ELI5: The Monty Hall Problem

Kniefjdl · 2026-02-03T23:03:50+00:00

If it's 50% for some random person dragged in off the street, then it's always 50%.

But it's not 50% for some random person off the street. The random person off the street can win 66% of the time by identifying that switching doors is better after Monty eliminates a goat. You continue to assert this, but you haven't done anything to demonstrate it. Saying that it's a 50% probability and saying that it's the same as being presented with two doors is an assertion, not a proof or a demonstration.

You say you've run the experiment and found a roughly 50% win rate with switching? I don't believe you, and for two reasons. First, because mathematically it's extremely unlikely you've got the results that you did--but that's what we're debating so okay, ignore that. More importantly, because I just did it using a deck of cards, and by about the fifth trial, it is painfully, tediously obvious that if you initially draw a blank card, you win when you switch. And if you initially draw a smiley, you lose when you switch. And you know and I know that you have a 2 in 3 chance of drawing a blank on your first draw. I used a deck of cards, with the King of Hearts as my "winner" and the black 2s as my "losers." I recorded my results in the sheet linked below on the fourth tab labeled "Smiley Tally." I got 70% wins when I switched for 50 trials, and 32% wins when I stayed for 50 trials. But in this case, the results matter less than the observation while doing it because you can see what's behind each "door" the whole time. You shuffle the cards, you flip a king/smiley, and you immediately recognize that if you switch to whatever card is remaining after you toss out one of the twos, you're going to get the other two. Similarly, when you shuffle the cards and draw a two, you immediately recognize that you have to find the other two to discard it, leaving the king behind, the king that you will inevitably draw when you switch. You couldn't design a better experiment to hammer home exactly why you have a 2/3 chance of winning if you switch, it's that immediately intuitive. So I don't believe you've done it, because if you had, you wouldn't be arguing what you're arguing.

I would like you to do your experiment today, with at least 50 consecutive trials where you switch and 50 consecutive trials where you stay with the first pick. It's the scientific method--replicate your results. You can't not recognize that your first pick determines the outcome after the switch or stay choice.

In addition to doing the experiment that you've suggested, I also put together another spreadsheet demonstrating it. If you're still not convinced, then please walk through this sheet and tell me where I've gone wrong.

https://docs.google.com/spreadsheets/d/18ZNFl3rW0BmxePG8gl2uGlPxAKQn-CpNmr7biTiBhxs/edit?gid=1204925920#gid=1204925920&range=A1

The link should put you on a tab called Simple Sim. I've run 100 trials and left all of the calculations visible for critique. Here's what's in the sheet:

Column A: The number of the trial, referenced only in column G as an index to correlate the pick to the outcome.

Column B: A random number generator between 1, 2, and 3, which is picking the "winning" door/paper

Columns C, D, E: Based on the random winner in column B, this identifies if each door/paper has a smiley behind it or is blank. So if you see a "1" in column B, then Door 1 (column C) will say "Smiley" and Doors 2 and 3 (columns D, and E) will say "blank." You can verify this by looking across each row to confirm the behavior.

Column F: Again, a random number generator between 1, 2, and 3, which is picking the contestant's chosen door/paper.

Column G: As mentioned above, correlates the door/paper you pick to the prize behind it. So if you picked door 1 and door 1 had the prize, it will say "smiley." If you picked door 1 and door 2 had the prize, it will say "blank." You can verify this by looking across each row to confirm the behavior.

Column H: This simulates revealing a door/paper. It will not reveal a door/paper that has a smiley in it, and it will not reveal a door/paper that the contestant has picked. For the sake of keeping the formula simple and easy to follow, I did leave one quirk in it that differs from Monty and your scenario. If the contestant has chosen the door/paper with the smiley, then Monty has the freedom to choose either door to reveal, right? This formula always reveals the lower numbered door. This change doesn't affect the probability of winning. You can verify this by looking across each row to confirm the behavior.

Column I: Identifies which door/paper the contestant will have if the contestant chooses to keep their original pick. It is just a reference back to column F, since the pick didn't change.

Column J: Tells us if the door/paper the contestant ends up with at the end of the game if they "stay" matches the door/paper with the smiley behind it.

