Have you ever heard of the Monty Hall Problem? Let’s explore this puzzling brain teaser and its solutions, as well as the insight into our lives it offers.
I can sleep at night again, now that I have resolved the Monty Hall Problem – one of life’s most perplexing mysteries. All is well with the universe once more.
What is the persistent question that for so long stole my peace of mind? It is the Monty Hall problem and the goat behind Door Number Three.
The so-called Monty Hall problem is a counter-intuitive statistics puzzle that goes as follows:
- You have to choose one of three doors. Behind one you will find a car; behind each of the others, you will find a goat.
- You pick Door #1, hoping for the car, of course.
- Monty Hall, the game show host, narrows your choices by opening Door #3 to reveal a goat.
- Then Monty offers you a choice: you can stick with your original door or switch to Door #2. What should you do?
Simple logic suggests that there is no advantage to switching doors. With the elimination of Door #3, your odds improve from one-in-three to even-money. It shouldn’t matter whether or not you switch: either way, you will still have a 50-50 chance.
But here human logic fails. By switching doors, you increase your odds from even money to two-thirds.
HERE’S WHY IT WORKS
Contrary to intuition, your original one-in-three chance of winning does not change just because one of the doors has been opened. Why not? Because the two doors you did not choose should be considered, statistically, as one choice — let’s call it Choice B — which is twice as likely as the one door you did choose, which we’ll call Choice A.
Therefore, once one of those two doors has been opened to reveal the goat, the two-to-one odds that the car lies hidden behind either of them become concentrated solely with the remaining door.
In other words, the probability of Choice B hasn’t changed, even though you’ve eliminated one of the doors. From a purely statistical point of view, this logic is sound.
HERE’S WHY IT SHOULDN’T WORK
Let’s take the same principle the Monty Hall Problem is based on to the extreme.
What if you had to choose from 100 doors? After you make your choice, Monty opens 98 of the remaining 99 doors to reveal 98 goats. According to statistics, you now have a 99% chance of winning by switching doors — virtually a sure thing.
But how could this be? Human reason rebels against the logic of raising your odds merely by eliminating incorrect choices. If I have two doors to choose between, how can the odds possibly be 99-to-one? Obviously, this is a statistical fiction that has no bearing on the real world.
Except that, it’s not.
AND HERE’S WHY IT REALLY DOES WORK
If you doubt that switching doors really does improve your odds, you can check your skepticism at the door. It’s been tested and proven. And it’s true.
But there’s more to the story.
Let’s change the rules of the Monty Hall Problem a bit. You’ve chosen your door, leaving two doors unopened. Now, Monty Hall opens one of the remaining doors at random. Behind it is the other goat. Should you change doors to improve your odds?
In this scenario, you might as well stay with your original choice; changing doors will make no difference whatsoever.
So, what’s the difference between the cases in the Monty Hall Problem?
In our first case, you don’t know behind which door the car awaits, but Monty does. By opening the door with the goat, he is not changing the odds; rather, he is using his prior knowledge to expose a wrong option. There is no chance that the car will be behind Door #2 because Monty already knows it isn’t there, which is why he opened that door.
Simply speaking, because Monty knows which door hides the car, he is able to manipulate the results and, consequently, the odds.
As a result, the chances that you have chosen the car remain only one-in-three, so you’re better off switching to the one door that now retains the original Choice B advantage of two-in-three odds.
However, if Monty had opened one door at random, the car might be behind that door. If it is, game over. If not, Door #2 becomes statistically irrelevant, and your odds improve to 50-50. Switch or don’t switch. Statistically, it makes no difference.
WHY DOES IT MATTER?
This distinction in the Monty Hall problem offers a profound insight into the way we lead our lives, even if we won’t be appearing on television game shows.
We like to think of our society as a meritocracy, in which our chances of success increase in proportion to our natural talent, education, savvy, integrity, hard work, and experience. And although sometimes they do, sometimes they don’t.
We’ve all heard these unpleasant truths, and we’ve certainly experienced some of them ourselves:
Leo Durocher: Nice guys finish last.
Clare Boothe Luce: No good deed goes unpunished.
Napoleon: If you wish to be a success in the world, promise everything, deliver nothing.
Ralph Waldo Emerson: The end of the human race will be that it will eventually die of civilization.
Agatha Christie: Where large sums of money are concerned, it is advisable to trust nobody.
Kin Hubbard: When a fellow says, “It is not the money but the principle of the thing,” it`s the money.
Anonymous: It’s not what you know; it’s who you know.
In a perfect world, we would all compete on an even playing field. But the world is far from perfect. There’s insider trading in the stock market, point-shaving and doping in competitive sports, evidence-tampering and procedural loopholes in jurisprudence, and a whole variety of manipulations in electioneering. We’re all familiar with the sad fact that life’s not fair.