The Monty Hall problem is one of the most enduring brain teasers in history. Made famous by US television gameshow host Monty Hall, it offers contestants a chance to win a car, through making a seemingly simple decision. But there’s a way to do it that brings you closer to driving it home.
A Car and Two Goats
The scenario is this: a contestant faces three doors. Behind one lies a car. Now the contestant picks a door, hoping to win it. Before the big reveal, the host, Monty Hall, opens one of the other doors to reveal a goat.
He then gives you a choice: Would you like to switch to the other unopened door, or stick with your initial choice?
Now mathematically, it does not appear to matter. That is because there are now two doors left, each with a 50-50 chance. Many contestants reason this way, and choose to stick to their original choice.
But as the gameshow reveals, majority of contestants end up being greeted by a bleating goat.
Were they just unlucky?
Not quite so, said then US-columnist Marilyn vos Savant, who offered this explanation.
Switch to Win
In a response to a reader who asked if he should switch to the other door, she said yes.
Savant urged the reader to think of it differently, as such:
Initially, their chances of picking the prize was one in three.
But after one of the goats is revealed, the contestant has a lot more information now than before.
That’s because Hall has prior knowledge of where the car is, and he uses it to choose which door to open. So before, the contestant had no information about any door, now they have partial information as to which door might contain the car – because of Hall’s move.
So, you must switch to get a better chance of winning!
In short, Savant says it’s the amount of information that the contestant now possesses that makes all the difference.
As the gameshow proved, two times out of three, contestants who switched won the car.
A Flurry of Objections
Despite the question’s sly simplicity, some of the world’s leading mathematics, including professors from MIT and renowned research institutions rejected her explanation, arguing the odds were still at 50-50.
Charles Reid from the University of Florida said, ‘May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again?’
Another Professor from Georgetown University went even further, lashing out at American education, ‘You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively towards the solution of a deplorable situation. How many irate mathematicians are needed to get you to change your mind?’
Some even blamed her gender, saying that ‘women look at math problems differently than men’.
In the end though, it was Marilyn who had the last laugh.
After a computer exercise that had confirmed her logic, majority of her critics conceded their mistake.
That included a mathematician at George Mason University, who sent her a public letter saying the episode was an ‘intense professional embarrassment’ and that he was now ‘eating humble pie’.
It’s All in the Mind
But Monty Hall himself felt it was the psychological effect of the question that made all the difference.
“[After I opened a door with a goat], they’d think the odds on their door had now gone up to 1 in 2, so they hated to give up the door no matter how much money I offered…The higher I got, the more [they] thought the car was behind [the other door]. I wanted to con [them] into switching there. That’s the kind of thing I can do when I’m in control of the game. You may think you have probability going for you when you follow the answer in her column, but there’s the psychological factor to consider.”- Monty Hall
In short, Hall believes it was the sting of regret that prevented one from switching, if it did not work out.
Convinced yet? Why don’t you try it out for yourself right here.