{"items": [{"author": "Jan", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345556782125757", "anchor": "fb-345556782125757", "service": "fb", "text": "I get it. And I can't believe how long that Wikipedia article is.", "timestamp": "1324388087"}, {"author": "Mike", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345564902124945", "anchor": "fb-345564902124945", "service": "fb", "text": "If you picked a dud to begin with, then when one of the two remaining is shown to be the other dud, the third has to be the prize. Only if you originally picked the prize could the third one be a dud. The odds are 2:1 that you picked a dud. Would that have convinced you?", "timestamp": "1324388962"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345565962124839", "anchor": "fb-345565962124839", "service": "fb", "text": "@Mike: I don't know. Now that I understand it, all sorts of explanations sound like they should have convinced me. I think the only way to find out if that is a good explanation is to try it on a bunch of people who've not seen the problem before.", "timestamp": "1324389068"}, {"author": "Marty", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345566292124806", "anchor": "fb-345566292124806", "service": "fb", "text": "It's the Monty Hall Problem again! Marilyn Vos Savant once tried to explain it, correctly, and got a lot of flack even from PhDs!", "timestamp": "1324389101"}, {"author": "Marty", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345567352124700", "anchor": "fb-345567352124700", "service": "fb", "text": "http://en.wikipedia.org/wiki/Monty_Hall_problem", "timestamp": "1324389180"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345568928791209", "anchor": "fb-345568928791209", "service": "fb", "text": "@Marty: I don't like the name \"Monty Hall Problem\" because this isn't how the game show worked: Monty Hall didn't allow switching doors.", "timestamp": "1324389341"}, {"author": "Marty", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345570358791066", "anchor": "fb-345570358791066", "service": "fb", "text": "Oh, I remember that, on his show \"Let's Make a Deal\", after the contestant chose one door, he would open another to show the goat, and then offered to let the contestant switch!", "timestamp": "1324389505"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345572722124163", "anchor": "fb-345572722124163", "service": "fb", "text": "@Marty: \"if you ever get on my show, the rules hold fast for you -- no trading boxes after the selection.\" -- Monty Hall, 1975
(via: http://www.letsmakeadeal.com/problem.htm)", "timestamp": "1324389765"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345579585456810", "anchor": "fb-345579585456810", "service": "fb", "text": "If you eliminate one door, the odds are 50/50 that Schrodinger is in the on you pick.", "timestamp": "1324390425"}, {"author": "Josh", "source_link": "https://plus.google.com/118273920476267337216", "anchor": "gp-1324390712559", "service": "gp", "text": "The thing that confuses me about this is the \"revealing information that is already known does not change the probability\" issue. Wikipedia says \"If the car is initially placed behind the doors with equal probability and the host chooses uniformly at random between doors hiding a goat (as is the case in the standard interpretation) this probability indeed remains unchanged, but if the host can choose non-randomly between such doors then the specific door that the host opens reveals additional information.\"\n
\n
\nWhat's up with that?", "timestamp": 1324390712}, {"author": "Marty", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345584952122940", "anchor": "fb-345584952122940", "service": "fb", "text": "Ha. I looked at some other YouTube episodes of Monty Hall, and the situations are indeed different from the way the problem is stated. The misnomer will, I bet (um\u2026), still remain the common name for this three-door problem.", "timestamp": "1324390893"}, {"author": "Jeff Kaufman", "source_link": "https://plus.google.com/103013777355236494008", "anchor": "gp-1324390902878", "service": "gp", "text": "@Josh\n That is confusing. I would ignore that part of the wikipedia article. That paragraph is helpfully labled as a \"source of confusion\", though.", "timestamp": 1324390902}, {"author": "Josh", "source_link": "https://plus.google.com/118273920476267337216", "anchor": "gp-1324393420793", "service": "gp", "text": "Well, but I find the \"no new information\" argument compelling (even though I know it's wrong). When I pick my door, I know that one of the doors I didn't pick has a goat; why does it matter whether we open that door or not? It doesn't tell me anything, so why does it change the probability that I picked the right door?\n
