.Rather, the goal of the assessment is to provide a precise description of what the student knows at a given moment.Assume that the collection Q of all possible questions that can be asked in probabilistic theory is available. This set Q contains the ten questions asked above and is called a domain of knowledge.

The knowledge of a student is represented by the subset of all those questions in Q that the student is capable of answering correctly (excluding lucky guesses and careless errors). This subset is named knowledge state.The knowledge state K is a subset of Q. So what you will see in the example is a Knowledge Structure which is a pair (Q;K) , where K is a collection of subsets of Q, containing at least the empty set and Q. Since some of the questions depend on knowledge, also relevant for other questions, a quasi-order on a set Q of questions can be established, this is called surmise relation. In our example, people who answer Item B1 correct, will also have knowledge about the Items A1 and A2.

I want to learn more!

A knowledge Structure of elementary algebra

Suppose that after a careful examination of the following 6 problems in a toy domain of knowledge.

A teacher establishes the following relations:

That means, that a person able to solve problem e is also able to solve problem b, since this is a predecessor of problem e.

If you want to display this surmise relations you can use a Hasse Diagram as shown below.

Testing your Knowledge State in Basic Probability Theory

B1: A box contains 10 marbles with the following colours: 2 yellow, 5 blue and 3 red. What is the probability that a randomly drawn marble is yellow or blue?

B2: Consider a standard deck containing 52 different cards. What is the probability of drawing a 9 in a black suit?

B3: Consider a standard deck containing 52 different cards. What is the probability of not drawing a heart?

B4: A box contains marbles that are red or yellow, small or large. The probability of drawing a red marble is .70, the probability of drawing a small marble is .40. Assume that the colour of the marbles is independent from their size. What is the probability of randomly drawing a marble that is yellow and small?

Here you see your knowledge state:

Look at your Test results

This is the most common knowledge state:

Compared to students that completed a statistics course the occurence of this knowledge state has a probability of 23 percent.