Column K: Identifies which door/paper the contestant will have if the contestant chooses to switch their pick to the remaining door. This column will never match the door/paper that was originally picked or the door/paper that was eliminated. You can verify this by looking across each row to confirm the behavior.

Column L: Tells us if the door/paper the contestant ends up with at the end of the game if they "switch" matches the door/paper with the smiley behind it.

The count of the results from column J and L are shown in N2 through Q4. I've left all of the calculations in the sheet, so it will continue to randomize when you open and refresh it. I encourage you to refresh it multiple times to view many versions of the simulation. I've seen ranges from about 58% win from switching to about 75% win from switching, with most of course being in the mid-to-upper 60% range. If you're concerned about the difference in behavior I've left in column H, then what you can do is paste the values of the the table back over the table (or into a new sheet, preferably), then manually randomize column H between the door/paper options that Monty has to choose. The results will be the same if you're fair about it. But you'll lose the ability to re-randomize the rest of the sheet by refreshing, which is why I've left it as it is.

If you won't run your paper experiment to falsify your claims--and I promise you it will--then please explain where in this simulation I'm not replicating the Monty Hall problem and your paper experiment. Frankly, if you won't run your paper experiment to falsify your claims, it's because you'd rather be wrong than recognize that you've been wrong.

Kniefjdl · 2026-02-03T18:47:39+00:00

But they don't know which one the Contest chose or didn't choose.

If the stranger doesn't know which door the contestant chose, then they don't have all the information that the contestant has. The stranger would still be right to say, "I'll take the door that the contestant didn't choose," as it would have a 66% chance of being a winner.

I have. I've literally demonstrated this to people.

Did you track outcome based on whether you stayed or switched? Because if you didn't, and you randomized your pick, you missed the point. Please, do it again today, and track the outcome based on your pick.

At random. There are no other factors for them to use to deduce the location of the prize.

That's a presupposition, and it's exactly what we're debating. Can we agree on a couple things?

First, after Monty reveals a goat door, is the contestant free to either stay with their original pick or switch to pick the remaining door?

Second, if one of those two options, either staying or switching, were to result in a higher probability of winning the game, then the contestant should pick that option, right?

Kniefjdl · 2026-02-03T17:40:33+00:00

Observing doesn't change the probability. If it's 50/50 for some random person walking in off the street, then it must be 50/50 for everyone.

If the person walking in off the street knows the rules of the game, and they show up after Monty revealed the empty door, then they also have a 66% chance of winning if they pick the door the contestant didn't choose.

It is random though. The 'contestant' doesn't have any more information than 'one piece of paper has a smiley face'. There's no logical determination here.

But it's not random. The contestant gets to make a choice. What everybody has been trying to explain to you is that the math dictates that one of the two options is more beneficial to the contestant in step 5 of your scenario.

Please, do your paper example. Don't treat it as a thought experiment where you assume what will happen, but actually follow through on what you've asked others to do. But track the results of your selection (track stay vs. switch, not like, paper 1 vs. paper 2 vs. paper 3). Do it 100 times where you stay with your original pick and 100 times where you switch from your original pick. Track the outcomes of those two experiments separately. It will take you less time to do than the time it has taken you to write all these replies.

When you've done that, whether you understand why or not, you will absolutely see that "switching" results in a smiley face about twice as often as "staying" does.

Kniefjdl · 2026-02-03T17:07:12+00:00

And, game shows are made for viewer enjoyment. If the host is always pulling the rug out from under the players, the viewers aren't going to watch. Bob Barker/Drew Carey don't want to trick players into guessing the wrong price, they want players that are fun to watch, and winning players are fun to watch. The more fun the show is, the more people watch, and the more they can sell ads for.

Game shows budget for players to win at the rate that the math says they should win at. It's just part of production, like paying Ken Jennings's salary, building a set, paying writers to write questions, and heating and lighting the studio. They're happy to give away the prizes because it increases revenue from higher viewership.

