Urn Problems) Urn Problems

Introduction to Urn Problems

In this tutorial, we will be using Urn Problems to illustrate different probability distributions. The general idea of an urn problem is that there is an urn filled with balls of different colours (here only two colours), from which a certain number of balls is drawn. One colour, here blue, is seen as a success. The drawing process can be restricted by certain rules, such as whether balls are sampled with or without replacement, and when to stop drawing balls. Depending on the rules applied to the general problem different probability distributions will result. In this tutorial, we will briefly introduce the binomial, hypergeometric, negative binomial and negative hypergeometric distributions.

Example: Set the number of balls in the urn and then draw with replacement. You can draw either by clicking on the ball or by randomly drawing balls. Play around till you feel comfortable and then move on to the other tabs.

Number of blue balls in the urn:

Number of red balls in the urn:

The Binomial Distribution

Drawing balls from the urn with replacement results in the so called binomial distribution. The number of drawn blue balls is counted. The probability of drawing k blue balls in n draws is given by

$$ P(\color{deepskyblue}{k}\,blue\,balls) \,=\,\binom{n}{ \color{deepskyblue}{k}} \cdot\left(\frac{\color{blue}{K}}{N} \right)^{\color{deepskyblue}{k}}\cdot \left(1-\frac{\color{blue}{K}}{N} \right)^{n-\color{deepskyblue}{k}}$$

where N represents the total number of balls (population size), K the total number of blue balls in the urn (successes in the population), n the number of balls drawn from the urn (sample size), and k the number of blue balls drawn from the urn (sample successes).

Number of blue balls in the urn:

Number of red balls in the urn:

Number of draws:

Number of blue balls in the urn:

Number of red balls in the urn:

The Hypergeometric Distribution

Drawing balls from the urn without replacement results in the so called hypergeometric distribution. The number of drawn blue balls is counted. The probability of drawing k blue balls in n draws is given by

$$ P(\color{deepskyblue}{k}\,blue\,balls)\,= \,\frac{\binom{K}{ \color{deepskyblue}{k}}\cdot \binom{N-K}{n- \color{deepskyblue}{k}}}{\binom{N}{n}}$$

where N represents the total number of balls (population size), K the total number of blue balls in the urn (successes in the population), n the number of balls drawn from the urn (sample size), and k the number of blue balls drawn from the urn (sample successes).

Number of blue balls in the urn:

Number of red balls in the urn:

Number of draws:

Number of blue balls in the urn:

Number of red balls in the urn:

The urn will automatically reset by changing the numbers of blue or red balls in the urn.

The Negative Binomial Distribution

Drawing balls from the urn with replacement until k blue balls (success balls) are drawn results in the so called negative binomial distribution. For simplicity we are using k = 1. This special case is also called geometric distribution. The probability of drawing n time until the first blue ball is drawn is given by

$$P(n\,draws\,until\,the\,first \,blue\,ball\,is\,drawn)\,= \,\binom{n-1}{\color{deepskyblue}{k}-1}\cdot \left(\frac{\color{blue}{K}}{N}\right)^ \color{deepskyblue}{k}\cdot \left( 1-\frac{\color{blue}{K}}{N} \right) ^{n-\color{deepskyblue}{k}}$$

where N represents the total number of balls (population size), K the total number of blue balls in the urn (successes in the population), n the number of balls drawn from the urn (sample size), and k the number of blue balls drawn from the urn (sample successes).

Number of blue balls in the urn:

Number of red balls in the urn:

Number of draws:

Number of blue balls in the urn:

Number of red balls in the urn:

The Negative Hypergeometric Distribution

Drawing balls from the urn without replacement until k blue balls (success balls) are drawn results in the so called negative hypergeometric distribution. For simplicity we are using k = 1. The probability of drawing n time until the first blue ball is drawn is given by

$$P(n\,draws\,until\,the\,first \,blue\,ball\,is\,drawn)\,=\, \frac{\binom{N-n}{\color{blue}{K-k}}}{ \binom{N}{K}}$$

where N represents the total number of balls (population size), K the total number of blue balls in the urn (successes in the population), n the number of balls drawn from the urn (sample size), and k the number of blue balls drawn from the urn (sample successes).

Number of blue balls in the urn:

Number of red balls in the urn:

Number of draws:

Number of blue balls in the urn:

Number of red balls in the urn:

The urn will automatically reset by changing the numbers of blue or red balls in the urn.

Do you think you got it? Try yourself!

Here you can check your knowledge about the distributions in this tutorial.

Which distributions calculate the probability of drawing k blue balls from the urn?

