The Monty Hall Paradox

The Monty Hall Paradox is a famous probability problem named after the host of the television game show "Let's Make a Deal," Monty Hall. The scenario goes as follows:

Contestant is presented with three doors. Behind one door is a valuable prize (such as a car), and behind the other two doors are less desirable prizes (such as goats).
The contestant selects one of the doors, but the door remains closed for the time being.
Monty, who knows what is behind each door, then opens one of the other two doors to reveal a less desirable prize (a goat). He does this regardless of the contestant's initial choice.
Now, the contestant is given a choice: stick with their original choice or switch to the other unopened door.
The paradox arises because the intuitive answer seems to be that it does not matter whether the contestant switches doors or not, as there are now only two doors left. However, probability theory demonstrates that the contestant's chances of winning the valuable prize increase significantly if they switch doors.

The reason behind this counterintuitive result lies in the fact that when the contestant initially picks a door, there's a 1/3 chance that they picked the prize door and a 2/3 chance that they picked a goat door. Monty's subsequent reveal provides additional information. By switching doors, the contestant essentially capitalizes on the 2/3 chance that they initially picked a goat door.

Despite its apparent simplicity, the Monty Hall Paradox has confounded and intrigued many people and serves as a classic example of how probability can defy common intuition.