Superfact 22: Suppose you’re on a game show, and you’re given the choice between three doors: Behind one door is a car; behind the other two doors there are goats. You want to pick the car. You pick a door, and the host, who knows what’s behind the three doors, opens another door revealing a goat. Now the question is, is it to your advantage to switch door choice? The answer is yes. And that is the surprising Monty Hall Problem.
It is quite common to argue that it does not matter. You don’t know what is behind the two remaining doors so it should be 50/50 right? In a test involving 228 people only 13% chose to switch. However, you should switch.
Monty Hall, the gameshow host of the Let’s Make a Deal television game show, knows where the car is, so he never chooses the door with the car. And by curating the remaining two doors for you, he raises the odds that switching is always a good bet. By switching your choice, you have a 2/3 chance of winning the car but if you stay with your original choice, you only have a 1/3 chance of winning the car.
So why is this a super-fact? First, we know it is true. It is mathematically proven and experimentally verified that switching door is the best choice. Secondly, this was widely contested and is still surprising to people. Finally, probabilistic thinking is the key to being rational and making good decisions. This fact is true, important and disputed and thus a super fact.
One way of viewing the situation is by noting that there is a 1/3 chance that the car is behind any door that the contestant picks and a 2/3 chance that the car is behind one of the other two doors.
If Monty opens one of the two doors that the contestant did not pick there is still a 1/3 probability that the car is behind the door the contestant picked and a 2/3 chance that the car is behind one of the other two doors. However, one of the doors that the contestant did not pick is now known to feature a goat. Therefore, the probability that the car is behind the other door is 2/3.
The table below is probably (no pun intended) a better way of illustrating the situation. In the table door 1 is the door designated to be the contestant’s first choice. Monty opens one of the remaining doors that has a goat behind it.
| Behind door 1 | Behind door 2 | Behind door 3 | Result if staying at door 1 | Result if switching to door offered. |
| Goat | Goat | Car | Wins goat | Wins Car |
| Goat | Car | Goat | Wins goat | Wins Car |
| Car | Goat | Goat | Wins Car | Wins goat |
There are various other ways of explaining the situation including Steven Pinker’s approach. It is easy to test this is real life and repeated experiments and simulations shown that if you switch you have a 2/3 chance of winning.
As an example of the controversy this probability puzzle caused was Marily Savant’s column in Parade Magazine. As a side note, Marilyn Vos Savant is the person who has the highest recorded intelligence quotient (IQ) as stated in the Guinness Book of Records. In response to a question regarding the Monty Hall game show problem she wrote that you should switch. She received letters from 10,000 readers disputing this, including 1,000 with PhDs. In the long run she prevailed.
To see the other Super Facts click here
Newstrade
Hi Thomas, I read this with interest. I would have thought switching doors was obvious, so I’m surprised to read it isn’t.
LikeLiked by 2 people
I’m thinking the same thing, Robbie – maybe I read this wrong? Also, now I’m wondering what odds are of me being in that situation lol
LikeLiked by 1 person
So you both saw it. That is amazing!
LikeLiked by 1 person
My expectations for myself are low 😂
LikeLiked by 1 person
Well, I wouldn’t be on a game show so I’d never be in this exact situation, but the principle applies to other situations. Maybe we are just the smart ones – wink!
LikeLiked by 2 people
da-AL, you are wondering how we survived, and I have to say that’s an interesting thought. Good decision making, especially in the modern world, require us to use the facts we have and figure out probabilities and as a psychologist noted, we are super bad at it. Our brains aren’t built for it, but we need it. This is just an example, but it is an example of how we get stuck on first impressions (must be 50/50), gut feelings, and does not think through deeply what is going on. The game show ran for decades without anyone figuring it out, and Marilyn starts a media storm by figuring it out. That’s how bad we are.
LikeLiked by 1 person
I am not surprised you say that, but this is a hard problem for most people. I also belonged to the 13% who would have switched but not because I knew why, but because I guessed that it was not the seemingly obvious answer. Incidentally, a Facebook article about Marilyn Vos Savant showed up in my feed after I posted this (FB seems to be spying on me). She had an IQ of 228, the world record, and she received a lot of blow back when she stated that you should switch. This was what the article said :
This woman holds the highest recorded IQ ever: an astonishing 228. Far surpassing Einstein (160-190), Hawking (160), and Musk (155). Yet, despite her brilliance, she faced ridicule for her response to a seemingly simple problem.
But she saw what no one else could.
Here’s her story:
Marilyn Vos Savant was far from an ordinary child.
By the age of 10, she had:
• Memorized entire books
• Read all 24 volumes of the Encyclopedia Britannica
• Achieved the highest recorded IQ of 228
She seemed destined for a life of genius.
But reality took a different turn.
“No one paid much attention to me—mostly because I was a girl. And I accepted that,” Marilyn Vos Savant once said.