\n
\nAh, well, I suppose that's the answer: It doesn't change the probability that I picked the right door, but identifying a wrong door increases the probability that the \nother\n door is the right door, from 1/3 to 2/3. Hmm.", "timestamp": 1324393420}, {"author": "Jeff Kaufman", "source_link": "https://plus.google.com/103013777355236494008", "anchor": "gp-1324393618008", "service": "gp", "text": "@Josh\n \"know that one of the doors I didn't pick has a goat; why does it matter whether we open that door or not? It doesn't tell me anything\"\n
\n
\nIt does tell you something: it tells you which of the other two doors you definitely don't want.\n
\n
\n\"It doesn't change the probability that I picked the right door, but identifying a wrong door increases the probability that the other door is the right door, from 1/3 to 2/3\"\n
\n
\nRight.", "timestamp": 1324393618}, {"author": "Jean", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345609338787168", "anchor": "fb-345609338787168", "service": "fb", "text": "The explanation from the article that made the most sense to me was \"Switching loses IFF the player initially picks the [prize], which happens with probability 1/3, so switching must win with probability 2/3.\"", "timestamp": "1324393630"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345609818787120", "anchor": "fb-345609818787120", "service": "fb", "text": "@Jean: I like that; I'll try it next time. Very succinct.", "timestamp": "1324393686"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345615385453230", "anchor": "fb-345615385453230", "service": "fb", "text": "So if the odds of your original pick remain one in three after one of the other two is eliminated, what if both of the other two are eliminated? Surely the odds can't still be only one in three then?", "timestamp": "1324394340"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345618015452967", "anchor": "fb-345618015452967", "service": "fb", "text": "@Paul: \"what if both of the other two are eliminated?\"
The normal rule is that one of the two you didn't pick will be eliminated, and that it won't be the one with the prize. I'm not sure what you're proposing instead. If you're proposing to always eliminate the other two, even if one of them has the prize, then the odds are still one in three for your initial choice.", "timestamp": "1324394623"}, {"author": "Marty", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345619958786106", "anchor": "fb-345619958786106", "service": "fb", "text": "Well @Paul, that wouldn't leave any choice, so it's not a real possibility. Probability only matters before a choice; after the answer is revealed, it's 100% probable that it's true. An alternative scenario might be, suppose there were 100 doors. You pick one, the host opens 98 of the others. Wouldn't it seem obvious that you're way more likely to win (99 to 1) if you switched to that one remaining door?", "timestamp": "1324394841"}, {"author": "Marty", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345621008786001", "anchor": "fb-345621008786001", "service": "fb", "text": "Jan, now you have a chance not to believe how long this thread gets.", "timestamp": "1324394947"}, {"author": "Robert", "source_link": "https://plus.google.com/117732328885787456164", "anchor": "gp-1324399099991", "service": "gp", "text": "This explanation convinces me (even though it still feels wrong intuitively). I think the source -- or at least \na\n source -- of the confusion is that the problem as stated downplays the fact that the unchosen door that gets opened is known by the opener to be a dud. That's the key piece of information; I think my initial inclination was to suppose that the \"revealed\" door \ncould\n have been the winner.", "timestamp": 1324399099}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345659728782129", "anchor": "fb-345659728782129", "service": "fb", "text": "Take Marty's example one step further. there are 100 doors with one prize. I pick a door. Then you pick a different door. The other 98 doors are eliminated. Are my odd of winning 1% and yours 99%?", "timestamp": "1324399136"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345665355448233", "anchor": "fb-345665355448233", "service": "fb", "text": "@Paul: in eliminating the other doors, what happens if one of them has the prize? If neither of us win in that case, we each have a 1% chance of winning and there's a 98% chance of no one