Setup

Doors is noting else than a vector representing the three doors in the Monty Hall problem. nbSimulations configures the number of trials (games) we simulate.

doors = 1:3
nbSimulations = 1000

Let’s simulate

Please note we use sample for randomly putting the prize behind a door and making the initial choice.

The if-else statement is for making sure the door without a prize is openend.

In winWithSwitch and winWithoutSwitch we store whether or not the final choice made resulted into a win when respectively switching and not switching doors.

switchedChoiceDoor means that you do not choose your initial choiceDoor and obviously you also want a different door then the one which was opened (i.e. showDoor).

In observedWinPropWithSwitch and observedWinPropWithoutSwitch we calculate the observed winning proportion up to the current number of trials conducted.

for (i in 1:nbSimulations) {
  prizeDoor = sample(doors, 1)
  choiceDoor = sample(doors, 1)
  
  if (choiceDoor != prizeDoor) {
    showDoor = doors[doors != prizeDoor & doors != choiceDoor]
  } else {
    showDoor = sample(doors[doors != choiceDoor], 1)
  }

  switchedChoiceDoor = doors[doors != choiceDoor & doors != showDoor]

  winWithSwitch[i] = switchedChoiceDoor == prizeDoor
  winWithoutSwitch[i] = choiceDoor == prizeDoor

  observedWinPropWithSwitch[i] = mean(winWithSwitch)
  observedWinPropWithoutSwitch[i] = mean(winWithoutSwitch)
}

Outcome of our simulation

## Number of trials: 1000
## Number of wins when switching (proportion): 666 (0.666)
## Number of wins when not switching (proportion): 334 (0.334)

Plotting observedWinPropWithSwitch and observedWinPropWithoutSwitch allows us to see that increasing the number of trials results into a better estimation of the probabilities.