# setwd("/var/www/shiny-server/tquant/LisbonApps/Group1/app") source("questions.R", local = TRUE) # load questions # Load packages library(shiny) library(shinydashboard) library("graph") library("Rgraphviz") # for plot library("paircompviz") library("pks") # for results ##### UI PAGE ##### ui <- dashboardPage(skin = "blue", dashboardHeader(title = "Knowledge Space Structure Test", titleWidth = 350), dashboardSidebar( # Sidebar Menu Names and Icons sidebarMenu( menuItem("Welcome!", tabName = "welcome",icon = icon("fort-awesome")), menuItem("Introduction", tabName = "introduction",icon = icon("gear")), menuItem("Examples", tabName = "examples",icon = icon("eye")), menuItem("Test", tabName = "test", icon = icon("graduation-cap")), menuItem("Results", tabName = "result", icon = icon("line-chart")))), dashboardBody( tabItems( # Welcome Page tabItem(tabName = "welcome", h2("Teaching Basics of the Knowledge Space Theory"), br(), fluidPage( includeScript("../../../Matomo-tquant.js"), fluidRow( column(7, img(src="idea.png", height = 250), br(), br(), br(), img(src="tquant.png", height = "100") ), column(5, box(title=h2("Development Team:"), solidHeader = TRUE, status = "primary", width = '100%', h2(img(src="oldenburg.png", width = "40"), " Nadine Jacobsen"), h2(img(src="oldenburg.png", width = "40"), " Franziska Klein"), h2(img(src="lisboa.png", width = "40"), " Ana Lapa"), h2(img(src="amsterdam.png", width="40"), " Alexandra Sarafoglou"), h2(img(src="padova.png", width = "40"), " Alessandro Silvestri") )) ) )), # Introduction Page tabItem(tabName = "introduction", box(title ="Theoretical Background: What are knowledge spaces?", status = "primary", solidHeader = TRUE, width = 12, p('So you may wonder what this knowledge space you just saw might be?'), img(src = "knowledgeState.png", height = 300, align = "right"), p('Knowledge spaces are a formal (mathematical) theory started in the eighties for the assessment of knowledge in human beings. The initial objective was to construct "an efficient machine for the assessment of knowledge". So what distinguishes the style from the Classical test theory and others? In KST no attempt is made to compute a numerical score for representing how much a student has learned.Rather, the goal of the assessment is to provide a precise description of what the student knows at a given moment.Assume that the collection', span('Q', style= 'color:blue'),'of all possible questions that can be asked in probabilistic theory is available. This set Q contains the ten questions asked above and is called a', span(' domain of knowledge.', style = 'color:blue')), p('The knowledge of a student is represented by the subset of all those questions in Q that the student is capable of answering correctly (excluding lucky guesses and careless errors). This subset is named knowledge state.The ' , span('knowledge state K', style = 'color:blue'),' is a subset of Q. So what you will see in the example is a', span('Knowledge Structure', style = 'color:blue'), 'which is a pair', span('(Q;K)', style= 'color:blue'),', where K is a collection of subsets of Q, containing at least the empty set and Q. Since some of the questions depend on knowledge, also relevant for other questions, a quasi-order on a set Q of questions can be established, this is called', span('surmise relation.', style= 'color:blue'), 'In our example, people who answer Item B1 correct, will also have knowledge about the Items A1 and A2.'), a('I want to learn more!', href ='https://drive.google.com/file/d/0B-lgUbL4jfs8S0hUaXBmOTNoLW8/view?pli=1'), br(),br()), img(src = "kst1.png", height = 375, width = 600, align = "center")), # Example Page tabItem(tabName = "examples", box(title = 'A knowledge Structure of elementary algebra', status = "primary", solidHeader = TRUE, width = 8, p('Suppose that after a careful examination of the following 6 problems in a toy domain of knowledge.'), img(src="problems.png",height = 200, align = "center"), p('A teacher establishes the following relations:'), img(src="surmiseRelation.png", height = 40, align = "center"), br(), br(), p('That means, that a person able to solve problem e is also able to solve problem b, since this is a', span('predecessor', style = 'color:blue'), 'of problem e.'), p('If you want to display this surmise relations you can use a', span('Hasse Diagram', style = 'color:blue'), 'as shown below.'), img(src="knowledgeStructureExample.png", height = 150, align = "center"))), # Testing Page tabItem(tabName = "test", fluidPage( fluidRow( column(8, h3("Testing your Knowledge State in Basic Probability Theory"), selectInput("b1", questions$b1, # question b1 choices = c(' ' = 'empty', '7/10' = 'correct', '1/3' = 'choice1', '2/7' = 'choice2', '1/5' = 'choice3'), width = '100%'), selectInput("b2", questions$b2, # question b2 choices = c(' ' = 'empty', '9/52' = 'choice1', '2/52' = 'correct', '2/9' = 'choice2', '9/52' = 'choice3'), width = '100%'), selectInput("b3", questions$b3, # question b3 choices = c(' ' = 'empty', '1/4' = 'choice', '3/4' = 'correct', '1/2' = 'choice2', '1/8' = 'choice3'), width = '100%'), selectInput("b4", questions$b4, # question b4 choices = c(' ' = 'empty', '40%' = 'choice1', '70%' = 'choice2', '50%' = 'choice3', '12%' = 'correct'), width = '100%'), uiOutput("block1")), column(4, box(title = "Here you see your knowledge state:", status = 'primary', solidHeader = TRUE, width = '100%', plotOutput("KST")), actionButton("KS.plot", "Plot my Results", icon = icon("refresh")) )) ), br(), br(), br(), br() # add empty space ), tabItem(tabName = "result", h1('Look at your Test results'), fluidPage( fluidRow( column(6, textOutput('knowledgeState'), br(), br(), textOutput('descriptiveYourState'), br(), br(), textOutput('descriptiveMostCommonState'), br(), br(), wellPanel( textInput('Name', placeholder = "Enter your name",""), actionButton("submit","Save Your Results"), uiOutput('table'), uiOutput('table2')) ), column(6, box(title = "This is the most common knowledge state:", status = 'primary', solidHeader = TRUE, width = '100%', plotOutput("KST2")), h4("Compared to students that completed a statistics course the occurence of this knowledge state has a probability of 23 percent.") ) ) ))) )) ###### SERVER ####### server <- function(input, output){ problem <- c("a1", "a2", "a3", "a4", "b1", "b2", "b3", "b4", "c1", "c2") state <<- rep(FALSE, 10) # will be updated data(probability) dat <- probability dat <- as.pattern(cbind(dat$b202, dat$b201, dat$b204, dat$b202, dat$b207, dat$b209, dat$b205, dat$b210, dat$b212, dat$b211)) # your knowledge state state.pattern <<- paste(as.character(as.numeric(state)), collapse = "") index <- which(names(table(dat)) %in% state.pattern) freq <- round(table(dat)[index]/504 * 100, 3) # questions source("block1.R", local = TRUE) # first round of questions; block b source("block0.R", local = TRUE) # second round of questions source("block2.R", local = TRUE) # second round of questions; block a source("block3.R", local = TRUE) # thrid round of questions; block c # plot source("KSplot.R", local = TRUE) go.plot <- eventReactive(input$KS.plot,{ # observe action Button hasse_Diag(state) }) output$KST <- renderPlot({ # plot diagram go.plot() }) output$KST2 <- renderPlot({ stateMostCommon <- rep(TRUE, 10) hasse_Diag(stateMostCommon) }) ################ # save results # ################ # Data <- reactive({ # # if (input$submit > 0) { # write.table(as.numeric(state), # file=paste("KStest_",input$Name, ".txt", sep =""))} # }) output$table <- renderText({ if (input$submit > 0){ c("Your Data has been saved! Thank you for your participation.") write.table(as.numeric(state), file=paste("KStest_",input$Name, ".txt", sep ="")) } }) output$table2 <- renderText({ if (input$submit > 0){ c("Your Data has been saved! Thank you for your participation.") } }) # Look at results output$knowledgeState <- renderText({ if(sum(state) == 0){ c("You can master 0 problems.") } else { # print knowledge state c("You can master the following problem(s):", problem[which(state == TRUE)]) } }) output$descriptiveYourState <- renderText({ if(length(index) == 0){ c("You are the first to display this knowledge state.") } else { # find your knowledge state paste("Compared to students that completed a statistics course the occurence of your knowledge state has a probability of", freq, "percent", sep=" ") } }) # output$descriptiveMostCommonState <- "Hi" } shinyApp(ui = ui, server = server)

