2.1 #8

library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.2.1     ✔ readr     2.2.0
✔ forcats   1.0.1     ✔ stringr   1.6.0
✔ ggplot2   4.0.3     ✔ tibble    3.3.1
✔ lubridate 1.9.5     ✔ tidyr     1.3.2
✔ purrr     1.2.2     
── 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 <- 100000
p_head <- 0.4

( replicate(N, sample(c('H','T'), 8, prob = c(p_head, 1-p_head), replace = TRUE), .as = flips)
  %>% mutate(num_heads = map_dbl(flips, \(x) sum(x=='H')),
             at_least_four_heads = map_lgl(flips, \(x) sum(x=='H') >=4 ))
  %>%  summarize(p_at_least_four_heads = mean(at_least_four_heads))
)

# A tibble: 1 × 1
  p_at_least_four_heads
                  <dbl>
1                 0.406

( replicate(N, sample(c('H','T'), 8, prob = c(p_head, 1-p_head), replace = TRUE), .as = flips)
  %>% mutate(last_two_heads = map_lgl(flips, \(x) all(tail(x, 2) == 'H')))
  %>% summarize(p_last_two_heads = mean(last_two_heads))
)

# A tibble: 1 × 1
  p_last_two_heads
             <dbl>
1            0.162