She attended a regular public school, left Washington University after two years to help run her parents’ business, and seemed destined for an ordinary life.
But in 1985, everything changed.
The Guinness Book of World Records listed her as having the “Highest IQ” ever recorded: 228.
Suddenly, Marilyn was thrust into the spotlight:
• Featured on the covers of New York Magazine and Parade Magazine
• Guest on Late Night with David Letterman
But she couldn’t have anticipated what lay ahead.
The Rise and the Question
Marilyn joined Parade Magazine to write the iconic “Ask Marilyn” column—a dream for someone with a passion for writing.
Yet, this dream turned into a nightmare with a single question in September 1990.
The Monty Hall Problem
Named after Monty Hall, the host of Let’s Make a Deal, the question went like this:
You’re on a game show.
There are 3 doors.
• 1 door hides a car.
• The other 2 hide goats.
You choose a door. The host opens another door, revealing a goat.
Should you switch doors?
Marilyn’s answer: “Yes, you should switch.”
The backlash was overwhelming. She received over 10,000 letters, including nearly 1,000 from PhDs, insisting she was wrong:
• “You are the goat!”
• “You blew it, and you blew it big!”
• “Maybe women look at math problems differently than men.”
But was she wrong?
The Math Behind the Answer
Consider the two possible scenarios:
You pick the car (1/3 chance):
• If you switch, you lose.
You pick a goat (2/3 chance):
• Monty reveals the other goat.
• If you switch, you win.
Switching gives you a 2/3 chance of winning.
Eventually, her answer was proven correct.
Vindication
MIT ran computer simulations confirming her logic.
MythBusters tested it and reached the same conclusion.
Some academics even apologized.
So why did so many fail to see the truth?
The Reasons People Got It Wrong
• They “reset” the scenario instead of recognizing the shifting probabilities.
• The simplicity of 3 doors obscured the underlying math.
• Many assumed each remaining door had a 50% chance.
Marilyn’s View
Marilyn blamed the compulsory schooling system for discouraging independent thinking. She argued that it:
• Creates passive learners
• Stifles exploration
• Hinders critical thinking
A Blessing and a Burden
Marilyn admits that her intellect often feels isolating—there’s no one to turn to when she needs answers.
Still, she sees her intelligence as a gift, not a curse.
Read more incredible stories: Stranger Things Drive You Insane
LikeLiked by 1 person
Goodness it’s a wonder our species has survived 😝
LikeLiked by 1 person
HI Thomas, very interesting about this woman and terrible about the abuse she faced when she was right. I never understand how other people thing so can’t understand why they wouldn’t obviously switch – haha! Mind you, I get told repeatedly by my work leadership that not everyone thinks like me.
LikeLiked by 1 person
Yes you are right Robbie. Being able to use the known facts to figure out a probability is key to so much decision making in life, and most of us are really bad at it. That’s why I think an example like this is important. We shouldn’t just use the first impression or a gut feeling but think things through well.
LikeLike
Most people believe whatever thought feels right without thinking it through well. That goes for me too even though I am working on it. That da-AL and you figured it out is kind of amazing. You are right, people were not treating her right. Someone with an IQ 228 does not have to be right all the time but if I have an IQ 100 and I disagree with an IQ 228 on a logical puzzle, that is a red flag. You would think people would at least be more humble about it and consider the possibility that they are wrong.
LikeLike
You are right, Thomas. People are very strange in their behaviour.
LikeLiked by 1 person
Fascinating and I hope I’ll remember to switch doors if I’m ever in that situation.
LikeLiked by 2 people
Ha ha yes me too.
LikeLike
I think I would choose to switch as well. Glad to know that is the correct answer. 🙂 Not that I’ll ever be on a game show…
LikeLiked by 1 person
That is great. It is not just about the game show though. Being able to use what facts you have to estimate a probability is the basis for almost all decision making in life and as psychologists have noted, we are very bad at it. Most people get stuck on what seems to make sense to them at first without thinking it through. Like the, it-must-be-50/50 trap in this case.
LikeLiked by 1 person
That makes a lot of sense.
LikeLiked by 1 person
Yes, if you think through it, then it does, but it is not an easy puzzle. It makes you understand that you have to think through things instead of relying on what at first seems right.
LikeLiked by 1 person
I’ve read about this before and I keep telling myself I have to remember it if I’m ever in a situation where it applies, but I’m pretty sure I would forget.
LikeLiked by 1 person
I feel the same. However, the principle of thinking things through instead of trusting instincts and first impressions applies is taught by this example, and that applies everywhere. Being able to use the known facts and formulate a probability, is the basis for all good decision making in life, but almost no one does this. Most of us are really bad at making decisions and deciding what is true. I think this example gives us some insights and it’s not just for a game show.
LikeLike