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What is the Monty Hall Problem?
The Monty Hall problem is a probability puzzle named after Monty Hall, the original host of the TV show Let’s Make a Deal. It’s a famous paradox that has a solution that is so absurd, most people refuse to believe it’s true.
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No.
1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No.
2?” Is it to your advantage to switch your choice? ~ (From Parade magazine’s Ask Marilyn column)
Should you Switch?
Believe it or not, it’s actually to your benefit to switch:
- If you switch, you have roughly a 2/3 chance of winning the car.
- If you stick to your original choice you have roughly a 1/3 chance of winning the car.
The Monty Hall problem
The Monty hall problem is one of the most famous problems in mathematics and in its original form goes back to a game show hosted by the famous Monty Hall himself.
The contestants on the game show were shown three shut doors. Behind one of these was a high value prize, such as a car. Behind the other two was a low value prize, such as a goat.
If the contestants opened the correct door then they won the prize, otherwise they won nothing.
The contestants were then asked to choose a door, and to tell the host which door they had chosen. This door remained shut for the time being. The host then opened a different door to reveal a goat behind it.
The contestants were then given a choice. They could stay with the door that they have chosen, or they could swap to the remaining unopened door.
The door they finally ended up with was then opened, to reveal the prize car, or maybe just a goat.
The question is: should the contestant change their choice of door or not?
The accepted answer is “yes”. In fact, by swapping doors you double your chances of winning the prize. This is surprising, which is why the problem has become so famous. This answer was given in the gambling film 21 and has also been advocated as a reason why you should make changes in your choice of love.
Car or goat?
However, it turns out that the accepted answer is not always correct and is an example of loose thinking. The answer to whether you switch doors or not depends entirely upon the host (and to some extent the contestant) and what they know, or don't know.
First let's assume that the host knows behind which door the car is. When you, the contestant, have picked a door the host will always choose to open a door with a goat, and you know that this is the case. This is a reasonable assumption: if the host chose to open the door with the car, the game would be over.
If you decide to stick with your door then you win if your initial guess was correct. The chance of this was 1/3 and the host opening a door with a goat doesn't change this: whatever your initial choice, they would always have responded accordingly, making sure they open a door with a goat.
If you swap doors, then you win if your initial guess was wrong. The probability of this was 2/3, and again this probability hasn't changed by the host revealing a goat.
What has changed, however, is the fact that this probability of winning is now tied solely to the third remaining door, the one you didn't pick initially and which the host didn't open.
This reasoning confirms the accepted answer to the Monty Hall problem: it pays to swap because your probability of winning is higher if you do.
What Is the Famous Monty Hall Problem?
If you’re old enough, you might remember a game show called Let’s Make a Deal hosted by a guy named Monty Hall. The show was a bit before my time, but one of the games from the show—or at least a variant of it—has stood the test of time to become one of the most debated math brain teasers ever. In honor of the show's host, it's called the Monty Hall problem.
What is this brain teaser about? Why is it so famous and so mind-bogglingly perplexing? And how can you break through the confusion and understand how to solve the Monty Hall puzzle once and for all? Those are exactly the questions we'll be talking about today!
What Is the Monty Hall Problem? Here's the situation: You're standing in front of three closed doors and you're told by Monty Hall that you will win whatever is behind one of the doors. What exciting prizes might you win? Behind one of the doors is a brand new car. And behind the other two? Goats.
I don't know about you, but I actually think it'd be kind of cool to have a goat…milk, cheese, yum. But apparently people were a lot more interested in the new set of wheels. So the name of the game for most people—and the one we'll be talking about today—is to guess which door is concealing the car.
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Understanding the Monty Hall Problem
The Monty Hall problem is a counter-intuitive statistics puzzle:
- There are 3 doors, behind which are two goats and a car.
- You pick a door (call it door A). You’re hoping for the car of course.
- Monty Hall, the game show host, examines the other doors (B & C) and opens one with a goat. (If both doors have goats, he picks randomly.)
Here’s the game: Do you stick with door A (original guess) or switch to the unopened door? Does it matter?
Surprisingly, the odds aren’t 50-50. If you switch doors you’ll win 2/3 of the time!
Today let’s get an intuition for why a simple game could be so baffling. The game is really about re-evaluating your decisions as new information emerges.
Play the game
You’re probably muttering that two doors mean it’s a 50-50 chance. Ok bub, let’s play the game:
Try playing the game 50 times, using a “pick and hold” strategy. Just pick door 1 (or 2, or 3) and keep clicking. Click click click. Look at your percent win rate. You’ll see it settle around 1/3.