# Block 0 # what happens after Block 0 output$block0.0 <- renderText({ # print knowledge state c("You can master 0 problems.") }) output$block0.1 <- renderText({ # print knowledge state c("You can master problem(s):", problem[which(state == TRUE)]) })

## Block 1 # what happens after block 1 output$block1 <- renderUI({ # all correct from block b if(input$b1 == "correct" & input$b2 == "correct" & input$b3 == "correct" & input$b4 == "correct"){ state[1:8] <<- TRUE fluidPage( selectInput('c1', label = questions$c1, # present all questions from c choices = c(' ' = 'empty', '16%' = 'choice1', '48%' = 'correct', '12%' = 'choice2', '2%' = 'choice3'), width = '100%'), selectInput('c2', label = questions$c2, choices = c(' ' = 'empty', '2%' = 'choice1', '30%' = 'choice2', '12%' = 'correct', '50%' = 'choice3'), width = '100%'), textOutput("block3.1")) # only three correct from block b } else if(input$b1 == "correct" & input$b2 == "correct" & input$b3 == "correct" & input$b4 != "correct" & input$b1 != "empty" & input$b4 != "empty") { state[1:7] <<- TRUE fluidPage( selectInput('c1', label = questions$c1, # present one question from block c choices = c(' ' = 'empty', '16%' = 'choice1', '48%' = 'correct', '12%' = 'choice2', '2%' = 'choice3'), width = '100%'), textOutput("block3.2")) } else if(input$b1 == "correct" & input$b2 == "correct" & input$b3 != "correct" & input$b4 == "correct") { state[c(1:4, 5, 6, 8)] <<- TRUE fluidPage( selectInput('c1', label = questions$c1, choices = c(' ' = 'empty', '16%' = 'choice1', '48%' = 'correct', '12%' = 'choice2', '2%' = 'choice3'), width = '100%'), textOutput("block3.2")) } else if(input$b1 == "correct" & input$b2 != "correct" & input$b3 == "correct" & input$b4 == "correct") { state[c(1:4, 5, 7, 8)] <<- TRUE fluidPage( selectInput('c2', label = questions$c2, choices = c(' ' = 'empty', '2%' = 'choice1', '30%' = 'choice2', '12%' = 'correct', '50%' = 'choice3'), width = '100%'), textOutput("block3.3")) } else if(input$b1 != "correct" & input$b2 == "correct" & input$b3 == "correct" & input$b4 == "correct" & input$b1 != "empty" & input$b4 != "empty") { state[c(2:4, 6, 7, 8)] <<- TRUE fluidPage( selectInput('a1', label = questions$a1, # present 1 question from block a choices = c(' ' = 'empty', '.75' = 'choice1', '.65' = 'choice2', '.40' = 'choice3', '.55' = 'correct'), width = '100%'), uiOutput("block2.1")) # only two correct from block b } else if(input$b1 == "correct" & input$b2 == "correct" & input$b3 != "correct" & input$b4 != "correct" & input$b3 != "empty" & input$b4 != "empty") { state[c(1:3, 5, 6)] <<- TRUE fluidPage( selectInput('a4', label = questions$a4, choices = c(' ' = 'empty', '.75' = 'choice2', '.35' = 'choice2', '.40' = 'choice3', '.65' = 'correct'), width = '100%'), uiOutput("block2.2")) } else if(input$b1 != "correct" & input$b2 == "correct" & input$b3 == "correct" & input$b4 != "correct" & input$b1 != "empty" & input$b4 != "empty") { state[c(2:4, 6, 7)] <<- TRUE fluidPage( selectInput('a1', label = questions$a1, choices = c(' ' = 'empty', '.75' = 'choice1', '.65' = 'choice2', '.40' = 'choice3', '.55' = 'correct'), width = '100%'), uiOutput("block2.1")) } else if(input$b1 != "correct" & input$b2 == "correct" & input$b3 != "correct" & input$b4 == "correct" & input$b1 != "empty" & input$b4 != "empty") { state[c(2:4, 6, 8)] <<- TRUE fluidPage( selectInput('a1', label = questions$a1, choices = c(' ' = 'empty', '.75' = 'choice1', '.65' = 'choice2', '.40' = 'choice3', '.55' = 'correct'), width = '100%'), uiOutput("block2.1")) } else if(input$b1 == "correct" & input$b2 != "correct" & input$b3 == "correct" & input$b4 != "correct" & input$b1 != "empty" & input$b4 != "empty") { state[c(1, 2, 4, 5, 7)] <<- TRUE fluidPage( selectInput('a3', label = questions$a3, choices = c(' ' = 'empty', '.40' = 'choice1', '.32' = 'correct', '1.20'= 