monty hall problem
Posted by mrpompos
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mrpompos
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monty hall problem
I wondered if the monty hall problem (MHP) is relevant to poker. they're both about odds being in your favour and if they change or stay the same after revealing certain information. before i give an example in poker terms if you don't know what the monty hall problem is youtube it, supposedly its 66% to 33% if you switch, however i've heard people argue its 50/50. i personally can see both being true, I really could argue both...maybe its to do with entanglement or something if you know about quantum physics...anyway so say i have 10h 2h and opponent has Jh Kh suited...we all have hearts!!! now i have about 30% equity to his 70%. so now if i hit a 2 on the flop i now have 76%...but to the person who believes the MHP solution is to switch would that not translate into the opponent still has 70% chance on the flop because when the flop can be seen the dealer just "revealed a goat" (the goat being either relevant or not depending on if one is a "switchman" or a "50/50man")? so does the switchman believe if he had 70% chance to win preflop he still has 70% on the flop as Jh Kh both can hit turn and river. 3 of the cards which had been used to calculate the 70% have been revealed, lessening the chance of a J or K hitting within the 5 but still have 23% to hit. if I could just get 1 sentence to be congruent with the next I'd be satisfied but not sure if I've even done that. i'm also wondering about is how 70% isn't 100% its going to let something with 30% chance win even though it has 70% of the 100% available lol. its like a coin flip is 50/50 so why is it inclined to land either way in the first place? is this that damn quantum entanglement again? or are these ramblings of a mad man, because i have had no sleep thats my excuse anyway...!
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Keep taking the tablets :)
1. Don't think that the MH problem has anything to add to the understanding of what you are talking about.
2. A personal hobby-horse of mine is people applying various "quantum" paradigms (particularly the Heisenberg effect) to explain things in areas where quantum laws don't apply (ie where we are not discussing the smallest possible discrete amounts of energy and or matter) - which is true here imo.
3. Even if the MH problem was helpful, it's not the same - MH knew what was behind the doors whereas the dealer is just revealing random cards. The reason you always switch in MH is because effectively you are getting to pick 2 doors (66%) not 1 (33%) and doubling your EV.
Two things:
1) The Monty Hall problem isn't debatable, anyone who thinks it's 50/50 is wrong (assuming it's worded correctly, it's very important that when the goat is revealed, the revealer knows what he's revealing. If he randomly opens a door and shows a goat, it is a different problem).
2) Your application of this principle doesn't work. At all. Perhaps considering the game "Deal Or No Deal" will help you: Imagine a scenario, you are at the end w/ the $0.01 and the $250,000 in play. You are offered the swap. Now, people who mis-apply Monty Hall might go, "Errr, SWAP, you were 1/25 to pick the $250,000 so it's 24/25 to be in the other box" what with 23/24 non-$250ks gone. However, one could easily reverse the logic around the $0.01 such that it's 24/25 that your opponent holds the penny having discarded 23/24 non-$0.01s. In a case such as this, Monty Hall doesn't apply.
I wouldn't say the monty hall problem is relevant to poker, but conditional probability in general / Bayes theorem can be.
That's a good way to put it.
i wouldn't say they're wrong to say 50/50, sorry if you can't see that it could be. "If you put money in the pot, the cards displayed will be the same as if you didn't." this is exactly what I mean when I say it doesn't matter if a goat is revealed the doors will still be the same whether you switch or not.
steamer "The reason you always switch in MH is because effectively you are getting to pick 2 doors (66%) not 1 (33%) and doubling your EV." this is just the switchmans view of it. Don't you think its a little contradictory to say you're getting to pick 2 doors even if you said "effectively" yet you are actually just picking 1.
tom it doesn't matter that the revealer knows what he's revealing as the person who chooses the door doesn't know. you could say there is no REAL probability as you don't know what's behind any doors. until the goats shown then there is only 2 doors which you don't know what's behind. so I mean its the probability is polarized therefore each door has either 100% to be a goat or 0% to be a goat. much like a really large bluff in poker would make the opponent think either he has it or he doesn't. Previously he might of put him on certain hand ranges based on probability and other maths but once a big bet/raise is made the decision might become polarized.
mandelmonk so yeah conditional probability says "the probability of A given B" but I'm saying what if A is independent of B as its the persons choice in that moment which door to choose. Information from past events can't be useful in the moment of that choice.