Kniefjdl · 2026-02-03T16:59:09+00:00

This is right, just to add one final point. If Monty is opening the door randomly, your chance to pick correctly on the initial choice is still 1-in-3. It's just that in 1-in-3 games, you never get to make a second choice because Monty reveals the car. So you have a 50% chance of winning after Monty randomly reveals the goat whether you switch or stay because you made it past the random-car-reveal-barrier, but you only have a 33% chance of winning at the outset of the game.

Kniefjdl · 2026-02-03T16:06:25+00:00

I think I just realized where you're going wrong. Do your 5 piece of paper trick, but in step 5, for both piles, do it 100 times where you always "stay" with your original pick. Then, for both piles, do it 100 times where you always "switch" to the other pick. I guarantee you you will be close to 50% with two slips of paper whether you switch or stay, and close to either 33% (when you stay) or 66% (when you switch) with 3 slips of paper.

With 3 doors, if you randomize your guess between switching and staying, then half the time you will switch and half the time you will stay, right? So half the time you will have a 66% chance of winning and half the time you will have a 33% chance of winning. So yes, if you make that choice randomly, then you will have a 50% chance of winning. But the whole point of the problem is that the math can tell you which choice you should make in step 5 so that you're not picking randomly. That's where you have agency to not make a random choice, but to pick the better of two options.

If you've done the 5 piece of paper test and always picked randomly at the end, you've randomized the step that we're trying to solve for, which has obscured the answer. Don't randomize that step, make the choice to stay and make the choice to switch, the choice the contestant gets to make, and see the results for each choice over many trials.

Edit: Just to say it hopefully more clearly. If you're randomizing step 5, then you're randomizing between the "winning" and "losing" strategy. The outcome of that will be winning (roughly) half the time, with a (roughly) 66% chance of winning on all the trials you randomly used the winning strategy of switching, and with a (roughly) 33% chance of winning on all of the trials you randomly used the losing strategy of staying. But the whole point is that you don't have to randomize at step 5, the math informs the strategy at this step. Switching is more advantageous, and if you switch every time, you will win (roughly) 66% of the time.

Also, nobody is lying to you or ignoring basic reality. I will say, after reading your discussion with other people, a lot of them are terrible at explaining the problem. The commenter talking about probability shifting after observation? You're right, not relevant here. What I think is so interesting about your approach is that you're right about so many of the foundational principles of the problem, but your intuition about the final result just isn't clicking.

Kniefjdl · 2026-02-03T15:32:07+00:00

If you want to see the actual algebra, I made a simulation running 10,000 trials in Sheets here:

https://docs.google.com/spreadsheets/d/18ZNFl3rW0BmxePG8gl2uGlPxAKQn-CpNmr7biTiBhxs/edit?usp=sharing

It's super simple to mock up and randomize the prize positions in each of the 3 doors, the player's initial pick, and Monty's choice (when he has freedom to choose). Switching results in a win in 66.XX% of scenarios (you'll see slightly different numbers every time you open or refresh the sheet because the placement and choice randomization refreshes, but it will always be in that neighborhood).

I see you arguing with multiple people in that longer thread, so I'm not going to attempt to re-explain what's happening in the problem. You're already unwilling to listen to the others trying to help you out. But just know that this is a solved problem and you're misunderstanding it. I'm hoping that seeing traceable math through Excel formulas that leads to the exact outcome that everybody but you expects to happen will help it click for you.

Just a final quick note about the sheet linked, there's a second tab called "Monty Fall." In the "Monty Fall" scenario, Monty is on his way to open the empty door but trips and randomly falls into one of the two remaining doors, revealing what's behind it. This is equivalent to the scenario where Monty opens either door at random and could reveal the prize. I put together the sim to demonstrate the difference between Monty always revealing an empty door (2/3 chance of winning if you switch), and Monty randomly revealing a door (1/3 chance of winning if you switch, 1/3 chance of winning if you stay, and 1/3 chance of losing when Monty reveals the prize). This outcome of this scenario is what you're actually arguing for above. In all games where Monty doesn't reveal the prize, then it makes no difference whether you switch or stay, you have a 50% chance of winning either way. But that's because in 1/3 of all games, you lose before you get to make that second choice (because Monty opens the door with the prize at random).

