The Monty Hall problem is a probability puzzle based on a game show scenario. The game show host, Monty Hall, presents the contestant with three doors. Behind one of the doors is a valuable prize, while the other two doors conceal goats. The contestant chooses one door, but before it is opened, Monty Hall, who knows what is behind each door, opens one of the other two doors to reveal a goat. The contestant is then given the option to switch their choice to the remaining unopened door or stick with their initial choice. The question is: should the contestant switch their choice or stick with their initial choice?
The answer is that the contestant should always switch their choice. This is because the probability of the prize being behind the initially chosen door is only 1/3, whereas the probability of it being behind one of the other two doors is 2/3. When Monty Hall opens one of the other doors to reveal a goat, it doesn't change the fact that the probability of the prize being behind one of the other two doors is 2/3. Therefore, by switching their choice, the contestant increases their chances of winning the prize from 1/3 to 2/3. This counterintuitive result can be proven mathematically, and has been confirmed by numerous simulations and experiments.