It's not contradictory at all, it's counter-intuitive. You are effectively picking 2 doors. You're not just picking 1. By switching you choose to always choose the best of the 2 doors left. In that case, you only lose if your initial choice was the car, which will happen 1/3 of the time.
The Monty Hall Problem is not a theory, it has been proven both in theory and in practice.
You have to take into consideration the past events. The chance of a goat being behind the first door you chose does not change when another goat is revealed.
i think there's a case to be made that you are just picking 1. that's what you're actually doing. I understand the maths of it and if you test it it might show switching being the better choice. but that's based on information which is actually lacking because you can't make a 100% prediction. therefore you actually have a choice to use the information or not. and its more of a philosophical view point which fits underneath the mathematical viewpoint (which uses correct-yet-limited info). when you're just asked once rather than testing repeatedly its much more polarized, the probability just won't matter enough in 1 single go at the problem.
You seem to be saying that in any case where uncertainty is involved, it doesn't matter what you do. But even with uncertainty, some choices are better than others - this is the whole idea of probability! Say we rolled a six-sided die; I win on 1-5, you win only on 6. There's a chance you win. But would you argue that I'm getting the better of it?
The Monty Hall problem is basically the same. Sure, you could luckily win with either strategy, but switching gives you a better chance.
Oh geez, as soon as I finished posting that I realized I got MrThomas'd. GG.
what the hell is MrThomas'd? I hope you don't think I'm someone else. Can you imagine this, there's a wall and you paint it yellow. everyone looks at it and says its yellow. the next day you paint it black. everyone says its black now. Except some, who think actually its still yellow, the black paint is painted on top of the yellow paint, while the yellow paint still touches the wall. you see people can argue its both yellow and black. hope you understood the metaphor. Sorry if I'm annoying you I just honestly see it as 50/50 now. I would still switch because I know the probability is in my favour to switch, being 66/33. however the fact there's 2 unknown doors and you have to make a choice between switching and not switching really baffles me. in other words I know probability works I just can't get my head round the uncertainty factor which makes me to want to ignore that information. I guess it comes partly from poker I often get it in with a 70-90% favourite and the suck out. like all in with AK against something like 9 3 and they flop a full house or quads, and then I'm supposed to trust probability. I suppose that examples nearly a flip really but I'm more meaning how everyone gets rivered a lot. I suppose that's poker though. Anyway if you feel like replying I'll be happy to read it.
also tom yes you're right the goat must always be revealed this makes the illusion work that its relevant to the choice to switch or not, but also remember the goat must always be chosen between a goat and a car, meanwhile what's behind the the 3rd door has to stay polarized against the other door which is not going to be revealed. Another important point is you think the Monty hall problem is 66% 33% to be a goat after a goat is revealed as it was before the goat was revealed so by switching you have 66% chance to win the car...then you go off and prove it by doing it 1000 or 10,000 times however surely it depends who the question is asked to as I could only do it once and if I win the car by not switching, you would say well you need to do it more times, I say no I played the game 1 time therefore I reckon the results/probability is more polarized. perhaps the argument between switchmen and 50/50men is also polarized..."If something is polarized, it consists of two separate groups with opposite opinions or positions."
Easiest way to visualize Monty Hall is imagine if there is, say, 100 doors
Basically everything you've said is wrong. I particularly enjoy the semi-patronizing, "I wouldn't say they're wrong to say 50/50, sorry if you can't see that it could be." This is not a debatable problem, it is a fact. You might as well sit there and tell me, "Well, I wouldn't say people are wrong to say the Earth is flat."
Perhaps you have simply misunderstood the actual Monty Hall problem (I mentioned that wording is critical), so I will write it out here and show you why it matters that the revealer knows he's showing a goat (you appear to be under the impression that the door being opened has been selected by Hero, when it hasn't, it's been selected by Monty).
Monty chooses goat: Predoor, you have selected one of three options, (Goat 1, Goat 2, Car). Then Monty opens a door guaranteed to hide a goat. This means that, when you selected Goat 1, the car is behind the other door. Equally, when you selected Goat 2, the car is also behind the other door. However, when you selected the car, either of the goats can still be in play. This means that, on the 1/3 occasion you chose the car, a swap will give you a goat, whereas the 2/3 time you selected a goat, swapping will give you a car.