winning.", "timestamp": "1324399672"}, {"author": "Tanner", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345666725448096", "anchor": "fb-345666725448096", "service": "fb", "text": "The thing that has always bothered me about this (sorry for entering late) is it works both ways. Let's say I pick door number 1, door number 2 opens, it's empty, and now the probabilities say I should switch to number 3. But had I originally picked number 3, then when number 2 opened apparently I should switch to number 1!", "timestamp": "1324399820"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345667575448011", "anchor": "fb-345667575448011", "service": "fb", "text": "I still don't see why eliminating the third door changes the odds on the unchosen door but not on the chosen door.", "timestamp": "1324399898"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345668235447945", "anchor": "fb-345668235447945", "service": "fb", "text": "@Paul: in deciding which of the other two doors to eliminate, the host, who knows where the prize is, is giving you information.", "timestamp": "1324399977"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345669088781193", "anchor": "fb-345669088781193", "service": "fb", "text": "@Tanner: we can assume, without loss of generality, that you always pick door #1. We can't assume that the game show host always opens #2. If #2 has the prize, they have to open #3.", "timestamp": "1324400071"}, {"author": "Jenny", "source_link": "https://plus.google.com/111419027123995253579", "anchor": "gp-1324400075011", "service": "gp", "text": "The most compelling explanation of this problem I ever heard was the classic \"consider the extremes\" approach. Let there be 100 doors instead of 3. You pick one door and Monty opens 98 other doors. You can decide whether to stick by your original answer or switch to the door he didn't open. Obviously you want to switch to that conspicuously closed door. Of course, if you use this explanation, you then have to convince your listener that Monty opening 98 doors (not just one) is the same problem. Also the article where Marilyn vos Savant correctly answers this question and then gets condescending mail from math professors around the country is amazing:\n
\nhttp://www.marilynvossavant.com/articles/gameshow.html", "timestamp": 1324400075}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345669148781187", "anchor": "fb-345669148781187", "service": "fb", "text": "Yes; he's telling you that it is in one of the other two, changing the odds on both of them to 50%.", "timestamp": "1324400077"}, {"author": "Tanner", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345671752114260", "anchor": "fb-345671752114260", "service": "fb", "text": "I was making a general point that in two scenarios where the car is behind the same door but the choosers pick two different doors, they both would receive the statistical benefit from switching, regardless of the fact that one of them actually has already chosen the correct door.", "timestamp": "1324400334"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345672802114155", "anchor": "fb-345672802114155", "service": "fb", "text": "Tanner: that's just crazy. Whether they switch or not, one is going to win and one is going to lose. How can there be any statistical advantage?", "timestamp": "1324400442"}, {"author": "Tanner", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345673218780780", "anchor": "fb-345673218780780", "service": "fb", "text": "That's my point.", "timestamp": "1324400482"}, {"author": "Tanner", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345674828780619", "anchor": "fb-345674828780619", "service": "fb", "text": "Or to put it another way: let's say there are two choosers. They each choose two different doors. The host opens up the door unchosen, and it's empty. Now each one of them has a door, and one of the doors has the prize. This \"statistical advantage\" to switching works for BOTH of them, so they \"should\" switch, and yet one of them loses the prize and one of them gains it, giving us 50/50 odds of winning from switching.", "timestamp": "1324400662"}, {"author": "Tanner", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345675245447244", "anchor": "fb-345675245447244", "service": "fb", "text": "... or so I would think, at least.", "timestamp": "1324400711"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345675355447233", "anchor": "fb-345675355447233", "service": "fb", "text": "@Paul: he's telling you more than that. There was originally a 2/3 chance that the prize was behind one of the two doors you didn't choose. Because the host is required to eliminate a dud, and you know there's