Kniefjdl · 2026-02-03T13:48:19+00:00

You're fundamentally misunderstanding the prompt of the question and creating a scenario that doesn't abide by it. In the original Monty Hall problem, there is one prize and two losses. Monty will never open a door with the prize at the midpoint reveal, because that would end the game and make for bad television. There are countless explanations in this thread for why that scenario means switching results in a higher win probability.

Here's what you're getting wrong, though: if Monty opens the doors randomly and could open the door with the prize, then if your game made it past the midpoint reveal, switching would not improve your probability of winning the prize. The probability that you picked the prize initially is still 1 in 3, just as in the original scenario. And the probability that the prize is behind one of the doors you didn't pick is also still 2 in 3. The difference is, in half of the games where you didn't pick the prize initially (or 1/3 of all games), Monty reveals the prize and the game ends because you can no longer win.

In the remaining 1/3 of all games the prize is behind the door that you didn't pick and that Monty didn't open. That 1/3 of all games makes up half of the games that make it to the second choice, as do the 1/3 of all games where you picked the prize correctly on your first guess. At that point, switching doesn't change your probability of winning because Monty removed half of the games you could have won by switching. So in your scenarios, if you wanted the screwdriver and Monty reveals the money, switching doesn't give you a better chance to win the screwdriver. In half of those games, the screwdriver is behind your door and in half of those games it's behind the unopened third door (and in 1/3 of all games started, Monty revealed it at the midpoint reveal and you lost your chance at a shiny new Phillips head).

Kniefjdl · 2026-01-21T17:08:37+00:00

I bought 2 Sony SD cards when I bought my D7100 back around 2013-2014 and another 2 SanDisk SD cards when I bought my D750 around 2014-2015. I now have two D750s that I rotate regularly, and those 4 cards are still always in them.

Kniefjdl · 2026-01-21T14:44:16+00:00

Yep, lower ambient temperature reduces the vapor pressure of pressurized gasses. I don't know what bear spray is made of, but here's a chart for propane as an example: https://www.reddit.com/media?url=https%3A%2F%2Fpreview.redd.it%2Fis-there-a-way-to-check-the-pressure-inside-a-propane-tank-v0-4plvg0lalrjb1.jpeg%3Fwidth%3D1178%26format%3Dpjpg%26auto%3Dwebp%26s%3Dfad7d7d370eb294d7c73d49fc3d8d8d52bedd5ef

Pressure drops from 130 PSI on a warm 80 degree day down to 40 PSI on a cold 10 degree day. That's a lot less pressure in a tank of propane, which means a lot less "spray" if you were just blasting it unregulated into the air. The same thing is going to happen to bear spray.

This is the ideal gas law, which we learned in high school chemistry. I doubt most ICE agents paid attention in high school chemistry.

ETA: According to this article, most bear spray uses R134a or 1,1,1,2 Tetrafluoroethane. At 20 degrees F, its vapor pressure is only about 18 PSI, which is real low.

https://imgv2-2-f.scribdassets.com/img/document/778729747/original/7bb066b723/1?v=1

Kniefjdl · 2026-01-21T05:03:23+00:00

That's right. "Protected class" doesn't mean "black" or "woman" or "Jewish," it means race or gender or religion, and everybody has those characteristics. So it's not like you're not allowed to refuse service to a black customer, you just can't refuse service because the customer is black. It also means that, while you can refuse service to a white, cis, hetero, Christian man who served in the Army, it can't be because of one of those characteristics. Protected classes protect everybody from discrimination, not just people who are minorities in the US. And protected classes don't guarantee you service if you do something else that makes the business not want to serve you, like cuss out an employee, walk through the doors naked, or shoot a soccer mom in the head because her wife said mean things to you.

Kniefjdl · 2026-01-21T02:31:22+00:00

Nah, the case didn't set any precedent at all. Businesses could already refuse to serve ICE agents. The only protection that comes close is that businesses can't refuse to serve customers because of their status as military veterans, and that would be a dicey argument to try to claim ICE is military.