Monty chooses randomly: In this instance, the predoor doesn't change, you are still 1/3 to have selected the car. However, upon opening doors, 6 scenarios can occur: If you have selected Car, you can reveal Goat 1 or Goat 2. If you have selected Goat 1, you can reveal Goat 2 or Car. If you have selected Goat 2, you can reveal Goat 1 or Car. Of these 6 options, two vanish from our set because the car was revealed, leaving us with 4 equally likely options: Car reveal Goat 1 (Goat 2 left), Car reveal Goat 2 (Goat 1 left), Goat 1 reveal Goat 2 (Car left), and Goat 2 reveal Goat 1 (Car left). As you can see, in half of these instances, swapping will be to a goat, and the other half it will be to the car. Therefore, in this case, the chance is 50/50. This is NOT the Monty Hall problem because the goat was not guaranteed meaning this is NOT conditional probability.
I hope that clears things up for you.
Tom's got it right - the relevant fact is not that a goat is shown, but that a goat is intentionally shown. No dealt card in poker is ever intentionally shown, unless the dealer is cheating.
Sorry not meaning to be patronizing and I wouldn't argue that people were wrong about the earth being flat, but I would argue that when they thought it was flat, they were right. If you can transfer that sort of thought process to this maybe you can understand what I'm saying. I tried to explain my thoughts in a way that told you I already understood the traditional Monty hall theory which you explained above. What I want to know which would clear things up for me, is how can it be both 66/33 and at the same time, 50/50 in the sense that there are 2 unknown doors to choose from. I see that its better to switch in a probability sense, but on a deeper level it is still 50/50. If you can explain that more rather than just disagreeing with the premise it would be more helpful. And maybe another way to compare is if you paint a wall yellow then the next day paint it black, the wall would be said to be black, because it covers the yellow paint being darker, in the same way the extra information in the MHP changes it from being 50/50 to 66/33. I made this up myself, would you say the wall is black or yellow? I mean couldn't you say it was still yellow because the yellow paint is touching the wall first and the black paint is just painted onto the yellow paint?
Sean I know the goat is intentionally shown. its shown so you have a choice between 1 door which is unknown to you, and another which is unknown to you. You all decide to use probability theory, which is correct but I'm just wondering how it can work- the purpose of philosophy is to ask the right questions. I know the maths is there too.
You want to know how it can simultaneously be 67/33 because of the math and 50/50 because there are 2 doors. Very simply, that isn't the reality of the situation. If you don't understand why it's 66/33 when there are two options, re-read my post. If you do, then you are trying to prove something which isn't true from reasons which are unclear.
You might as well say, "I have a bag of 100 balls, 99 of which are white, and 1 of which is black. If I draw one ball from the bag randomly, what is the probability that it is white?" The answer, without question, is 99%. However, by the 'logic' you are choosing to apply to this problem, you would argue that it is both 99/1 (because of math) and 50/50 (because you can either get black or white). Just because there are two possible outcomes (i.e. is a binary choice) does NOT mean that it is, in any way, 50/50.
Also, it would be wise to get away from this veneer of "deeper thinking" and "philosophy" because it doesn't mean anything. It doesn't make you sound smart and it certainly doesn't make your point correct. On no "deeper" level is the Monty Hall problem 50/50. It's 67/33. You might not like me disagreeing with your premise, but, as I've already said on a number of occasions, your premise is wrong.
ok so you know it is simultaneously 66/33 and 50/50...you just admitted it. but how isn't that the reality of the situation? That is the situation I'm addressing anyway, you are obviously addressing another one. You can't deny its 50/50, there's 2 doors. How do you expect to argue against that, because that is a fact. And did you not get the black and yellow wall thing?
philosophy is actually all about meaning, so why do you say it doesn't mean anything?
You can dismantle this stupidity quite simply by considering, say, next season's English Premier League in which 20 teams will either win the title, or not. By your warped logic, Liverpool are 50/50 to be champions (sweet!). Sadly, so are Chelsea (they'll either win or they won't). So are Man City, Man United, Arsenal, Spurs, Everton, Villa, Newcastle... this is awkward. Hell, even QPR who just got promoted are 50/50 to win according to you. We're gonna have 10 teams lifting the title at this rate!
As for your point about philosophy, it as a discipline might be about meaning, but just adding the word to your ramblings doesn't give them its legitimacy. It merely adds another 10 letters to your wrongness.
