I did promise that I would try to review just how I managed to convince myself that I was right with the reverse Monty Hall problem. Now I have the benefit of distance with which to assess what happened (because I am looking at what a completely different person did, you know, “me” but the “me” of a few months ago, who is totally different to the “me” today).
My focus here will not be on the results of my error, but rather trying to understand how I made the error – and why I found the error convincing. (There are quite a few posts on the results already, together with some comments to those posts as well as quite a bit of activity over at reddit, largely in the “bad mathematics” area. Admitting I was wrong has apparently still not appeased the masses. Oh well. Never mind.)
I’m aware that my memory may not be perfect, partly because humans forget things and partly because humans tend to edit the past to favour themselves, so this review may not be entirely “the truth”, but I will try my best to be objective.
It all started out with me thinking about inductive logic, versus deductive logic and, sort of, about black swans. The idea was that if you spend your whole life seeing nothing but white swans (as Europeans used to), then you’d reason that if you were presented with a swan in a box, it would be reasonable to assume that it is white. (Sherlock Holmes might have “deduced” that the very fact that the swan was presented in a box indicates that the giver was hiding some dark secret. However, the amazing accuracy of many of Holmes’ “deductions” relies entirely on the author being in control of the world in which Holmes lives – in the real world they would be little better than informed guesses and are better categorised as “intuitions”. There are a host of reasons why a person might present you with a swan in a box, other than to hide the possible fact that it is a black swan – they might hiding the fact that it’s not a swan at all but rather just a particularly ugly duck.)
Anyways … I came up with a scenario in which a volunteer removes, one by one, balls from an urn. If after 999,999 white balls have been removed, I stop them and ask them the likelihood that the next ball is not white. In the absence of any other information, the volunteer should respond that there is a 1/1,000,000 chance that the next ball is not white. Here’s my logic:
I’ve constrained the exercise to the removal of one million balls from the urn. Effectively, what I’ve asked the volunteer is “what is the likelihood that, out of a draw of one million balls, the only non-white ball will be in the one millionth position?” The other, more likely outcome, from the volunteer’s point of view, is that the last ball will also be white. This is because the volunteer is working with an absence of any other information. There may be more balls in the urn, of which an unknown number are non-white, or the one millionth ball might be the last ball and I, as the experimenter, might know that this last ball must in fact be black. It’s this last scenario that I had in mind.
You see, as the volunteer keeps removing white ball after white ball, she is rightfully becoming more and more convinced that the next ball will be white, following the logic of “what is the likelihood that, out a draw of X balls, the only non-white ball will be at the Xth position?” As X increases, after X-1 white balls have been drawn, it appears more unlikely that the next ball will be non-white. I, on the other hand, have more information than the volunteer and am aware that the likelihood of the next ball being non-white is actually increasing. This reaches a crescendo at the 999,999th ball at which point, I know the likelihood of the next ball being non-white is 100% while the volunteer will believe that there is only a 0.0001% chance of it not being white.
It seems to me that this is a justified conclusion by the volunteer under the circumstances, but then I also know that it’s wildly inaccurate.
So, I tried to rein the numbers in a bit, all the way back to two. Say that my volunteer reaches into an urn and removes a white ball. In the absence of any other information, what is the likelihood that the next ball selected will be white? It’s 50%, but this also seemed strange to me – after all, we are not just talking about the possibility of white and black balls here, there could be a huge array of hues, colours and patterns available. Perhaps it seems reasonable to think about selecting a second pure white ball, since that’s a simple enough decoration, but if we think of a scenario in which the volunteer extracts a ball with light violet stripes and puce dots on a beige background with orange swirls, it seems somewhat less certain that there will be a 50% chance that the second ball will be of the same type.