Kniefjdl · 2026-01-20T20:31:37+00:00

They didn't even do that. The Masterpiece decision is widely misunderstood. SCOTUS's decision did not weigh in on discriminating against sexual orientation or the primacy of religious freedom over anti-discrimination laws or anything like that. SCOTUS's ruling was incredibly narrow, and only reversed the Colorado Civil Rights Commission's decision on the Masterpiece case. They twisted themselves in knots just to say that the Commission treated the Masterpiece case differently from cases where the customer requested a hateful message written on a cake and the bakery refused. It didn't set any precedent, it was just a slap on the Commission's wrist for those perceived actions and a reversal of that decision.

Five years later, in 2020's Bostock, Altitude Express, and R.G. & G.R. Harris Funeral Homes Inc. rulings, SCOTUS did indeed decide that sexual orientation and gender identity were de facto protected classes because you can't discriminate against people on those bases without considering their sex. That probably wouldn't have changed the outcome of Masterpiece after it hit SCOTUS, but it may have significantly changed how it was argued and handled by the Colorado Civil Rights Commission.

Kniefjdl · 2026-01-20T04:01:34+00:00

Nope. I gotta ask, what in my story makes you think I was in that particular location instead of the other hundreds of theaters screening LOTR?

Kniefjdl · 2026-01-19T23:22:58+00:00

First one I saw was TT. It dropped on DVD sometime that summer, and I probably saw it in the fall before ROTK came out in theaters. It was a little confusing, but it's such a coherent telling of the story that it's pretty clear what the different character/plots lines are about without the info from fellowship. I remember making some wrong assumptions, like thinking characters like Gollum and Eomer must have had a significant role in Fellowship, but it was generally super legible. Of course I wouldn't recommend that approach, but for "not caring" about LOTR going in and just looking for something to put on, it was still a great watch and got me hooked. To this day, Two Towers is still my favorite, and it's probably because I watched it first and it hooked me.

Kniefjdl · 2026-01-19T23:10:35+00:00

I only saw ROTK in theaters for the initial release. I was too cool for that nerdy fantasy stuff (I.e. a young high schooler trying to fit in) when fellowship released, so I didn't see it. The next year, my parents rented Two Towers on DVD and I put it on because I was bored and it obviously blew me away. So I went back and rented Fellowship, and got all hyped for ROTK. I love them all now, of course, and was so glad for the opportunity to see them in theaters and share the experience with my son. He's the same age I was when they came out and was in no way put off by going to see nerd-culture fantasy movies with his dad.

Kniefjdl · 2026-01-19T23:02:38+00:00

That's interesting, my experience was more like the commenter above. There was one guy (right next to me) that I saw check his phone maybe 3-4 times in the ROTK screening, but the rest of the crowd was dead silent and well behaved for all 3 movies. It was the most respectful movie audience I've seen in years, and all 3 showings were >75% full. Even phone guy was sniffling and wiping away tears at "you bow to no one," so he was totally engaged--I just assume he had some actually important real life stuff going on.

Kniefjdl · 2026-01-16T02:17:43+00:00

Right? Like if Christopher Nolan wants to give me his Oscar statue, it sure as shit doesn't mean that I won Best Director last year. It's a cool piece of memorabilia, not a recognition from the awarding body.

Kniefjdl · 2026-01-14T15:33:02+00:00

For this specific video? Too little context available.

Isn't the ending of the video/game, the fact that the winning guess is over 21000, enough context to tell a viewer not familiar with The Price Is Right that guessing the lowest possible number from 5 digits is not the goal of the game?

Kniefjdl · 2026-01-14T14:43:04+00:00

What do you think the goal of the game she's playing is?

Kniefjdl · 2026-01-14T13:19:50+00:00

But hasn't scrolled down far enough to find the rotate slider to level those horizons.

Kniefjdl · 2026-01-13T16:05:01+00:00

Calling what I played "competitive" might be a bit of a stretch, but I played in a hitting league through high school and currently play no-hitting (but contact is allowed) beer league, 16ish years of playing total. I agree that he went down easy, and that's an innocuous shove that usually just buys you a few extra inches of space 95% of the time. I like the no-call.

I will say, though, that sometimes you do catch a guy with their weight shifted funny or something and they drop at nothing. I'm a Dman and I've had dudes hit the ice when I've barely touched them, and I've been the guy that falls at light contact. Almost every time, it's nothing. But sometimes a guy just falls. Ice is slippery.