Binomial Distribution Hypergeometric Distribution Negative Binomial Distribution Negative Hypergeometric Distribution

Which distributions calculate the probability of n draws until the first blue ball is drawn?

Binomial Distribution Hypergeometric Distribution Negative Binomial Distribution Negative Hypergeometric Distribution

For which distributions are balls drawn with replacement?

Binomial Distribution Hypergeometric Distribution Negative Binomial Distribution Negative Hypergeometric Distribution

For which distributions are balls drawn without replacement?

Binomial Distribution Hypergeometric Distribution Negative Binomial Distribution Negative Hypergeometric Distribution

Please select the letter representing the following descriptions in the formula.

Number of blue balls in the urn

Total number of balls in the urn

Number of blue balls drawn from the urn

Total number of balls drawn from the urn

Please select the distributions belonging to the formula.

$$P\,=\,\frac{\binom{N-n}{K-1}}{ \binom{N}{K}}$$

$$P\,=\,\binom{n-1}{k-1} \cdot\left(\frac{K}{N} \right)^{k}\cdot \left(1-\frac{K}{N} \right)^{n-k} $$

$$ P\,=\,\frac{\binom{K}{k} \cdot\binom{N-K}{n-k}}{\binom{N}{n}}$$

$$P\,=\,\binom{n}{k} \cdot\left(\frac{K}{N} \right)^{k}\cdot \left(1-\frac{K}{N} \right)^{n-k} $$

There are eight blue and four red balls in the urn. You draw ten balls.

Please select the distributions belonging to the plots.

This is a

There are seven blue and five red balls in the urn. You draw until the first blue ball is drawn.

Please select the distributions belonging to the plots.

This is a

There are three blue and thirteen red balls in the urn. You draw sixteen balls.

Please select the distributions belonging to the plots.

This is a

There are five blue and five red balls in the urn. You draw five balls.

Please select the distributions belonging to the plots.

This is a

There are four blue and twelve red balls in the urn. You draw until the first blue ball is drawn.

Please select the distributions belonging to the plots.

This is a

Richard has an urn and offers a game to Tom. If Tom draws five or more red balls, he wins. They won't replace balls drawn from the urn. He draws nine times. Tom only wants to play the game, if his chance to win is greater than 0.5.

Use the app to get the result.

Manuel has an urn with seven balls. Four are blue and three are red. Manuel draws without replacement.

Use the app to get the result.

Select the alternatives for which Tom would play the game

5 blue and 5 red balls

10 blue and 5 red balls

5 blue and 10 red balls

7 blue and 8 red balls

9 blue and 8 red balls

Select the right statements

The probability of drawing five balls until the first blue ball is drawn is zero.

The probability of drawing three balls until the first blue ball is drawn is 0.06

The probability of drawing three balls until the first blue ball is drawn is 0.69.

The probability that the first ball drawn is blue is 0.57.

Sarah has an urn with elven blue and nine red balls. She draws without replacement.

Use the app to get the result.

Lara has an urn with seven blue and eight red balls. Lara wants to know how often she has to draw to get at least one blue ball with a probability of 0.7

Use the app to get the result.

Select the true statements

The probability of drawing three blue balls with five draws is 0.42

It is most likely to draw six or seven blue balls out of ten balls.

The probability for drawing one blue and one red ball is 0.62

It most likely to draw two blue balls out of four.

How often does Lara have to draw?

At least two times.

One time is enough.

At least three times.

Samuel has an urn with six blue and five red balls. He draws five times.

Use the app to get the result.

Mia has three blue balls and two red balls. She puts four of them in a bag.

Use the app to get the result.

Choose the right statements

The probability to draw three blue balls with replacement is 0.34

It is most likely to draw two blue balls without replacement

The probability of drawing the first blue ball with replacement in draw five is 0.02

The probability of drawing the first blue ball without replacement in draw five is 0.02

Select the right statements

The probability for two blue balls in the bag is 0.6.

There are at least two red balls in the bag.

It is most likely that two blue balls are in the bag.

