AE 10: Probability

Learning goals

• Introduce probabilities and how we can use them to understand categorical data
• Create a contingency table using pivot_wider() and kable()
• Use a contingency table to explore the relationship between two categorical variables.

Introduction

library(tidyverse)
library(knitr)

sta199 <- read_csv("sta199-fa21-year-major.csv")

For this Application Exercise, we will look at the year in school and majors for students taking STA 199 in Fall 2021. The data set includes the following variables:

• section: STA 199 section
• year: Year in school
• major_category: Major / academic interest.
  • For the purposes of this AE, we’ll call this the student’s “major”.

Definitions

• The probability of an event tells us how likely an event is to occur, and it can take values from 0 to 1, inclusive. It can be viewed as
  • the proportion of times the event would occur if it could be observed an infinite number of times.
  • our degree of belief an event will happen.
• An event is the basic element to which probability is applied, e.g. the result of an observation or experiment.
  • Example: A is the event a student in STA 199 is a sophomore.
• A sample space is the set of all possible outcomes. Each outcome in the sample space is disjoint or mutually exclusive meaning they can’t occur simultaneously.
  • Example: The sample space for year is {First-year, Sophomore, Junior, Senior}

Exercise 1

Let’s take a look at the majors. Note that we have categorized majors so that each student can only be in one major category.

• What is the sample space for major? You can use code to identify the sample space.

sta199 %>% 
  distinct(major_category) %>% 
  kable

major_category
other
pubpol only
stats only
compsci only
undecided
stat + other major
econ only

• Let’s make a table that includes the majors, the number of students in each, and the associated probabilities.

sta199 %>% 
  count(major_category) %>%
  mutate(prop = n / sum(n)) %>%
  kable

major_category	n	prop
compsci only	40	0.1619433
econ only	15	0.0607287
other	98	0.3967611
pubpol only	38	0.1538462
stat + other major	36	0.1457490
stats only	10	0.0404858
undecided	10	0.0404858

• What is the probability a randomly selected STA 199 student is a “pubpol only” major?

sta199 %>% 
  count(major_category) %>%
  mutate(prop = n / sum(n)) %>%
  filter(major_category == "pubpol only")

## # A tibble: 1 × 3
##   major_category     n  prop
##   <chr>          <int> <dbl>
## 1 pubpol only       38 0.154

• What is the probability a randomly selected STA 199 student is studying statistics?

sta199 %>% 
  mutate(stats_any = ifelse(major_category == "stats only" | major_category == "stat + other major", 1, 0)) %>%
  summarize(mean_stats = mean(stats_any))

## # A tibble: 1 × 1
##   mean_stats
##        <dbl>
## 1      0.186

• What is the probability a randomly selected STA 199 student is not a “pubpol only” major?

sta199 %>% 
  mutate(not_pub_pol = ifelse(major_category == "pubpol only", 0, 1)) %>%
  summarize(mean_not_pub_pol = mean(not_pub_pol))

## # A tibble: 1 × 1
##   mean_not_pub_pol
##              <dbl>
## 1            0.846

Exercise 2

Now let’s make a table looking at the relationship between year and major.

sta199 %>%
  count(year, major_category)

## # A tibble: 23 × 3
##    year       major_category         n
##    <chr>      <chr>              <int>
##  1 First-year compsci only           8
##  2 First-year econ only              6
##  3 First-year other                 39
##  4 First-year pubpol only           22
##  5 First-year stat + other major    26
##  6 First-year stats only             7
##  7 First-year undecided              5
##  8 Junior     compsci only           7
##  9 Junior     econ only              3
## 10 Junior     other                 12
## # … with 13 more rows

We’ll reformat the data into a contingency table, a table frequently used to study the association between two categorical variables. In this contingency table, each row will represent a year, each column will represent a major, and each cell is the number of students have a particular combination of year and major.

To make the contingency table, we will use a new function in dplyr called pivot_wider(). It will take the data frame produced by count() that is current in a “long” format and reshape it to be in a “wide” format.

We will also use the kable() function in the knitr package to neatly format our new table.

sta199 %>% 
  count(year, major_category) %>%
  pivot_wider(id_cols = c(year, major_category),#how we identify unique obs
              names_from = major_category, #how we will name the columns
              values_from = n, #values used for each cell
              values_fill = 0) %>% #how to fill cells with 0 observations 
  kable() # neatly display the results

year	compsci only	econ only	other	pubpol only	stat + other major	stats only	undecided
First-year	8	6	39	22	26	7	5
Junior	7	3	12	4	1	0	0
Senior	2	0	5	1	1	0	0
Sophomore	23	6	42	11	8	3	5

• How many students in STA 199 are first-years and in the “econ only” majors category?
• How many students in STA 199 are in the “other” major category?

Exercise 3

For each of the following exercises:

1. Calculate the probability using the contingency table above.

2. Then write code to check your answer using the sta199 data frame and dplyr functions.

• What is the probability a randomly selected STA 199 student is a sophomore?

sta199 %>% 
  count(year) %>%
  mutate(prop = n / sum(n)) %>%
  filter(year == "Sophomore")

## # A tibble: 1 × 3
##   year          n  prop
##   <chr>     <int> <dbl>
## 1 Sophomore    98 0.397

• What is the probability that a randomly selected STA 199 student is a “compsci only” major?

sta199 %>% 
  count(major_category) %>%
  mutate(prop = n / sum(n)) %>%
  filter(major_category == "compsci only")

## # A tibble: 1 × 3
##   major_category     n  prop
##   <chr>          <int> <dbl>
## 1 compsci only      40 0.162

• What is the probability that a randomly selected STA 199 student is a sophomore or a “compsci only” major?

sta199 %>% 
  mutate(comp_soph = ifelse(year == "Sophomore" | major_category == "compsci only", 1, 0)) %>%
  summarize(mean_stats = mean(comp_soph))

## # A tibble: 1 × 1
##   mean_stats
##        <dbl>
## 1      0.466

• What is the probability that a randomly selected STA 199 student is a sophomore and a “compsci only” major

sta199 %>% 
  mutate(comp_soph = ifelse(year == "Sophomore" & major_category == "compsci only", 1, 0)) %>%
  summarize(mean_stats = mean(comp_soph))

## # A tibble: 1 × 1
##   mean_stats
##        <dbl>
## 1     0.0931

Resources

• Notes on pivot_wider and pivot_longer
  • Click here for slides
  • Click here for video