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
Perhaps the best way to think of probability is in terms of information. If two phenomenon are independent, what that really means is that one doesn’t give you any information about the other.
So now let’s consider the doors. You pick a door and Monty then gives you a bit of information you didn’t have before: one of the two doors the prize isn’t behind.
Since you have more information in the ‘second game’ than the ‘first game’, they cannot be independent of one another. That information must *somehow* alter your odds.
In addition to what others have said, it becomes easier to think of this as a single game if you imagine the Monty Hall problem with 100 doors instead of just three.
You choose Door 1.
Host opens all doors except 1 and 55.
The likelihood that you chose the one correct door out of 100 is unlikely. This likelihood doesn’t change even if a bunch of dud doors get opened. When you chose door number one, it will forever be a 1 in 100 choice. If you swap to door 55, it becomes a 1 in 2 choice.
The higher the number of doors, the more obvious this becomes. You can have one billion doors but the host will still open all but your first choice and one other one. It would be nearly impossible to pick the correct number out of a billion, but far, far more possible if the host narrows things down to just two numbers.
Best intuitive explanation I’ve ever had from a friend of mine.
Same game, but 1000 doors.
You pick one.
The host then removed 998 of the doors as having nothing behind them.
Then they ask if you want to change your mind.
With this new information, how confident are you that you picked the only door that has a prize?
You know now that one of the doors left has the prize. The host have you that information. And it’s very useful information.
Are you confident that the one you picked before… When there are a hundred doors, is the actual door?
One other explanation – never mind Monty opening a door.
You pick a door. Then Monty says do you want the door you picked, or do you want the other two doors?
Which would you pick in those circumstances?
Here’s a slightly different way to look at it: After you pick your first door, *Monty is compelled to show you a goat*. If your first door was a goat (2/3) you win by switching. If your first door was a prize (1/3) you lose by switching.
When Marilyn Vos Savant started the big controversy on this years ago, she used language that made it sound as though Monty *just happens* to show you a goat, like it was some sort of random act on his part. Not so. He ALWAYS has to show you a goat. Two times out of three, when your first pick is a goat, Monty guarantees a win by showing you the other one.
The easiest way to change your intuition for the game is to imagine there are 100 doors, and after you pick one, the host closes 98 others. Seems pretty obvious you should always switch now, yes?
Staying only wins if you guessed right on your first guess which is 1/3 chance. Switching always wins if you guessed wrong on your first guess, which is a 2/3 chance.
If they *randomly* chose one of the other doors to show you, then the odds would be 50/50. But also, the randomly chosen door has 1/3 odds of having the car. Since they *intentionally* show a goat, then that changes the odds.
When you pick the first door. There is a 1/3 chance you’re right. If you stay with it you win 1/3 of the time. That would mean, 2/3 of the time YOU SHOULD SWITCH, and you will win. 🙂
Let’s pretend that instead of opening a door and showing you a goat, Monty just says, “Hey, instead of the *one* door you already picked…you can switch to the other *two* doors.”
Clearly two doors gives you better odds of winning than one door does. Specifically 2/3 chance.
And of those two other doors…one of them definitely has a goat behind it (because there is only one winning door)
So ask yourself…..why would opening that door and showing the goat change those odds?
The answer, of course, is that it doesn’t…. which is why you should always switch.
Imagine if there were 1 million doors. One car, and 999,999 goats. You pick one, and Monty Hall opens 999,998 doors. Are you gonna keep your one in a million chances or take the sure thing. Obviously, its only a slight edge to switch with only 3 doors.
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