Now reset and play it 20 times, using a “pick and switch” approach. Pick a door, Monty reveals a goat (grey door), and you switch to the other. Look at your win rate. Is it above 50% Is it closer to 60%? To 66%?
There’s a chance the stay-and-hold strategy does decent on a small number of trials (under 20 or so). If you had a coin, how many flips would you need to convince yourself it was fair? You might get 2 heads in a row and think it was rigged. Just play the game a few dozen times to even it out and reduce the noise.
Understanding Why Switching Works
- That’s the hard (but convincing) way of realizing switching works. Here’s an easier way:
- If I pick a door and hold, I have a 1/3 chance of winning.
- My first guess is 1 in 3 — there are 3 random options, right?
If I rigidly stick with my first choice no matter what, I can’t improve my chances. Monty could add 50 doors, blow the other ones up, do a voodoo rain dance — it doesn’t matter.
The best I can do with my original choice is 1 in 3. The other door must have the rest of the chances, or 2/3.
The explanation may make sense, but doesn’t explain why the odds “get better” on the other side. (Several readers have left their own explanations in the comments — try them out if the 1/3 stay vs 2/3 switch doesn’t click).
Understanding The Game Filter
Let’s see why removing doors makes switching attractive. Instead of the regular game, imagine this variant:
- There are 100 doors to pick from in the beginning
- You pick one door
- Monty looks at the 99 others, finds the goats, and opens all but 1
Do you stick with your original door (1/100), or the other door, which was filtered from 99? (Try this in the simulator game; use 10 doors instead of 100).
Monty Hall Problem — from Wolfram MathWorld
The Monty Hall problem is named for its similarity to the Let's Make a Deal television game show hosted by Monty Hall. The problem is stated as follows. Assume that a room is equipped with three doors. Behind two are goats, and behind the third is a shiny new car. You are asked to pick a door, and will win whatever is behind it.
Let's say you pick door 1. Before the door is opened, however, someone who knows what's behind the doors (Monty Hall) opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door (i.e., the door which neither you picked nor he opened).
The Monty Hall problem is deciding whether you do.
The correct answer is that you do want to switch. If you do not switch, you have the expected 1/3 chance of winning the car, since no matter whether you initially picked the correct door, Monty will show you a door with a goat.
But after Monty has eliminated one of the doors for you, you obviously do not improve your chances of winning to better than 1/3 by sticking with your original choice.
If you now switch doors, however, there is a 2/3 chance you will win the car (counterintuitive though it seems).
The Season 1 episode “Man Hunt” (2005) of the television crime drama NUMB3RS mentions the Monty Hall problem.
The problem can be generalized to four doors as follows. Let one door conceal the car, with goats behind the other three. Pick a door . Then the host will open one of the nonwinners and give you the option of switching.
Call your new choice (which could be the same as if you don't switch) . The host will then open a second nonwinner, and you must decide for choice if you want to stick to or switch to the remaining door.
The probabilities of winning are shown below for the four possible strategies.
The above results are characteristic of the best strategy for the -stage Monty Hall problem: stick until the last choice, then switch.
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Don’t Switch! Why Mathematicians’ Answer to the Monty Hall Problem is Wrong
The Monty Hall problem is one of those rare curiosities – a mathematical problem that has made the front pages of national news. Everyone now knows, or thinks they know, the answer but a realistic look at the problem demonstrates that the standard mathematician’s answer is wrong.
The mathematics is fine, of course, but the assumptions are unrealistic in the context in which they are set.
In fact, it is not clear that this problem can be appropriately addressed using the standard tools of probability theory and this raises questions about what we think probabilities are and the way we teach them.
The Monty Hall problem hit the headlines in 1990, when Craig F.
Whitaker of Columbia, Maryland, asked Marilyn vos Savant: ‘Suppose you’re on a game show, and you’re given the choice of three doors: behind one door is a car; behind the others, goats. You pick a door, say No.
1, and the host, who knows what’s behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, ”Do you want to pick door No. 2?”’ Is it to your advantage to take the switch?’
Vos Savant wrote a column called ‘Ask Marilyn’ in the popular magazine Parade, in which she responded to readers’ questions. According to the Guinness Book of Records, at the time she was the woman with the highest IQ in the world.
Vos Savant responded to Whitaker in her column of 9 September 1990: she said you should switch and that you double your chances of winning if you do. The result was a torrent of criticism and abuse – much of it from mathematicians – such as:
- ‘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?’ (Charles Reid, PhD, University of Florida)
- ‘You blew it, and you blew it big! … There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!’ (Scott Smith, PhD, University of Florida)
- ‘You made a mistake, but look at the positive side. If all those PhD’s were wrong, the country would be in some very serious trouble.’ (Everett Harman, PhD, US Army Research Institute)