Stat 450 Problem Set 1

Packages

library(tidyverse)

Warning: package 'ggplot2' was built under R version 4.5.2

Warning: package 'tibble' was built under R version 4.5.3

Warning: package 'tidyr' was built under R version 4.5.3

Warning: package 'dplyr' was built under R version 4.5.3

Warning: package 'stringr' was built under R version 4.5.3

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.2.1     ✔ readr     2.1.5
✔ forcats   1.0.0     ✔ stringr   1.6.0
✔ ggplot2   4.0.0     ✔ tibble    3.3.1
✔ lubridate 1.9.4     ✔ tidyr     1.3.2
✔ 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

library(purrrfect)


Attaching package: 'purrrfect'

The following objects are masked from 'package:base':

    replicate, tabulate

N <- 10000

Problem 8

An unfair coin has a 40% chance of landing heads. The coin is flipped 8 times.
a. What is the probability that at least half of the flips are heads?

p_head <- 0.4

one_trial <- \() sample(c('H','T'), 8, prob = c(p_head, 1-p_head), replace = TRUE)
over_half <- \(x) sum(x == 'H') >= 4

(
  replicate(N, one_trial(), .as = flips)
  |> mutate(at_least_half = map_lgl(flips, over_half))
  |> summarize(prob_at_least_half = mean(at_least_half))
)

# A tibble: 1 × 1
  prob_at_least_half
               <dbl>
1              0.403

What is the probability the last two flips are both heads?

last_two <- \(x) x[7] == 'H' & x[8] == 'H'

(
  replicate(N, one_trial(), .as = flips)
  |> mutate(last_two_heads = map_lgl(flips, last_two))
  |> summarize(p_last_two_heads = mean(last_two_heads))
)

# A tibble: 1 × 1
  p_last_two_heads
             <dbl>
1            0.156

Problem 9

Simulate the last warmup problem. A professor has written 6 possible exam questions, of which a student has thoroughly studied 4. Four questions are selected at random for the exam. What is the probability the student can solve all four of the problems on the exam?

#Functions used
selection <- rep(c('S','NS'), c(4,2))
count_S <- \(x) sum(x == 'S')
one_test <- \() sample(selection, 4, replace = FALSE)

#Replicating and getting result
(replicate(N, one_test(), .as = tests)
  |> mutate(correct_q = map_int(tests, count_S))
  |> mutate(all_good = correct_q == 4)
  |> summarize(percent_all = mean(all_good))
)

# A tibble: 1 × 1
  percent_all
        <dbl>
1      0.0618