Game One: You are given a choice of three doors. You pick number one. The host opens one of the other two doors, having been given instructions that, if you pick the car, the host is to open one of the other doors, and if you pick a goat, the host opens the other door with a goat. Stalemate. It is a predetermined outcome.
Game Two: The prior game’s outcome stands. The new choice you have is do you keep door number one, or do you switch?
How do you have a 2/3 chance of winning if you switch?
In: 107
They’re not entirely separate; Game One gave you information that you can use in Game Two.
Let’s imagine a different game. Suppose there are two coins: one is a normal fair coin, the other is a two-headed coin. The host picks one of the two coins at random and then flips it. Your job is to guess whether it’s the two-headed coin.
If the flipped coin lands “tails”, then the answer is obvious: it’s not the two-headed coin. But what if it lands “heads”? In this case, you should guess that it probably *is* the two-headed coin, because the two-headed coin is twice as likely to land “heads” as the fair coin. You have a two-thirds chance of being right.
(25% of the time, the host will pick the fair coin, it’ll land “tails”, and you’ll correctly guess that it’s the fair coin. 25% of the time, the host will pick the fair coin, it’ll land “heads”, and you’ll wrongly guess that it’s the two-headed coin. 25% of the time, the host will pick the two-headed coin, it’ll land “heads”, and you’ll rightly guess that it’s the two-headed coin. And 25% of the time, the host will pick the two-headed coin, it’ll land on the other “heads”, and you’ll rightly guess that it’s the two-headed coin. So you’ll be right 50/75 of the times that you guess “it’s the two-headed coin”. 50/75 = 2/3.)
This is what’s going on in the Monty Hall problem. Suppose you picked Door #1 and the host opened Door #3. You now know that the car is not behind Door #3. It was originally equally like to be behind Door #1 or Door #2. However, if it were behind Door #1, there would have been only a 50% chance of the host opening Door #3; if it were behind Door #2, there would have been a 100% chance of the host opening Door #3. So, just as you should guess that the coin which landed “heads” is the two-headed coin, you should guess that the car is behind Door #2.
Latest Answers