at least one dud, the combined chance that the prize is behind the other two doors is still 2/3 even after he opens one of them to show that it is empty. So it has a 2/3 chance of being behind the remaining other door.", "timestamp": "1324400724"}, {"author": "Tanner", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345675762113859", "anchor": "fb-345675762113859", "service": "fb", "text": "But there's also a 2/3 chance it was behind the opened door and the one you chose, isn't there?", "timestamp": "1324400766"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345676522113783", "anchor": "fb-345676522113783", "service": "fb", "text": "@Tanner: \"The host opens up the door unchosen, and it's empty.\"
This is the problem. This can only happen 1/3 of the time. A critical requirement here is that the host must *always* open an unchosen door not containing the prize. In your modified version the host cannot do that.", "timestamp": "1324400837"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345677685447000", "anchor": "fb-345677685447000", "service": "fb", "text": "That's the fallacy. There's no reason why the odds of only one of the remaining doors should change and not the other.", "timestamp": "1324400944"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345677992113636", "anchor": "fb-345677992113636", "service": "fb", "text": "@Paul: \"That's the fallacy\"
What is?", "timestamp": "1324400974"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345682505446518", "anchor": "fb-345682505446518", "service": "fb", "text": "You're treating the two uneliminated doors differently.", "timestamp": "1324401389"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345698865444882", "anchor": "fb-345698865444882", "service": "fb", "text": "@Paul: \"You're treating the two uneliminated doors differently\"
And rightly so. The elimination tells you nothing about the door you chose because you already knew one of the other two would be eliminated. You didn't know *which* of the other two would be eliminated, however, which means you learned something about the remaining door.", "timestamp": "1324403052"}, {"author": "Tanner", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345703532111082", "anchor": "fb-345703532111082", "service": "fb", "text": "I am ducking out of this conversation but intend to return with empirical evidence later.", "timestamp": "1324403512"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345705572110878", "anchor": "fb-345705572110878", "service": "fb", "text": "But it was never behind the eliminated door. Opening it doesn't change that. Why should the unselected remaining door get all the benefit of that information?", "timestamp": "1324403723"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345706982110737", "anchor": "fb-345706982110737", "service": "fb", "text": "If the results in practice skew that way, it may be because it is good for ratings to have winners, so they encourage you to switch if you've chosen wrong.", "timestamp": "1324403871"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345712265443542", "anchor": "fb-345712265443542", "service": "fb", "text": "@Paul: \"if the results in practice skew that way\"
If you sit down and try this yourself, you will find that the results are 2:1 in favor of switching. Procedure:
1) Acquire three identical opaque cups and a penny
2) Put the penny under a cup
3) Shuffle them until you don't know where the penny is
4) Choose a cup
5) Look under the other two. If one has the penny eliminate (and put aside) the other. If neither has the penny eliminate one at random.
6) Tally a point for \"keep\" if the penny is under your originally chosen cup, a point for \"switch\" if it's under the remaining unchosen cup
7) Repeat 1-6 about 20 times.
This is a little strange because you're simulating both the contestant and the host, but it should be fine as long as you can actually lose track of the penny in step 3.", "timestamp": "1324404388"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345715798776522", "anchor": "fb-345715798776522", "service": "fb", "text": "Your step 5 is the kicker. Once you know where the penny is all randomness is off.", "timestamp": "1324404764"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345722335442535", "anchor": "fb-345722335442535", "service": "fb", "text": "There are three cups and a penny under a random cup. I choose one. I lift up one of the remaining two. At this point there are three equally likely scenarios:
1: The penny is under the cup I selected.
2: The penny is under the other unlifted cup.
3: The penny has just been exposed.
