2.1 Simulation Study

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?

# message: FALSE
library(tidyverse)

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library(purrrfect)

Attaching package: 'purrrfect'

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

    replicate, tabulate

Problem 8:

p_heads = 0.4
n_flips = 8

one_omega = \() sample(c("H", "T"), n_flips, replace = TRUE, prob = c(p_heads, 1 - p_heads))

N = 10000

sim8 = tibble(omega = base::replicate(N, one_omega(), simplify = FALSE)) %>%
  mutate(
    n_heads = map_int(omega, \(flips) sum(flips == "H")),
    last_two_heads = map_lgl(omega, \(flips) all(tail(flips, 2) == "H"))
  )

sim8 %>%
  summarise(prob_at_least_half = mean(n_heads >= 4))

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

sim8 %>%
  summarise(prob_last_two_heads = mean(last_two_heads))

# A tibble: 1 × 1
  prob_last_two_heads
                <dbl>
1               0.158

Problem 9:

studied = rep(c(TRUE, FALSE), c(4, 2))

one_omega = \() sample(studied, 4, replace = FALSE)
studiedcount = \(omega) sum(omega)

N = 100000

sim9 = tibble(omega = base::replicate(N, one_omega(), simplify = FALSE)) %>%
  mutate(ycount = map_int(omega, studiedcount))

sim9 %>%
  summarise(prop_all_studied = mean(ycount == 4))

# A tibble: 1 × 1
  prop_all_studied
             <dbl>
1           0.0659