This Simple Puzzle Tricks Mathematicians -- Monty Hall Problem in 5 Levels

In an "Ask Marilyn" column, Marilyn vos Savant correctly solved the puzzle now known as the Monty Hall Problem. Around 10,000 people wrote in to say she was wrong, including many mathematicians! Researchers have even found that pigeons tend to learn the optimal strategy faster than humans during repeated trials. Let's explore the Monty Hall problem in 5 levels, from simulations of the game through a complete solution with conditional probabilities.

00:00 Introduction
00:33 The Puzzle and Simulations
01:50 'Simple Solution'
03:10 The 10 Doors Game
04:19 The Unequal Doors Game
05:40 Conditional Probability Solution

[MorganZ]

My understanding of the Monty Hall problem is this: Due to the rules of the show and the host's knowledge (the host knows what is behind every door, and he MUST reveal a goat door after the player's choice), the initial probability is not changed after the goat reveal. So your initial door still has its original one-third chance. Therefore, the other two-thirds chance is congealed behind the remaining door.

If the host didn't know what was behind the doors and could accidentally pick the car himself, then all of the "50/50 after the host door is a goat" people would be correct.

[jaykemper]

The 10-door problem solidified it for me. Initially, you have a 1/10 chance of getting it right. Which means there is a 9/10 chance it's under the remaining door. Scale that up to 1, 000, 000 and it becomes VERY obvious.

[Unchained_Alice]

I think of it as - If you stay the same you choose 1 of the 3 doors. If you switch you choose the other 2 that you didnt pick at the start.
Makes so much more sense in my head then.

[TedHopp]

A very similar problem comes up in the card game bridge, where the solution goes by the name Principle of Restricted Choice. The situation is that you are the declarer and the opponents hold two equivalent cards in a suit (say, the queen and king of spades). On one trick, an opponent plays one of those cards. The Principle of Restricted Choice says that the chance that the same player has the other card has decreased; that is, one should play as if that player's choice was restricted.

[rolandet]

Thanx for the excellent explanation of a funny old 'problem'!
Compact enough for those who have a basic understanding of probability already, and sufficiently detailed to get all others curious enough to undertake further research if required 👍!
🙂👻

[pl]

great breakdown good visuals lovely narrated

[NicholasAngelidis]

Dr Sean, it would be great if you did a video on conditional probability at some point! your explanations are very intuitive!

[sresnic]

What can really blow your mind is that the exact same action can have multiple probabilities depending on prior knowledge.

Imagine two people playing the game simultaneously. Player A plays normally while Player B is held off stage in a soundproof booth.

A makes his initial selection and one of the other doors is opened to reveal a goat. B is then brought on stage (not knowing which door A selected) and is told to pick a door.

Player A is then asked if he wants to switch his initial pick.

There are only 2 doors left and simply opening one of them will reveal if A or B or both won the grand prize. However, that one action has three different probabilities of success.

Player A has a 1/3 chance of winning if he doesn’t switch.

Player A has a 2/3 chance of winning if he switches.

Player B has a 1/2 chance of winning.

[oafkad]

I think the big thing people miss is the host knows the door results. Its like rolling a weighted die, it still looks like a D6 but your odds are anything but 1 in 6.

[crispyandspicy]

"I pick door 1!"
*Door 3 opens, revealing the car*
"I think I'll switch, that has the best chances"

[mstmar]

There's a variation on the monty hall problem where there's equal chance of winning with staying than switching. basically after you pick a door, the host flips a coin if you pick a goat, and you lose automatically if heads turns up and are asked if you want to switch on tails. if you get to the point where you're asked to switch, you now have a 50% chance of being on the goat and 50% chance of being on the car. the total odds are 1/3 of being on car, .5*2/3 = 1/3 of being on a goat and losing automatically, and .5*2/3 of being on a goat and being asked to switch.

[nickronca]

Think of it like this: If you choose a door with the intention of staying then it does not matter what order the doors are revealed in. If you pick something at the beginning with the intention of running with your initial pick, then the only way you win is if your initial pick is correct which happens 1/3 of the time. It does not matter what order they are revealed in. The ONLY thing the order of reveal changes is when the probability of winning goes from 1/3 to 0 or 1.

[ratatouille]

I usually only find these kind of videos like 8 months after they are posted. hello empty comment section!

[TheCosmicMedicineMan]

The Monty Hall problem may seem complex, but it's actually quite straightforward. With three doors, your initial chances of winning are 33.3%. When one door is removed, you’re left with only two doors, which gives you a straightforward 50/50 choice. Regardless of differing opinions, the math remains the same. The only real uncertainty comes from the fact that, if this were a television show, the outcome might be manipulated for entertainment purposes.

[lovishnahar]

ty sir, im glad u made video on probability

[sjoerd]

It took a while, but the biggest hurdle was realizing why the odds of having pick the first door in the first place do not change when the host opened a door (out of 3), It is because the host knows exactly which door of open (100% probability that he will open the door/s he wants to open). Kind of obvious but hey, it wasn't.

[LilBurntCrust]

You should switch because you gained new knowledge and one door that you could possibly have chosen is shown that it isn’t the objective prize thing the basically desired outcome, and now it would just be a two thirds chance instead of at the start where it was 1/3 or 33.333…% so that’s why you should switch in the problem, you see that there are 3 doors so that probably should stay the same but it doesn’t since new knowledge was gained and a door possibility was taken away narrowing your choice, same for any number of doors in this hypothetical (thought) experiment.

Edit: I just kind of seemed to understand this when I heard of the problem for the first time and thought that there was something more to it so I watched through most of a Vox video about the problem so yeah, that was just it, very simple but it was weirdly dividing for many groups of people in the fields of statistics, mathematics and probability alike..

[ingiford]

Best way to think about it is expend it from 3 to a hundred choices, and use the same rules. If that is not enough say a million choices, etc.

[jumeme]

Really really really love your videos! Cant wait to see you blow up!

[fixipszikon]

So to keep the original probability for the second selection, what matters is that in case you pick a goat first, they must keep the car in play.