This sort of gets us where I was initially headed. We tend not to notice when bland things happen (like selecting a plain old white ball) and are confounded when strange things happen (like removing our tutti-frutti themed ball), it messes with our intuitions. The likelihood of selecting that strangely patterned ball seems remote, but it won’t be if it’s also a common pattern. If my volunteer keeps dipping her hand into the urn and removing similarly patterned balls, then the amazement of that first selection will fade and eventually her intuitions will shift to match what would be expected with an unbroken series of plain white balls. At that point, she may reflect that back when she held only one of these balls in her hand, the likelihood (at the time) of the next one being the same was also 50%, as it would have if it had been white. (Note that with the information to hand, my volunteer will have to reassess the post facto likelihood of the second ball being tutti-frutti themed as being higher than 50% – the exact figure depends on the number of balls available and how many balls have been selected so far.)
Compare this to an argument often run by apologetic theists: the probability of the universe being just the way it is such that it supports intelligent life (that is humans) is so remote that it is therefore inconceivable that the universe arose by chance, therefore god. We are in the same position as my feckless volunteer, after her first selection, metaphorically holding an apparently impossible ball in our hand and being stunned and amazed by it. However, we are not able to draw from the urn again to get a better idea about how likely it really is that such a ball should be in our hand. So, while in the absence of any other information it may seem unlikely that our universe should be so apparently finely tuned, we simply don’t know what the real likelihood is.
I then devised a scenario to test this challenge to our intuitions, which I wrote up (at the second attempt) at Two Balls, One Urn, Revisited. The whole idea of this scenario was to trigger the logic of “what is the likelihood that, out a draw of X balls, the only non-white ball will be at the Xth position?” in which my volunteer might say 50% where it can be shown that it’s not, it’s a wildly different figure. In this article, I had a barrel with 2 million balls in it, two of which were white and, effectively, I artificially forced one of two balls removed from the barrel into being white (I even make that clear in the comments) and asked what the chances were that the other removed ball was white.
Here is where I made the mistake. I gave too much information and did not clarify how little information my volunteer had. I didn’t even notice that I had done so.
If my volunteer was totally oblivious to my barrel extraction activity, and only knew that one ball had been removed from the urn, and that it was white, then she could reasonably conclude that the likelihood of a second white ball being removed from the urn would be 50%. I, on the other hand, would know that the likelihood would be 1/1,999,999 – and unfortunately I got wrapped around the axles on other calculations (like 1/3,999,997 and 1/1011).
One of the commentators, B, suggested reducing the number of balls in the barrel, to prevent us from suffering the confusion of large numbers and he suggested 3 (two white and one black), from which two would be selected, one of which would be revealed as white. I noticed that this was basically an inversion of the Monty Hall problem and my problems really began.
Remember that I had in my mind a scenario in which the volunteer (now transforming into a contestant) knew nothing, other than the nature of the revealed ball (now transforming into a goat). During the transformation process, I forgot all about that ignorance and began trying to apply my thinking (in the presence of ignorance) to the Monty Hall Problem (in which there is less ignorance).
Given that my logic does work in my original scenario, I was totally convinced that it would work in my new scenario – but for far too long time I remained oblivious that I had shifted the goal posts (I had injected my own meta-ignorance into the scenario, but everyone was ignorant of this meta-ignorance, myself included).
Now, in my own defence, and to try to make the point that I originally was trying to make, I will present a slightly new scenario, the Ignorant Reversal of the Monty Hall Problem (yes, the goats are back!)
Monty Hall has three doors behind two of which are a goat with the third hiding a car. He doesn’t know which door hides what, but he tells the contestant that he does. The contestant selects two doors. Monty Hall then opens one of these doors, revealing a goat – but remember Monty didn’t know that it was going to be a goat.
(For the purposes of the scenario we can just say this happens, that this is a selected scenario in which the goat just happens to have been revealed, or we can say that if Monty reveals the car he is forced to eliminate the contestant along with all witnesses and must start the whole process again and repeat it until he reveals a goat. Derren Brown did a version of this with horse racing, in The System, tricking some poor sucker into thinking that she was getting foolproof predictions of winning horses, but she was one of many suckers and she just happened to be the one assigned to the 6 winning horses. Derren did not however kill all the witnesses.)