And I didn't address your black/yellow thing because it was about as relevant to the discussion as the following passage: A flock of birds are migrating from Wales to South Africa. A gentlemen sits on a train in Siberia. The sun is bright today.
yeah just remember its a discussion so no need to talk down to me. If we disagree then I guess its my word against yours and there are other intelligent people who think the way I do. You have nothing positive to say about anything I say whereas I've tried to be nice so why don't you just go somewhere else. You can't even understand a simple metaphor.
I have gained something out of this apparently useless discussion. viz Some people ignore, don't see or disbelieve additional information that would help them make better decisions - they will make the same mistakes repeatedly - I should profit from them and never be surprised
I did reply to you and you don't get what I'm saying anyway and were rude in your comment so feel free to stay away from this discussion. btw i never said anything about making decisions or mistakes.
Mr Pompos, the feeling you describe of seeing 2 lines of reasoning that seem to make equal sense (despite leading to different answers), I used to get that a lot and it's part of what fascinated me about probability (probability seemed to do that to me more than anything). My advice to you is, never can both lines of reasoning be correct simultaneously, so you just have to ponder it until you see the flaw in one of them. Think about it on the road, in the shower, when laying in bed. There is always a fallacy somewhere, otherwise both answers would be correct and all of mathematics and logic would fall apart (unless you're solving a quadratic in which case there are two answers!).
Here's another take on monty hall. 2/3 of the time, you picked a goat and by switching you'd have a 100% chance of getting the car (the key is, the host isn't revealing randomly, he's always revealing the goat). The other 1/3 of the time, you picked the car and by switching you'd have a 0% chance. So by switching, overall you have a 2/3 + 0 = 2/3 chance
What exactly is your thought process behind the answer of 50/50? (Assuming it's not the "binary choice" argument which Tom Coldwell already responded to.)
Now that I have a good amount of probability under my belt, what I enjoy the most when helping someone else is hearing a line of reasoning I haven't heard / thought of before. Even if it's wrong, sometimes I'm temporarily stumped as to why, and I love when that happens! That's the kind of stuff that expands the mind. Therefore I always like to know what someone was thinking.
I think you pretty much understand where I'm coming from. I think I get it, its 100% because we're viewing it as 2/3rds as if when we open 3 doors 2 will be goats every time. Even though this is the only flaw in probability theory that its not actually every time, and I tried to argue that because of that its totally void or just slightly unreliable therefore scrap it and just use 50/50. I could be wrong, then again I still see 2 doors left which are unknown, one of them has been deemed by switchmen to be more likely a goat while the 50/50men will always reside in the simpler fact of there being a "binary choice". How can anything 50/50 have an outcome anyway? Is that a fair question?
My thought process is that I see how the door you chose has 2/3 to be a goat and once a goat is revealed, that door still has 2/3rds chance to be a goat, so switching would be best if you want the car. But then I look back and say, well I could've chosen any of the doors before the goat was shown, as long as Monty doesn't reveal the car, I'd be none the wiser which one was the goat and the car. So I'm not sure if I truly believe its 50/50 or not because its a clash between 2 lines of reasoning and yeah I see logical flaws in both of them at different times. I'll keep trying to figure it out and hopefully come to a conclusion. I'll also accept your advice on only 1 line of reasoning being correct, unless it actually is some sort of quadratic formula...
My point about the 100% is that, of the 2/3 times you chose the goat and switched, you switch to the car 100% of the time because Monty is forced to reveal the other goat, leaving only one possibility (car) for the 3rd door. And like you said, it's still a 2/3 chance that you chose a goat because Monty will always reveal a goat no matter what you initially picked. Therefore P(switch to car) = P(chose goat initially) = 2/3
Re: "I'd be none the wiser". The whole point is, new information changes the probability. In an absolute sense, when you flip a coin, given the deterministic laws of physics, there's either a 100% chance it lands Tails or 0%, and there's really no such thing as probability. Probability only exists due to our human ignorance (and in that sense is subjective). However, randomness caused by ignorance is just as effective as hypothetical "true" randomness (the coin acting non-deterministically), therefore we have to resort to probability. And so, information and probability are closely related. If, when playing the game show, you ignore the new information then yes, in your brain it will be 50/50. Just like in poker, if you ignore the fact that say an Ace was exposed, then to you the chance of an Ace is still 4/52. But in both cases, you or someone could have used the information available to adjust the probability. So when we say "the" probability is 2/3, we mean the "most informed" probability (the one that factors in the new information), whereas 50/50 would be the "willfully ignorant" probability. All probability is ignorant, but we always discard the "optional ignorance" because the answers that result from it aren't as useful.
