Do you increase your chances of getting a car by opening the door you originally chose or by opening the other unopened door, or is the probability of winning the same either way?
While it is not intuitive, one increases one’s chance of winning a car by always switching. This has puzzled, and continues to puzzle people, including mathematicians.
Simply put, one has a 2/3 chance of winning the car when switching doors, while not switching gives only a 1/3 chance. Experimental simulations validate the analytical arguments.
Here is an intuitive argument. Assuming random placement, there are three possible choices each having a 1/3 (33.3%) probability of occurring. Clearly, if you do not choose to switch, the results will be the exact opposite.
| Case 1 (1/3) | Case 2 (1/3) | Case 3 (1/3) | |
| Contestant 1st Choice | Door with car | Door with goat 1 | Door with goat 2 |
| Monte will show | Door with goat1 or goat2 | Door with goat 2 | Door with goat 1 |
| Choice after switch | Door with other goat | Door with car | Door with car |
| Results | Lose | Win | Win |