There are at least two blue balls in the bag.

show with app

app.R

library(shiny)
library(shinydashboard)
library(ggplot2)

#=========================ui========================================
ui <- dashboardPage(skin = "blue",
  dashboardHeader(
    title = tagList(
      tags$span(
        class = "logo-mini", 
        HTML("<span style='font-size:8px'>Urn Problems</span>)")
      ),
      tags$span(
        class = "logo-lg", "Urn Problems"
      )
    )
  ),
                    
# ====================== Sidebar ========================
  dashboardSidebar( 
# create a sidebar
    sidebarMenu( 
# create sidebar menu items
      menuItem(
        "Introduction",
        tabName = "Introduction",
        icon = icon("info-circle") 
      ),
      menuItem(   
        HTML("Binomial <br> &nbsp &nbsp &nbsp &nbsp Distribution"),
        tabName = "Binomial",
        icon = icon("menu-right", lib = "glyphicon"),
        menuSubItem( HTML("The Binomial <br> &nbsp &nbsp &nbsp &nbsp 
                          Distribution"), tabName = "Binomial_dist"),
        menuSubItem("The urn", tabName = "Binomial_urn")
      ),
      menuItem(
        HTML("Hypergeometric <br> &nbsp &nbsp &nbsp &nbsp Distribution"),
          tabName = "Hypergeometric",
          icon = icon("menu-right", lib = "glyphicon"),
          menuSubItem(HTML("The Hypergeometric <br> &nbsp &nbsp &nbsp
            &nbsp Distribution"), tabName = "Hyper_dist"),
          menuSubItem("The urn", tabName = "Hyper_urn")
        ),
        menuItem(
          HTML("Negative Binomial <br> &nbsp &nbsp &nbsp &nbsp Distribution"),
          tabName = "Geometric",
          icon = icon("menu-right", lib = "glyphicon"),
          menuSubItem( HTML("The Negative Binomial<br> &nbsp &nbsp &nbsp &nbsp 
                            Distribution"), tabName = "Geo_dist"),
          menuSubItem("The urn", tabName = "Geo_urn")
        ),
        menuItem(
          HTML("Negative Hypergeometric <br> &nbsp &nbsp &nbsp 
               &nbsp Distribution"),
          tabName = "Negative_Hypergeometric",
          icon = icon("menu-right", lib = "glyphicon"),
          menuSubItem(HTML("The Negative Hypergeometric <br>
                           &nbsp &nbsp &nbsp &nbsp Distribution"),
                      tabName = "NegHyp_dist"),
          menuSubItem("The urn", tabName = "NegHyp_urn")
        ),
        menuItem("Your turn!", tabName = "Exercise", 
                 icon = icon("pencil-square-o"))
      )
    ),
                    