Scenario three never occurs in the game, leaving the other two equally likely scenarios.", "timestamp": "1324405441"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345723495442419", "anchor": "fb-345723495442419", "service": "fb", "text": "@Paul: that's not the game. You don't look under one alternate cup; you look under both. You're pretending to be the game show host deciding which door to open. The host must open a door, and can't open the door with the prize behind it.", "timestamp": "1324405591"}, {"author": "Todd", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345725388775563", "anchor": "fb-345725388775563", "service": "fb", "text": "Jeff, I think the fact that you're still getting an argument about this, despite the fact that there is a large body of evidence (empirical and theoretical) available that demonstrate the accuracy of your position, serves as excellent evidence for your *other* proposition- namely, that people's intuitions make them terrible at understanding this problem.", "timestamp": "1324405783"}, {"author": "Todd", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345726112108824", "anchor": "fb-345726112108824", "service": "fb", "text": "And Paul, you could always try running the test with a \"host\" to accurately simulate the situation if you don't want to test it by yourself because that feels like cheating.", "timestamp": "1324405851"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345728665441902", "anchor": "fb-345728665441902", "service": "fb", "text": "But if the host knows, then the host is cheating.", "timestamp": "1324406138"}, {"author": "Todd", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345730892108346", "anchor": "fb-345730892108346", "service": "fb", "text": "The host isn't playing. Are you sure you understand the ground rules?", "timestamp": "1324406391"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345731555441613", "anchor": "fb-345731555441613", "service": "fb", "text": "Yes, and apparently the host is ecouraging people to switch in order to boost the ratings, as I suggested above.", "timestamp": "1324406459"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345731705441598", "anchor": "fb-345731705441598", "service": "fb", "text": "@Paul: the host knows because that's the way the game works. Wikipedia: \"After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one at random. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door.\"", "timestamp": "1324406476"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345732275441541", "anchor": "fb-345732275441541", "service": "fb", "text": "(goat means no prize, goats being less desirable than cars)", "timestamp": "1324406543"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345732335441535", "anchor": "fb-345732335441535", "service": "fb", "text": "My point exactly. It's good for ratings to have winners, so if you've chosen wrong he gives you a chance to switch.", "timestamp": "1324406549"}, {"author": "Todd", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345732538774848", "anchor": "fb-345732538774848", "service": "fb", "text": "Paul, even if your theory is true (that the host wants ratings-boosting switching), surely running an experiment in which someone plays host for you would eliminate that issue, since that person would have no such consideration.", "timestamp": "1324406577"}, {"author": "Todd", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345733728774729", "anchor": "fb-345733728774729", "service": "fb", "text": "Also, bear in mind that you ALWAYS have the option to switch, regardless of whether you've chosen correctly or not (in fact you don't know if you've chosen correctly until after you decide whether you're going to switch).", "timestamp": "1324406713"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345734338774668", "anchor": "fb-345734338774668", "service": "fb", "text": "Yes, and as I described above, once you eliminate the spurious scenario where the host exposes the penny (or car) the odds are 50/50 between the remaining two.", "timestamp": "1324406776"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345734788774623", "anchor": "fb-345734788774623", "service": "fb", "text": "@Paul: \"It's good for ratings to have winners\"
We're not talking about an actual game show. [1] This is a probability puzzle that could be implemented in a game show (or tested on your own with pennies and cups).
\"if you've chosen wrong he gives you a chance to switch\"
From the problem statement on wikipedia \"Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors\". You're given a chance to switch whether you have chosen wrong or not.
\"once you eliminate the spurious scenario where the host exposes the penny (or car)\"
This never happens because \"the door he opens must have a goat behind it\".
[1] Monty Hall's game show, \"Let's Make a Deal\", did not work this way. Contestants were not given a chance to switch. The problem started being called the \"Monty Hall\" problem by someone who thought it was similar to the show. (See, Marty?)", "timestamp": "1324406832"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345735712107864", "anchor": "fb-345735712107864", "service": "fb", "text": "\"This never happens because \"the door he opens must have a goat behind it\".\"
Yes, as I said, the spurious scenario is eliminated.", "timestamp": "1324406940"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345736242107811", "anchor": "fb-345736242107811", "service": "fb", "text": "@Paul: Do you still think the experimental procedure I proposed above doesn't match the problem statement?", "timestamp": "1324407002"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345737135441055", "anchor": "fb-345737135441055", "service": "fb", "text": "You're explaining away the suprious one third of the time where the penny is exposed and the results are spoiled. You're claiming it as a win when it requres a redo.", "timestamp": "1324407109"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345738692107566", "anchor": "fb-345738692107566", "service": "fb", "text": "@Paul: please reread the relevant part of the problem, from wikipedia:
\"After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one at random. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door.\"
The way the problem is defined, the host never exposes the prize when opening a door. They know what's behind all three of the doors, it's only the contestant who doesn't know.", "timestamp": "1324407299"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345740095440759", "anchor": "fb-345740095440759", "service": "fb", "text": "In other words, you're specifically excluding a double blind experiment, because you know that a double blind experiment would yield other results. I could probably prove the existence of a Flying Spaghetti Monster under those rules.", "timestamp": "1324407453"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345742275440541", "anchor": "fb-345742275440541", "service": "fb", "text": "@Paul: the information that the contestant gets from the host when the host, knowing what's behind all the doors, opens a door without the prize is crucial to the problem. It would be a different problem if you somehow made it double blind.", "timestamp": "1324407687"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345747195440049", "anchor": "fb-345747195440049", "service": "fb", "text": "So it's all a rhetorical trick. You assume the host is impartial and playing fair, but he's actually cheating in your favor. \"How many were walking to St. Ives?\"", "timestamp": "1324408207"}, {"author": "Marty", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345749932106442", "anchor": "fb-345749932106442", "service": "fb", "text": "Oh geez. Make up a table with all the possible outcomes, and count the ones in your favor for any particular scenario, divide by the total, to get your probability.", "timestamp": "1324408501"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345750412106394", "anchor": "fb-345750412106394", "service": "fb", "text": "@Paul \"the host is actually cheating in your favor\"
I don't see how it's cheating: the host follows the rules of the game show, which specify opening doors in a certain manner.
