The intuition of most students tells them that each of the doors, the chosen door and the unchosen door, are equally likely to contain the prize so that there is a 50-50 chance of winning with either selection. This, however, is not the case. The probability of winning by using the switching technique is 2/3 while the odds of winning by not switching is 1/3. The easiest way to explain this to students is as follows. The probability of picking the wrong door in the initial stage of the game is 2/3. If the contestant picks the wrong door initially, the host must reveal the remaining empty door in the second stage of the game. Thus, if the contestant switches after picking the wrong door initially, the contestant will win the prize. The probability of winning by switching then reduces to the probability of picking the wrong door in the initial stage which is clearly 2/3.
Despite a very clear explanation of this paradox, most students have a difficulty understanding the problem. It is very difficult to conquer the strong intuition which most students have in this case. As a challenge to students who don't believe the explanation, an instructor may ask the students to actually play the game a number of times by switching and by not switching and to keep track of the relative frequency of wins with each strategy. An applet has developed which allows students to repeatedly play the game and keep track of the results. The applet is given below.
Within the applet, the computer plays the role of the host. Upon loading the applet, students are asked to pick a door by clicking the mouse on the proper region. After the initial pick, the computer reveals an empty door by displaying a picture of a donkey behind one of the numbers. The students then have the option of staying with their initial selection or switching to the remaining door. The computer keeps track of the number of times the game is played with each strategy and the number of times the game is won with each strategy. This information is displayed at the bottom of the applet after each game. Using empirical techniques, the student is then able to see the strategy with the highest probability of winning as the relative frequencies converge to the true probabilities. An instructor may use this game to motivate the ideas of repeated trials as a means for investigating a random phenomena.