# =========================== tagStyle ==================================
    dashboardBody(
      includeScript("../../../Matomo-tquant.js"),
      tags$script(HTML("$('body').addClass('sidebar-mini');")),
      tags$head(tags$style(
          HTML(".main-header .logo {
               font-weight: bold;
               font-size: 22px;}")
        )
      ),
      tags$head(tags$style(
          HTML(".content-wrapper {
               background-color: #FFFFFF;}")
        )
      ),
# change colors of the slider
      tags$style( 
        HTML(".js-irs-0 .irs-single, 
             .js-irs-0 .irs-bar-edge
             {border: blue; 
             background: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-0 .irs-bar 
             {background: blue; 
             border-top: blue;
             border-bottom: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-1 .irs-single, 
             .js-irs-1 .irs-bar-edge
             {border: red; 
             background: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-1 .irs-bar 
             {background: red; 
             border-top: red;
             border-bottom: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-2 .irs-single, 
             .js-irs-2 .irs-bar-edge
             {border: blue; 
             background: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-2 .irs-bar 
             {background: blue; 
             border-top: blue;
             border-bottom: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-3 .irs-single, 
             .js-irs-3 .irs-bar-edge
             {border: red; 
             background: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-3 .irs-bar 
             {background: red; 
             border-top: red;
             border-bottom: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-4 .irs-single, 
             .js-irs-4 .irs-bar-edge
             {border: blue; 
             background: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-4 .irs-bar 
             {background: blue; 
             border-top: blue;
             border-bottom: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-5 .irs-single, 
             .js-irs-5 .irs-bar-edge
             {border: red; 
             background: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-5 .irs-bar 
             {background: red; 
             border-top: red;
             border-bottom: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-6 .irs-single, 
             .js-irs-6 .irs-bar-edge
             {border: blue; 
             background: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-6 .irs-bar 
             {background: blue; 
             border-top: blue; 
             border-bottom: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-7 .irs-single, 
             .js-irs-7 .irs-bar-edge
             {border: red; 
             background: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-7 .irs-bar 
             {background: red; 
             border-top: red; 
             border-bottom: red}"
        )
                      ),
      tags$style(
        HTML(".js-irs-8 .irs-single, 
             .js-irs-8 .irs-bar-edge
             {border: blue; 
             background: blue}"
        )
      ),
      tags$style(
        HTML( ".js-irs-8 .irs-bar 
             {background: blue; 
             border-top: blue;
             border-bottom: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-9 .irs-single, 
             .js-irs-9 .irs-bar-edge
             {border: red; 
             background: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-9 .irs-bar 
             {background: red; 
             border-top: red;
             border-bottom: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-10 .irs-single, 
             .js-irs-10 .irs-bar-edge 
             {border: blue; 
             background: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-10 .irs-bar 
             {background: blue; 
             border-top: blue;
             border-bottom: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-11 .irs-single, 
             .js-irs-11 .irs-bar-edge 
             {border: red; background: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-11 .irs-bar 
             {background: red;
             border-top: red;
             border-bottom: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-12 .irs-single, 
             .js-irs-12 .irs-bar-edge
             {border: blue; 
             background: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-12 .irs-bar 
             {background: blue; 
             border-top: blue;
             border-bottom: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-13 .irs-single, 
             .js-irs-13 .irs-bar-edge
             {border: red; 
             background: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-13 .irs-bar 
             {background: red; 
             border-top: red;
             border-bottom: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-14 .irs-single, 
             .js-irs-14 .irs-bar-edge
             {border: blue; 
             background: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-14 .irs-bar 
             {background: blue; 
             border-top: blue;
             border-bottom: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-15 .irs-single, 
             .js-irs-15 .irs-bar-edge
             {border: red; 
             background: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-15 .irs-bar 
             {background: red; 
             border-top: red; 
             border-bottom: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-16 .irs-single, 
             .js-irs-16 .irs-bar-edge
             {border: blue; 
             background: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-16 .irs-bar 
             {background: blue; 
             border-top: blue;
             border-bottom: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-17 .irs-single, 
             .js-irs-17 .irs-bar-edge
             {border: red; 
             background: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-17 .irs-bar 
             {background: red; 
             border-top: red;
             border-bottom: red}"
        )
      ),
      tags$style(
        HTML(".js-irs-18 .irs-single, 
             .js-irs-18 .irs-bar-edge
             {border: blue;
             background: blue}"
        )
      ),
      tags$style(
        HTML(".js-irs-18 .irs-bar 
             {background: blue; 
             border-top: blue;
             border-bottom: blue}"
        )
      ),
      tags$style(".fa-times {color:#c40505}"),
      tags$style(".fa-check {color:#50ea0e}"),
      tags$style(".fa-repeat {color:#0000c9}"),
      tags$style(".fa-reply {color:#0000c9}"),
      tags$head(tags$link(rel = "stylesheet", 
          type = "text/css", href = "style.css")
      ),
                      
# =========================== Introduction ============                
      tabItems(
        source(file.path("Layout", "Intro.R"), local = TRUE)$value,
                        
# ====================== Binomial ========================================      
        source(file.path("Layout", "Binom.R"), local = TRUE)$value,
        source(file.path("Layout", "Binom2.R"), local = TRUE)$value,

# ====================== Hypergeometric ===================================
        source(file.path("Layout", "Hyper.R"), local = TRUE)$value,
        source(file.path("Layout", "Hyper2.R"), local = TRUE)$value,
     
# ======================== Geometeric =====================================
        source(file.path("Layout", "Geo.R"), local = TRUE)$value,   
        source(file.path("Layout", "Geo2.R"), local = TRUE)$value,  

#==================================== Negative Hypergeometric =================
        source(file.path("Layout", "Neg_Hyp.R"), local = TRUE)$value,
        source(file.path("Layout", "Neg_Hyp2.R"), local = TRUE)$value,

#==================================== Exercise =================
        source(file.path("Layout", "Exercise.R"), local = TRUE)$value
      )
    )
)


#=============================================================================
#====================================== server ===============================
#=============================================================================

server <- function(input, output, session) {
  
# session$onSessionEnded(stopApp)        
# ==================================== Introduction ==========================
  source(file.path("Calculation", "Intro.R"), local = TRUE)$value
  
# ================================ The Binomial distribution ===================
  source(file.path("Calculation", "Binom.R"), local = TRUE)$value
  
# ==================================== Hypergeometric ==========================
  source(file.path("Calculation", "Hyper.R"), local = TRUE)$value
  
# ===================================== Geometric =============================
  source(file.path("Calculation", "Geo.R"), local = TRUE)$value
  
# ======================================= Negative Hypergeometric =============
  source(file.path("Calculation", "Neg_Hyp.R"), local = TRUE)$value 
  
#==================================== Exercise =================
  source(file.path("Calculation", "Exercise.R"), local = TRUE)$value
}

shinyApp(ui = ui, server = server)