'choice2', '.50' = 'choice3'), width = '100%'), textOutput("block2.3")) } else if(input$b1 != "correct" & input$b2 != "correct" & input$b3 == "correct" & input$b4 == "correct" & input$b1 != "empty" & input$b4 != "empty") { state[c(2:4, 7, 8)] <<- TRUE fluidPage( selectInput('a1', label = questions$a1, choices = c(' ' = 'empty', '.75' = 'choice1', '.65' = 'choice2', '.40' = 'choice3', '.55' = 'correct'), width = '100%'), uiOutput("block2.1")) } else if(input$b1 == "correct" & input$b2 != "correct" & input$b3 != "correct" & input$b4 == "correct") { state[c(1:4, 5, 8)] <<- TRUE fluidPage( # finished K(b1, b4) textOutput("block0.1") ) # only one correct from block b } else if(input$b1 != "correct" & input$b2 != "correct" & input$b3 != "correct" & input$b4 == "correct" & input$b1 != "empty" & input$b4 != "empty") { state[c(3, 4, 8)] <<- TRUE fluidPage( selectInput('a1', label = questions$a1, # present 2 questions from block a choices = c(' ' = 'empty', '.75' = 'choice1', '.65' = 'choice2', '.40' = 'choice3', '.55' = 'correct'), width = '100%'), selectInput('a2', label = questions$a2, choices = c(' ' = 'empty', '1/3' = 'choice1', '12/30' = 'correct', '1/2' = 'choice2', '8/30' = 'choice3'), width = '100%'), textOutput("block2.4")) } else if(input$b1 != "correct" & input$b2 != "correct" & input$b3 == "correct" & input$b4 != "correct" & input$b1 != "empty" & input$b4 != "empty") { state[c(2, 4, 7)] <<- TRUE fluidPage( selectInput('a1', label = questions$a1, choices = c(' ' = 'empty', '.75' = 'choice1', '.65' = 'choice2', '.40' = 'choice3', '.55' = 'correct'), width = '100%'), selectInput('a3', label = questions$a3, choices = c(' ' = 'empty', '.40' = 'choice1', '.32' = 'correct', '1.20'= 'choice2', '.50' = 'choice3'), width = '100%'), textOutput("block2.5")) } else if(input$b1 != "correct" & input$b2 == "correct" & input$b3 != "correct" & input$b4 != "correct" & input$b1 != "empty" & input$b4 != "empty") { state[c(2, 3, 6)] <<- TRUE fluidPage( selectInput('a1', label = questions$a1, choices = c(' ' = 'empty', '.75' = 'choice1', '.65' = 'choice2', '.40' = 'choice3', '.55' = 'correct'), width = '100%'), selectInput('a4', label = questions$a4, choices = c(' ' = 'empty', '.75' = 'choice2', '.35' = 'choice2', '.40' = 'choice3', '.65' = 'correct'), width = '100%'), textOutput("block2.6")) } else if(input$b1 == "correct" & input$b2 != "correct" & input$b3 != "correct" & input$b4 != "correct" & input$b1 != "empty" & input$b4 != "empty") { state[c(1, 2, 5)] <<- TRUE fluidPage( selectInput('a3', label = questions$a3, choices = c(' ' = 'empty', '.40' = 'choice1', '.32' = 'correct', '1.20'= 'choice2', '.50' = 'choice3'), width = '100%'), selectInput('a4', label = questions$a4, choices = c(' ' = 'empty', '.75' = 'choice2', '.35' = 'choice2', '.40' = 'choice3', '.65' = 'correct'), width = '100%'), textOutput("block2.7")) # none correct from block b but none empty } else if(input$b1 != "correct" & input$b2 != "correct" & input$b3 != "correct" & input$b4 != "correct" & input$b1 != "empty" & input$b4 != "empty") { fluidPage( selectInput('a1', label = questions$a1, # present all questions from block a choices = c(' ' = 'empty', '.75' = 'choice1', '.65' = 'choice2', '.40' = 'choice3', '.55' = 'correct'), width = '100%'), selectInput('a2', label = questions$a2, choices = c(' ' = 'empty', '1/3' = 'choice1', '12/30' = 'correct', '1/2' = 'choice2', '8/30' = 'choice3'), width = '100%'), selectInput('a3', label = questions$a3, choices = c(' ' = 'empty', '.40' = 'choice1', '.32' = 'correct', '1.20'= 'choice2', '.50' = 'choice3'), width = '100%'), selectInput('a4', label = questions$a4, choices = c(' ' = 'empty', '.75' = 'choice1', '.35' = 'choice2', '.40' = 'choice3', '.65' = 'correct'), width = '100%'), textOutput("block2.8")) } else { fluidPage( textOutput("block0.0")) } })