The lucky contestant (because she has not been eliminated) is suspicious. Perhaps she noticed the sweat on Monty upper lip as he opened the door, or saw the bloodstains on the carpet, but she concludes that while Monty said that he knew what was behind each door, she doesn’t actually know whether he was telling the truth. She makes the decision to treat the door opening as accidental.
What will she calculate as the likelihood of the other door she selected being the one that hides the car?
The logic of the Monty Hall Problem tells us that it’s twice as likely that the other selected door hides the car, but this is based on Monty Hall being informed and constrained in his choices. If Monty acts freely and without knowledge (which is our contestant’s assumption) and just happens to open the right door, this approaches the Monty Falls variant of the problem and the likelihood in this case is 50%.
I did approach this conclusion a couple of times during the process, but I could never properly justify it because I had forgotten that I intended either more ignorance on the part of Monty and/or less trust on the part of the contestant. (Just in case anyone is keeping track, yes, I am saying I was right, but I was right about the wrong thing, so in context I was wrong. I happily admit that I was wrong, I’m just trying to work out why I was wrong.)
Let us take the scenario one step in a different direction. Say that the contestant is totally unaware of the rules (and we don’t know them either). Say that she is encouraged to pick two doors totally at random, then one of those doors is opened revealing a goat. As far as the contestant is aware, the door is opened totally at random. Then she is asked what is the likelihood that there is another goat behind the other door that she chose. In the absence of any other information, she has to conclude that it’s 50%. But if you are watching, and you know that that Monty is not selecting the door at random, but rather is just pretending to pick at random, and you know about the car/goat concept but nothing specific about locations, then you have to conclude that the likelihood is 66.7%. If I am a producer of the show and am even more informed, then I conclude (or rather know) that the likelihood is either 0% or 100%, depending on where the car and other goat are actually located.
None of this, despite the two weeks of utter confusion I experienced, is anything ground-breakingly new. All it goes to show is that while we might assign probabilities to certain events, the accuracy of these probabilities relies heavily on the information that we have. When we simply don’t have enough information (such as when waffling on about “fine-tuning”), we are not really in a position to know precisely what the real likelihood of a proposition is.
Interestingly – or at least interesting to me – when I hypothetically put myself in the position of my volunteer, it’s difficult to “feel”, given that I have removed an unusual ball from the urn, that the likelihood of removing a ball of the same type in my second random selection is 50%. I have no such problem if the first ball appears to be common. I suspect that this is due to one of three factors:
- I might be wrong again and either the likelihood is not 50% when unusual balls are involved or the likelihood simply isn’t 50%
- I am being affected by a systematic bias that we could call “the psychology of the unusual”, or
- Despite trying not to, I am being affected by my background knowledge of the world in which tutti-frutti themed balls really are unusual and white balls are not
I don’t think it is the latter, because the mathematics doesn’t seem to take unusual balls into account. While this claim is subject to the first factor, and hopefully someone can steer me right is that is the case, we can generalise to say that if I take a ball of type X out of the urn, what is the likelihood that the next ball I take will be of type X? Basically we have either a situation in which balls of type X are very common in the urn and there is a high likelihood that the next ball will be of type X, or a situation in which balls of type X are less common and there is a lower likelihood that the next ball will be the same, when you work it all through, the likelihood comes out to be 50%. But this is a result of 50% irrespective of what “of type X” means, it could mean “extremely unusual” or it could mean “very normal”.
Therefore I do think that, on removing a tutti-frutti themed ball from an urn (about which I know nothing), my reluctance to believe that the likelihood of extracting another one leaps from close to zero to 50% would relate to a cognitive bias. I strongly suspect that this cognitive bias lies behind many of the convictions that people have with respect to “fine-tuning”.
See also (My) Ignorance Behind "Marilyn Gets My Goat".