The setting of this puzzle is the television game show "Let's Make a Deal!", which aired in the early 1970s and was emceed by a man named Monty Hall. You do not need to have ever seen this show to think about this puzzle
On a stage there are three garage sized doors, labeled number 1, number 2 and number 3. Behind one of the doors is a spectacular prize, let's say a luxury car (if you don't like the car, substitute the prize of your dreams). Behind the other two doors are dud prizes, say goats (if you happen to like goats, substitute something you really would not care to win).
You guess which door has the car, trying to guess correctly. Having no other information on which to decide, you pick randomly, say door number 1. Monty Hall then opens door number 3, revealing a goat. At this point Monty gives you the choice to switch your choice to door number 2, or keep your original guess. Once you have chosen, you win whatever is behind the door you have chosen.
The question is: should you stay with your original guess, or should you switch to the other door to increase your chances of winning, or does it matter?
It is a very curious fact that your strategy does matter, i.e., you can increase your chances of winning by pursuing some strategy. It is not an even chance that you will win either way, as the intuition may suggest.
This is a subtle problem to work out, and thinking about it can be frustrating. One way to get evidence about a probability is to model it with an experiment. Play the Monty Hall game and record the results to find the likelihood of winning with each strategy (original choice vs. switching to the other door). Have a friend play the role of the contestant, and you play the role of Monty. After playing the game a few times, you can switch roles.
Have Monty "hide the car" by randomly choosing a number from one to three. You may do this by rolling a die (for example, rolling a 1 or a 4 means door 1, 2 or 5 means door 2, 3 or 6 means door 3) or use a computer. (For example, in Excel, type '=RANDBETWEEN(1,3)' in an empty cell in a fresh spreadsheet. In Maple, type
> with(stats);to load the statistics package, then type
> random[discreteuniform[1,3]]();at the Maple prompt). The contestant then picks a door, using any method, random or not. If the contestant picks the door with the car, Monty flips a coin (or uses the random generator) to pick either of the other two doors to reveal. If the contestant has picked a goat, Monty reveals the other goat, though Monty should go through the motion of flipping the coin anyway so as not to give away the fact that the contestant picked a goat. (Monty always reveals a goat--never the car, and never the door chosen by the contestant). Now the contestant decides to stay with the original choice or switch to the other remaining (unrevealed) door. The game is over. Record which strategy was used by the contestant, and whether the contestant won the car or not. Run this game 20 times or more for each strategy.
Now can you say which strategy is better and why?
Here is a link where you can play the Monty Hall game online.