## Block 2 # what happens when we ask questions in block a # ask one question # ask a1 output$block2.1 <- renderUI({ if(input$a1 == "correct"){ state[1] <<- TRUE} fluidPage( selectInput('c2', label = questions$c2, choices = c(' ' = 'empty', '2%' = 'choice1', '30%' = 'choice2', '12%' = 'correct', '50%' = 'choice3'), width = '100%'), textOutput("block3.3")) }) # ask a4 output$block2.2 <- renderUI({ if(input$a4 == "correct"){ state[4] <<- TRUE} fluidPage( selectInput('c1', label = questions$c1, choices = c(' ' = 'empty', '16%' = 'choice1', '48%' = 'correct', '12%' = 'choice2', '2%' = 'choice3'), width = '100%'), textOutput("block3.2")) }) # ask a3 output$block2.3 <- renderText({ if(input$a3 == "correct"){ state[3] <<- TRUE} # print knowledge state c("You can master problem(s):", problem[which(state == TRUE)]) }) # ask two questions # ask a1 and a2 output$block2.4 <- renderText({ if(input$a1 == "correct"){ state[1] <<- TRUE} if(input$a2 == "correct"){ state[2] <- TRUE} # print knowledge state c("You can master problem(s):", problem[which(state == TRUE)]) }) # ask a1 and a3 output$block2.5 <- renderText({ if(input$a1 == "correct"){ state[1] <<- TRUE} if(input$a3 == "correct"){ state[3] <<- TRUE} # print knowledge state c("You can master problem(s):", problem[which(state == TRUE)]) }) # ask a1 and a4 output$block2.6 <- renderText({ if(input$a1 == "correct"){ state[1] <<- TRUE} if(input$a4 == "correct"){ state[4] <<- TRUE} # print knowledge state c("You can master problem(s):", problem[which(state == TRUE)]) }) # ask a3 and a4 output$block2.7 <- renderText({ if(input$a3 == "correct"){ state[3] <<- TRUE} if(input$a4 == "correct"){ state[4] <<- TRUE} # print knowledge state c("You can master problem(s):", problem[which(state == TRUE)]) }) # ask a1, a2, a3 and a4 output$block2.8 <- renderText({ if(input$a1 == "correct"){ state[1] <<- TRUE} if(input$a2 == "correct"){ state[2] <<- TRUE} if(input$a3 == "correct"){ state[3] <<- TRUE} if(input$a4 == "correct"){ state[4] <<- TRUE} # print knowledge state if(sum(state) == 0){ c("You can master 0 problems.") } else { c("You can master problem(s):", problem[which(state == TRUE)])} })

