Monty Hall problem

To change or not to change doors?

Most people familiar with the Monty Hall problem find it counter-intuitive that we should always accept the offer to change doors in the game, as that gives us a ~66% probability of success in the long run (vs. ~33% if we don’t change doors). You can read the Wiki article linked below to understand the logic behind it, but you can also simulate 300 attempts (x-axis) at the game to see how never changing vs. always changing doors affects the probability of success (y-axis).

The function in the code below, mHall(), is a simple simulation of the game (no arguments needed).

Code

mHall = function(){
  
  options = c("money", "goat", "goat")
  doors = sample(x = options, size = 3)
  names(doors) = c("A", "B", "C")
  
  firstChoice = readline(prompt = "Which door do you choose: A, B, or C? ")
  
  if(doors[firstChoice] == "money"){
    badDoors = doors[!names(doors) %in% firstChoice]
    newOffer = sample(x = badDoors, size = 1)
    secondChoice = readline(prompt = paste("I can tell you that door ", names(newOffer), 
                                           " has a ", 
                                           doors[names(newOffer)], 
                                           ". Would you like to change your answer (y or n)? ", 
                                           sep = ""))
    if(secondChoice == "y"){
      newChoice = doors[!names(doors) %in% c(firstChoice, names(newOffer))]
      return(paste("Here's what's behind door ", names(newChoice), ": ", 
                   newChoice[[1]], sep = ""))
    } else {
      return(paste("Here's what's behind door ", firstChoice, ": ", 
                   doors[[firstChoice]], sep = ""))
    }
  } else {
    notChosen = doors[!names(doors) %in% firstChoice]
    goodDoor = doors[names(which(notChosen == "money"))]
    badDoor = doors[names(which(notChosen == "goat"))]
    secondChoice = readline(prompt = paste("I can tell you that door ", names(badDoor), 
                                           " has a ", doors[names(badDoor)], 
                                           ". Would you like to change your answer (y or n)? ", 
                                           sep = ""))
    if(secondChoice == "y"){
      newChoice = goodDoor
      return(paste("Here's what's behind door ", names(newChoice), ": ", 
                   newChoice[[1]], sep = ""))
    } else {
      return(paste("Here's what's behind door ", firstChoice, ": ",
                   doors[[firstChoice]], sep = ""))
    }
  }
  
  return(doors[firstChoice])
}