Monty Hall
On the Monty Hall game show, the contestant was shown three closed doors. Behind one of the doors was a car, and behind the other two, were goats. The host (Monty Hall) knew what was behind each door (The contestant did not!). The contestant was asked to choose a door. Then the host opened one of the other two doors to reveal a goat. Next the contestant had to choose whether to
1. Stay with the door he/she had chosen and recieve the prize behind it
or
2. Switch to the other closed door and recieve the prize behind that door.
It is assumed that the contestant wants to win the car!
In this game, if you are the contestant you can choose between two possible fixed strategies; either you stay with the original choice or you switch. Contemplate the following questions:
What are your chances of winning the car if your strategy is to stay with the original choice?
What are your chances of winning the car if you switch to the other closed door?
Is there a greater probability of winning the car if you switch to the other closed door?
Check what your friends think! Usually there is some disagreement on the answers to the above questions. These are questions that challenge our intuition. The solution of the problem is given below. If you are not convinced, you have the opportunity to play the game many times using one of the simulations given below(I personally prefer the one with the pigs!)
I would suggest playing the game fifty times (or more) with each strtegy. Record the results in a table as follows:
| Strategy :Stay | Strategy : Switch | |||
| Game | Result | Game | Result | |
| 1 2 3 4 5 etc.. |
win lose win lose lose etc.. |
1 2 3 4 5 etc |
win win win lose lose etc... |
|
The record the relative frequency of wins (winning the car) in each case according to the formula:
Relative frequency of wins = #wins/#games
This will give an estimate of the probability of winning for each strategy, switching and staying with the original choice. (The simulation with the pigs below keeps track of the relative frequencies).
Simulations
Monty Hall 2 (pigs)
Solution To Monty Hall Problem