Experiment 4: The game as a simulation

A hundred games are a start, but no proof yet. R can play the game itself, as often as you like: the car goes behind a random door, the player picks a random door, and the host opens a door that is neither chosen nor hides the car. You build this in two steps: first a single game as a function, then a simulation that calls this function a thousand times and turns the results into a data set that looks like Let's Make a Deal: 100 games with three doors, only ten times as large.

Step 1: One game as a function

For chance there is sample(). sample(1:3, 1) draws one of the numbers 1 to 3. There is one pitfall, and it is exactly at the host: if only one door is left for him, say door 3, then sample(3, 1) does not draw the 3 but a number from 1:3, because R reads a single number as "from 1 up to here".

sample(3, 1)

That is why you draw safely from a vector doors with doors[sample(length(doors), 1)]: what is drawn is a position, not a value.

Write a function play_game() that plays one game. It gets "stay" or "switch" and returns the game as a table with one row, with the same columns as the 100 games plus won. setdiff(a, b) gives the values of a that are not in b; with it you find the doors the host may open and the door you switch to.

play_game <- function(decision) {
  doors <- 1:3
  prize_door <- sample(doors, 1)
  contestant_choice <- sample(doors, 1)
  # Which doors may the host open, and which one does he open?
  # Which door does the player end up with?
  # Has the player won?
  monty_opens <- NA # replace this with your door
  won <- FALSE # and this with your result
  tibble(prize_door, contestant_choice, monty_opens, decision, won)
}

play_game("switch")

play_game <- function(decision) {
  doors <- 1:3
  prize_door <- sample(doors, 1)
  contestant_choice <- sample(doors, 1)
  options <- setdiff(doors, c(prize_door, contestant_choice))
  monty_opens <- options[sample(length(options), 1)]
  if (decision == "switch") {
    final_choice <- setdiff(doors, c(contestant_choice, monty_opens))
  } else {
    final_choice <- contestant_choice
  }
  won <- final_choice == prize_door
  tibble(prize_door, contestant_choice, monty_opens, decision, won)
}

play_game("switch")

Step 2: The simulation as a data set

Write a function simulate_games(n) that plays n games and returns them as one data set with one row per game. The decision should be random, stay or switch. Two tools help: lapply(x, f) calls f for every value of x and collects the results in a list, and bind_rows() stacks a list of tables into one. Simulate 1,000 games with it and compute how often staying and how often switching wins.

# Copy your function play_game() from step 1 here.

decisions <- sample(c("stay", "switch"), 5, replace = TRUE)
decisions

play_game <- function(decision) {
  doors <- 1:3
  prize_door <- sample(doors, 1)
  contestant_choice <- sample(doors, 1)
  options <- setdiff(doors, c(prize_door, contestant_choice))
  monty_opens <- options[sample(length(options), 1)]
  if (decision == "switch") {
    final_choice <- setdiff(doors, c(contestant_choice, monty_opens))
  } else {
    final_choice <- contestant_choice
  }
  won <- final_choice == prize_door
  tibble(prize_door, contestant_choice, monty_opens, decision, won)
}

simulate_games <- function(n) {
  decisions <- sample(c("stay", "switch"), n, replace = TRUE)
  bind_rows(lapply(decisions, play_game))
}

simulation <- simulate_games(1000)
simulation |>
  group_by(decision) |>
  summarize(games = n(), rate = mean(won))

Think about it: why does won in the end not depend on which door the host opens?

Take the data set with you

Save your simulation as CSV and as RDS. The box then offers both files for download; on your own computer they are in the working directory, and there write_csv() from the tidyverse works instead of write.csv() too.

# Copy play_game() and simulate_games() here.

simulation <- simulate_games(1000)
write.csv(simulation, "monty_hall_simulation.csv", row.names = FALSE)
saveRDS(simulation, "monty_hall_simulation.rds")

Which of the two files keeps that won is a logical value and decision text? Read both back in and compare them with glimpse().