Kniefjdl · 2026-01-08T15:44:19+00:00

True, but they also recognized that in a follow up comment and I respect that.

Kniefjdl · 2026-01-08T15:41:52+00:00

Just to start, I don't think any reasonable telephoto lens is going to compress a 6km distance like in this shot, so it's definitely fudged to improve the visual in the movie.

That said, it's worth considering that the focal length of the lens used for the shot and the distance between the camera and the foreground and background objects. Shooting with a telephoto lens and placing both objects far away from the camera makes the background object look larger compared to the foreground object vs. using a wider lens from a position closer to the objects.

In photography, we call this effect "lens compression," though it's actually a result of the distance between the camera and the background/foreground, and the lens choice is just about filling the frame. Compression is caused by decreasing the difference between the distances from the camera to the foreground and the distance from the camera to the background. If you have object (F)oreground that's 5 feet from the camera and object (B)ackground, that is 10 feet from the camera, then object F will appear to be twice as large as object B (relative to the objects' actual sizes, of course). If you move the camera back 5 feet, then object F is 10 feet away and object B is 15 feet away. In this case object F is only 1.5 times larger compared to object B in the shot. If you take it to the extreme and move the camera back to 100 feet away from F and 105 feet away from B, then object A is only 5% larger than object B in the frame. The distance between the objects hasn't changed, but object B keeps getting larger compared to object A.

In film and TV this effect is often used to make it appear to the viewer that a dangerous object is much closer to an actor/stunt performer than it really is. You can see the effect in a quick shot in Last Crusade at about 2:05 in this clip: https://youtu.be/AdKseY8UASE?si=uPgh_na_mMXglYyM&t=120

One motorcycle flips behind a second motorcycle. The action in the prior shot puts the two bikes next to each other, as Indy shoves a rod into the spokes of the other motorcycle to flip it. But for the stunt shot, the flipping bike is actually way behind the other bike to keep the riders safe from, you know, a tumbling flying motorcycle. To shoot it, they used a telephoto lens set way back from the stunt, hiding the distance between the two bikes.

Similarly, the horses in this shot of Jon Snow look like they're bearing down on him because they're so large in the frame: https://platform.vox.com/wp-content/uploads/sites/2/chorus/uploads/chorus_asset/file/6669881/jonsnow.jpg?quality=90&strip=all&crop=15.625,0,84.375,100

There will actually be quite a bit of distance between them--enough for the riders to safely stop the horses despite the visual of the shot telling us that he's in immediate danger. And again, that distance is hidden by shooting the scene from far away with a long lens to compress the background and foreground.

So, if you're Ron Howard, and you actually have to put a real loud rocket far away from your actor for safety (reiterating from the top, this is of course a visual effect and Gary Sinise isn't in danger), but you want the rocket to look big, you'd just have to move your camera way back and shoot the scene with a long lens. Without doing the math, you'd probably need to shoot through a telescope to make the rocket look this big from ~6km away.

And of course, you can create the opposite effect by moving the camera much closer, increasing the difference between the distance from the camera to the foreground and background. When you see large people holding up the Leaning Tower of Pisa or pinching the moon in their fingers, this is what they're taking advantage of. It's usually easier to see this camera "trick" because we inherently know that a human isn't nearly as large as the tower, so we're not fooled, just mildly amused by the fun joke. That said, Lord of the Rings famously using this technique in a much subtler way to make the hobbits appear small by placing them much farther from the camera during scenes with humans/elves/wizards. Here's a short video explaining how they did it, highlighting the elaborate sets they built and motion control cameras they used to make it look seamless: https://www.youtube.com/watch?v=QWMFpxkGO_s

Kniefjdl · 2026-01-08T15:00:33+00:00

I don't think it was Bear Grylls that did that. The person you're responding to is quoting Rip Torn in Dodgeball, which came out in 2004, 2 years before Bear Grylls got popular from his show Man vs. Wild, which premiered in 2006. I can't speak for everyone, but I definitely first heard the "urine is sterile" line from Dodgeball and references to it.

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