Math Problems 10/2/26

Question 1

Construct a bar graph for the given data below.

# install.packages("tidyverse")
library(tidyverse)

## Warning: package 'lubridate' was built under R version 4.5.2

## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr     1.1.4     ✔ readr     2.1.5
## ✔ forcats   1.0.1     ✔ stringr   1.5.2
## ✔ ggplot2   4.0.0     ✔ tibble    3.3.0
## ✔ lubridate 1.9.4     ✔ tidyr     1.3.1
## ✔ purrr     1.1.0     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

q1_data <- data.frame(Color = c("red","orange","green","gray","purple","yellow"),
                      Number = c(4,3,4,1,6,2))
ggplot(q1_data,aes(x = factor(Color),y = Number)) +
  geom_col() +
  scale_fill_identity() +
  labs(title = "Sample Data",
       x = "Color",
       y = "Number") +
  theme_gray()

Question 2

Which of the following is a factor of 315?

A. 8

B. 9

C. 10

D. 11

# install.packages("tidyverse")
library(tidyverse)
q2_data <- data.frame(Choice = LETTERS[1:4],
                      Number = 8:11)
correct_answer <- q2_data %>%
  mutate(Factor = 315 %% Number == 0) %>%
  filter(Factor == TRUE) %>%
  pull(Choice)
cat("The correct answer is:",correct_answer,"\n")

## The correct answer is: B

Question 3

For a group of seven people, what are the chances that any of them celebrate their birthday on the same day of the week?

set.seed(123) # for reproducibility
N <- 1e5 # 100,000 trials
simulation <- replicate(N, {
  birthdays <- sample(1:7,size = 7,replace = T) # weekdays 1 to 7
  any(duplicated(birthdays))
})
cat("The probability is:",mean(simulation),"\n")

## The probability is: 0.9939