@Marty \"Make up a table with all the possible outcomes\"
I think the disagreement with Paul was over which outcomes to include in the table.", "timestamp": "1324408549"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345751908772911", "anchor": "fb-345751908772911", "service": "fb", "text": "\"I think the disagreement with Paul was over which outcomes to include in the table.\" That was precisely the point of contention, as well as whether the rules are misleading in that regard.", "timestamp": "1324408686"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345752218772880", "anchor": "fb-345752218772880", "service": "fb", "text": "By the way, I think this was fun and I hope everyone else does too.", "timestamp": "1324408724"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345753168772785", "anchor": "fb-345753168772785", "service": "fb", "text": "There are three doors: behind one is a car, behind another is a goat, and behind the third there is a cat that may or may not be alive. Which do you choose?", "timestamp": "1324408837"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345755168772585", "anchor": "fb-345755168772585", "service": "fb", "text": "While you might think the issue is just the problem statement being confusing, if you do away with the problem statement and just have choices and rewards, people still do badly. (Relative to pigeons as a control.) Any ideas why?", "timestamp": "1324409050"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345758645438904", "anchor": "fb-345758645438904", "service": "fb", "text": "I think that people overlook the fact that, in selecting a door, they're exempting that door from contention for elimination. I realized the effect of that about halfway throuigh the discussion, and was playing for fun after that.", "timestamp": "1324409375"}, {"author": "Todd", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345758688772233", "anchor": "fb-345758688772233", "service": "fb", "text": "Jeff, it might actually be fair to say that the host IS cheating in the contestant's favor, and that that is WHY the contestant should switch. The whole point is that the host gives you information, which biases your choice. If Paul sees that as cheating, it doesn't really change the underlying structure.", "timestamp": "1324409378"}, {"author": "Todd", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345759625438806", "anchor": "fb-345759625438806", "service": "fb", "text": "Trolling should not be allowed on Jeff's posts. Don't ruin one of the few places where serious and meaningful discussion can take place in a social network =(", "timestamp": "1324409463"}, {"author": "Marty", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345761478771954", "anchor": "fb-345761478771954", "service": "fb", "text": "Sorry, I misunderstood. Easy to do, apparently!", "timestamp": "1324409623"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345764285438340", "anchor": "fb-345764285438340", "service": "fb", "text": "I hope I'm not the troll here. I thought I was being thought provoking.", "timestamp": "1324409894"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345767255438043", "anchor": "fb-345767255438043", "service": "fb", "text": "@Todd: aw, thanks.", "timestamp": "1324410185"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345775782103857", "anchor": "fb-345775782103857", "service": "fb", "text": "@Paul: if you did actually understand where we disagreed but were pretending not to, that would be trolling.", "timestamp": "1324411033"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345776392103796", "anchor": "fb-345776392103796", "service": "fb", "text": "Sorry", "timestamp": "1324411093"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345779692103466", "anchor": "fb-345779692103466", "service": "fb", "text": "I was playing like a rhetorical exercise, where you make the best arguments you can for a given position in the hope of drawing the best from the other side. It's not like the outcome has any consequences for any of us. I agree with Todd's view of Jeff's posts, and I apolgize if anyone felt I was abusing the forum.", "timestamp": "1324411402"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345780892103346", "anchor": "fb-345780892103346", "service": "fb", "text": "@Paul: the problem is, I was interested in the question of what actually convinces someone who is sure this is not how the problem works. The perspective of someone who is only pretending to not know how the problem works is less interesting.", "timestamp": "1324411521"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345782592103176", "anchor": "fb-345782592103176", "service": "fb", "text": "Sorry. After all my extraneous noise, my comment