## Block 3 # what happens after block 3 output$block3.1 <- renderText({ if(input$c1 == "correct"){ state[9] <<- TRUE } if(input$c2 == "correct"){ state[10] <<- TRUE} # print knowledge state c("You can master problem(s):", problem[which(state == TRUE)]) }) output$block3.2 <- renderText({ if(input$c1 == "correct"){ state[9] <<- TRUE} # print knowledge state c("You can master problem(s):", problem[which(state == TRUE)]) }) output$block3.3 <- renderText({ if(input$c2 == "correct"){ state[10] <<- TRUE} # print knowledge state c("You can master problem(s):", problem[which(state == TRUE)]) })

#generate matrix data <- matrix(data = FALSE, ncol = 10, nrow = 10) # 1-5-9, 1-6-10, 1-7-10 data[2, 5] = data[5, 9] = TRUE data[2, 6] = data[6, 9] = TRUE data[2, 7] = data[7, 10] = TRUE # 2-5-9, 2-8-9 data[4, 7] = data[7, 10] = TRUE data[4, 8] = data[8, 10] = TRUE #3-6-10 data[1, 5] = data[5, 9] = TRUE # 4-8-9, 4-7-10 data[3, 8] = data[8, 10] = TRUE data[3, 6] = data[6, 9] = TRUE hasse_Diag = function(vec) { vec[which(vec == TRUE)] = "darkseagreen3" vec[which(vec == FALSE)] = "white" v = c("A1", "A2", "A3", "A4", "B1", "B2", "B3", "B4", "C1", "C2") hasse(data, v, ebg = "black", vcol = "navy", vbg = vec, vsize = .75, main = " ", compress = TRUE) }

# Questions and answers # block b questions <- data.frame( b1 = c( 'B1: A box contains 10 marbles with the following colours: 2 yellow, 5 blue and 3 red. What is the probability that a randomly drawn marble is yellow or blue?'), b2 = c('B2: Consider a standard deck containing 52 different cards. What is the probability of drawing a 9 in a black suit?'), b3 = c('B3: Consider a standard deck containing 52 different cards. What is the probability of not drawing a heart?'), b4 = c('B4: A box contains marbles that are red or yellow, small or large. The probability of drawing a red marble is .70, the probability of drawing a small marble is .40. Assume that the colour of the marbles is independent from their size. What is the probability of randomly drawing a marble that is yellow and small?'), # block a a1 = c('A1: A bag contains 5-Cent, 10-Cent and 20-Cent pieces. The probability of drawing a 5-Cent piece is .20, that of drawing a 10-Cent piece is .45 and that of drawing a 20-Cent piece is .35. What is the probability that the coin randomly drawn is a 5-Cent or a 20-Cent piece?'), a2 = c('A2: A box contains 30 marbles with the following colours: 8 red, 10 black and 12 yellow. What is the probability that a randomly drawn marble is yellow?'), a3 = c('A3: In a school the probability of observing a male pupil is 40% and the probability of observing a righthanded pupil is 80%. Assume that the hand writing is independent from the gender. What is the probability of observing a pupil that is male and right-handed?'), a4 = c('A4: A bag contains 5-Cent, 10-Cent and 20-Cent pieces. The probability of drawing a 5-Cent piece is .35, that of drawing a 10-Cent piece is .25 and that of drawing a 20-Cent piece is .40. What is the probability that the coin randomly drawn is NOT a 5-Cent piece?'), # block c c1 = c('C1: A box contains 20 marbles. 10 marbles are red, 6 are yellow and 4 are black. 12 marbles are small and 8 are large. Assume that the colour of the marbles is independent from their size. What is the probability of randomly drawing a marble that is red or yellow and that is small?'), c2 = c('C2: In a garage there are 50 cars. 20 are black and 10 are diesel. Assume that the colour of the cars is independent from the fuel. What is the probability that a car is not black and it is diesel?') )

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