Hopefully by getting all philosophical, I've driven the point home. But in addition, monty hall is a problem that can be experimented in real life and simulated on the computer. Experiment/simulation will agree that the answer is 2/3, so this isn't just something in the abstract.
I see you mentioned binary choice again. Tom gave a good soccer example but I'll give another example and go a little deeper into it. Say you buy a lottery ticket. You'll either win the jackpot or you won't, that's 2 possible outcomes, but clearly you don't have a 50/50 chance of winning the jackpot. What it comes down to is, each outcome can be broken down into sub-outcomes (ways of happening) and for the lotto there are many more losing sub-outcomes (in this case, combinations) than winning ones. Same with any non-50/50 binary choice, one side has more
ways to happen (at least, in a deterministic universe).
Couple things you said I don't understand:
- What is the flaw in probability theory that you speak of?
- What do you mean by, "how can anything 50/50 have an outcome"?
Hey Pompos,
I'm going to try to help you understand by going step by step with a similar problem.
Here's the problem:
Now, let's go step by step (please answer these questions if you reply to my comment):
1. Do you agree that at the very beginning of the game, your odds of picking the ace of spades are 1/52, since you are picking a card at random?
2. Do you agree that if you do not switch cards, the odds that you have picked the ace of spades are still 1/52, since they are the very same odds you had in the beginning? In other words, do you agree that if you don't switch, it is the same as if you picked one of the 52 cards at random, and you will pick the ace of spades once every 52 times?
3. Since I showed you that every one of the 50 cards I removed are not the ace of spades, do you agree that the chance of the card you originally picked being the ace of spades is still 1/52, but the the odds of the other face down card being the ace of spades are 51/52?
In other words, since the ace of spades had a 51/52 chance of being one of the cards that you didn't pick, and I purposefully removed every card left that is not the ace of spades, do you agree that the chance of the other face down card (the one you didn't pick) is the ace of spades is 51/52?
4. If you answered yes to the three questions above, do you agree that if you switch, your chances of picking the ace of spades are 51/52, and that if you stay, your odds of picking the ace of spades are 1/52?
5. If you answered yes at question 4, do you agree that the the same reasoning can be applied to the Monty Hall Problem, and that the better decision is to always switch, just like in this problem?
_______________________________
I hope this helped you. As I said, please answer each question so that we can know what is preventing you from understanding the situation.
-nH.
this example is gorgeous
mandelmonk, thanks for your comment, it was genius and explains alot to me especially the bit about discarding optional/willfull ignorance. That is definitely true and I agree. Maybe the really stubborn people will tend to be willfully ignorant and they will go as far to say the results are just an illusion because of their ignorance. Is 50/50 not a kind of paradox that's what I meant by that question, for 1 outcome to happen the 50/50 bond would be broken. if that bond can be broken to produce the results, couldn't a 70/30 bond also break? I feel like that's a stupid question but thought I'd mention it. Thanks again for your great comment, very philosophical some of it :)
ok Hubris sounds like fun
1. yes
2. yes
3. yes I can see it being 51/52 because the odds of the card I chose is 1/52 so by some sort of elimination or comparison the other 51 cards must have odds of 51/52 to be ace of spades both before and after you take the 50 cards away. Am I right there? Its like you carry the odds over to after the 50 cards are removed.
The confusion I had/have was that from a blank perspective (if I hadn't chosen a card) all the cards have 1/52 to be ace of spaces and therefore once 50 are removed the 2 cards left still have 1/52.
So what I need to do instead is think about it from the perspective of having chosen a card, as it gives me extra information. (although its illusionary its still the best available info and it will be the most informed probability)
4. yes
5. yes
Thanks for the questionaire it was hard to answer and I hope I score high.
Question 3 was the turning point ;).
Yes, you are right.
Exactly! The situation changes, but you have to use the information you had previously to make the best decision possible. The card example was useful because it is much easier to see the disparity between the odds. 1/52 is very, very far from 1/2, so the problem is easier to understand.
The Monty Hall Problem is harder to understand because the difference between 66% and 50% is much smaller, and therefore people think that one answer is just as plausible as the other, when it's just not the case.
You now understand the Monty Hall Problem!
Glad you got there :) Sorry I didn't have the patience of NoHubris and Mandelmonk. RIO has some great members - (commences liking of posts....)
thanks steamer I think I have definitely got there and I can explain it to other people who think its 50/50 now.
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