\"I think that people overlook the fact that, in selecting a door, they're exempting that door from contention for elimination,\" may have been my only valid contribution.", "timestamp": "1324411676"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345820118766090", "anchor": "fb-345820118766090", "service": "fb", "text": "As partial penance for my trolling, perhaps I can explain why I was so sure I was right at first and what convinced me that I was wrong, since that was Jeff's more fundamental question to begin with. I began so convinced that I thought the original question was \"Why do people think that switching DOES help?\" My intuition, which (modesty aside) is usually prettly relable, was not budged by any of the explanations. It was only when I began running the experiments in my head and began to encounter unexpected results that I began to doubt. I continued running the experiment, looking for the flaw that I thought must be in the experiment rather than in my initial assumption. (Meanwhile I continued to advocate my postion in the thread.) I began to sense that I was wrong, but couldn't understand the logic of it and pressed the discussion hoping that it would be spelled out more clearly. I thought I might have found the flaw I was looking for when I realized that the host knows where the car (or penny) is, so it was not a \"double blind\" test. Eventually, at about the point I dragged the Spaghetti Monster into it, I realized that the \"double blind\" analogy was false. The host is not picking one of doors at random; he's picking one of the two goats at random, except that 2/3rds of the time you have taken one of them out of contention by choosing its door. I'm sorry if I offended anyone.", "timestamp": "1324415340"}, {"author": "Todd", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345864318761670", "anchor": "fb-345864318761670", "service": "fb", "text": "People learning and changing their minds- it's why you deserved the compliment, Jeff =) And lest that come across as condescending or arrogant towards Paul, rest assured that I speak from personal experience as well in this, just not regarding this particular thread.", "timestamp": "1324419779"}, {"author": "Kiran", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345900915424677", "anchor": "fb-345900915424677", "service": "fb", "text": "Paul's summary--that the trickster is choosing a *goat* rather than a door, and that if a goat is behind your door he *must* choose the other goat--is actually helpful. (IIUC, you had a 2/3 chance of choosing a goat, which means you had the same chance of coercing the host.) Perhaps it the problem were phrased that way, people would find it less troublesome.", "timestamp": "1324423959"}, {"author": "George", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345908318757270", "anchor": "fb-345908318757270", "service": "fb", "text": "I agree with J-bar, the most convincing argument is to consider a huge number of doors and opening all but one of them (of course we assume the doors that get opened are selected to guarantee the prize isn't revealed). Honestly, I don't see why this problem is so troubling to people. There are many other probabilistic problems I find much more troublesome. I think if one just computes the relevant quantities carefully with a tree diagram it is relatively clear.", "timestamp": "1324424803"}, {"author": "Todd", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345970815417687", "anchor": "fb-345970815417687", "service": "fb", "text": "George- b/c it's not intuitive. Careful computation isn't intuition.", "timestamp": "1324431453"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345981485416620", "anchor": "fb-345981485416620", "service": "fb", "text": "@George: \"I think if one just computes the relevant quantities carefully with a tree diagram it is relatively clear.\"
Do you find it strange that pigeons don't need to make tree diagrams to learn that they should adopt the 'switch' strategy? While (the tested) people never learn at all?
