Welcome to the famous Monty Hall problem! You're on a game show with three doors. Behind one door is a valuable prize, behind the other two are nothing. You choose door 1. The host, who knows what's behind each door, then opens door 3 to reveal it's empty. Now you have a choice: stick with door 1, or switch to door 2. What should you do?
Let's analyze the initial probabilities. When you first choose door 1, each door has an equal one-third probability of having the prize. This means there's a one-third chance the prize is behind your chosen door, and a two-thirds chance it's behind one of the other two doors combined. This is the key insight that makes switching advantageous.
Now the host opens door 3, revealing it's empty. This is crucial: the host always knows where the prize is and will never open the door with the prize. By opening door 3, the host hasn't changed the probability that your original choice is correct - it's still one-third. However, all the probability that was split between doors 2 and 3 now concentrates on door 2 alone, giving it a two-thirds probability.
Let's look at what happens if you play this game many times. Computer simulations and mathematical analysis both confirm that if you always stay with your original choice, you'll win about 33% of the time - exactly one-third. But if you always switch, you'll win about 67% of the time - exactly two-thirds. This means switching literally doubles your chances of winning!
So the answer to the Monty Hall problem is clear: always switch! This counterintuitive result demonstrates how our intuition can mislead us in probability. The key insight is that the host's action of opening an empty door doesn't change your original choice's probability, but concentrates all the remaining probability onto the other unopened door. Remember: switching doubles your chances from one-third to two-thirds. When faced with the Monty Hall problem, always switch doors!