http://www.ncbi.nlm.nih.gov/.../PMC30.../pdf/nihms288435.pdf", "timestamp": "1324432268"}, {"author": "George", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=345996492081786", "anchor": "fb-345996492081786", "service": "fb", "text": "No, I don't find it odd. People are very bad at doing explicit probabilistic reasoning, but they can be as good or better than pigeons at selecting actions to maximize rewards. I bet if you abstracted the problem for the humans so they had to pick action A or B in the scenario after seeing state X or Y they would quickly learn to \"switch\" but they wouldn't know it corresponded to the hypothetical game show scenario.", "timestamp": "1324433964"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=346012952080140", "anchor": "fb-346012952080140", "service": "fb", "text": "@George: \"if you abstracted the problem for the humans so they had to pick action A or B in the scenario after seeing state X or Y they would quickly learn to \"switch\" but they wouldn't know it corresponded to the hypothetical game show scenario.\"
I believe that's exactly what they tried in the paper I linked. And what they found was that \"human participants showed that humans failed to adopt optimal strategies, even with extensive training\" (though \"birds adjusted their probability of switching and staying to approximate the optimal strategy\").", "timestamp": "1324435808"}, {"author": "Tanner", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=346118142069621", "anchor": "fb-346118142069621", "service": "fb", "text": "... so the end results of my computer program:
Statistics:
Games played: 10000
Wins: 5017
Losses: 4983
Number of games where the player's choice changed: 4973
Of those games, wins: 3316
Winning percentage with switching: 66.68007239091092 %
Number of games where the player stayed with his original choice: 5027
Of those games, wins: 1701
Winning percentage with staying: 33.83727869504675 %
... I am duly chagrined.", "timestamp": "1324450760"}, {"author": "Kiran", "source_link": "https://www.facebook.com/jefftk/posts/345544475460321?comment_id=346144215400347", "anchor": "fb-346144215400347", "service": "fb", "text": "Jeff's question is about what makes this so confusing to humans, not why bird brains are better at solving such problems (which leads to interesting speculation about the minds of birds and their ancestors, such as whether they have a cognitive subsystem which makes tree diagrams intuitive.) Let me rephrase my (hopefully correct) understanding of this in language inspired by pioneers of evolutionary psychology Leda Cosmides and John Tooby, who discussed a similar issue related to the Wason selection task.
http://en.wikipedia.org/wiki/Wason_selection_task
The chance that I pick the car are 1/3. IOW, once in three tries I chose correctly, which allows the host to trick me out of the car I rightly deserve by opening a *random* door, at which time (since I read about the so-called MHP on Facebook) I switch and lose the car I rightly deserved. But 2/3 of the time--twice in three tries--I choose a goat, and thus force the evil host to show me the *other goat*. I switch and win the car.
Leaving aside the question of whether two cars and one goat are more valuable to me than two goats and one car, 2/3 of the time I can force the host to give me more information, presumably because he knows I'll have him beaten by thugs if he shows me the car. Since my chances of coercing the host are greater than my chances of being tricked--IOW, I can force him to show me the other goat more often than not--I should behave as if I have the upper hand and therefore switch.
Part of what might make this confusing to humans is that (at least) two things are unclear from (and perhaps even intentionally obscured by) the formal description of the problem. First, 2/3 of the time, I can force the host to give me more information. Second and more important, the host is constrained as to the nature of the additional information. He *must* show me a goat (not just open a door!) and if I pick a goat he must show me the *other* goat. If my chance of picking a goat is higher than that of picking a car, so is my chance of getting to see the other goat.
I probably picked a goat!
Understanding that is important. It means the goat I see is probably the other goat. Because there are more goats than cars, I probably picked a goat. The host definitely showed me a goat. So now I probably know where both goats are. And that means the car is probably behind the door I didn't choose. If I'm *likely* to have gained some information, I should act *as if* I gained some information, and the only action short of becoming a bicycling vegetarian is to change doors.
Now, getting back to the question of cars v. goats... I'm surprised Chris Lahey hasn't weighed in on this issue.", "timestamp": "1324456104"}]}