i am obsessed with the relationship between meaning and context. Rabbit. Several years ago, at the time when i was first being born into the loveliness of clear observation and thinking, i decided to deeply consider "meaning" as a phenomenon. i broke it down like this: the meaning of words, the meaning of a bird chirping in a forest, the meaning of a fish swimming away from a shark. Did all of those scenarios have "meaning"? What did it mean for something to have a "meaning"? Was there something in common about the different meanings those scenarios had if indeed they had them? Just what was the meaning of meaning? One of the things i discovered within that inquiry is that meaning and context arise together. There is no meaning apart from context; and that fact fascinates me.
You, dear reader, are also obsessed with the relationship between meaning and context. Any of your countless searches for meaning, whether large, or lowercase like the looking up of a word in the dictionary, are at one-and-the-same-time searches for context. Those of you who have forgotten the word "rabbit" after the opening sentence of this article have done so because of a lack of meaning due to an absence of relevant context. Those of you who remembered the word did so because of the seeming irrelevancy between the word "rabbit" and the context of meanings that constitute this article thus far. Those of you who laughed did so because "context rabbit." is fucking hilarious in the context of a heady, boring, or dry opening sentence, or funny in the context of too much information and a reluctance to read another blog post. In that case, i suggest running through the forest as soon as possible... Not like a jogger thinking, "I'm running because it's good for me... By golly, that air sure does smell fresh... Is that cedar.... or pine...." , but like a kid pretending to be a mad warrior-hunter leaping over logs at topspeed barefoot. If you felt anger on the heels of context rabbit you did so in the context of being somewhat of a dick (and should see suggestion above). "Is 'context. Rabbit.' a typo? Is this bastard just trying to be clever? What does it mean?"
Context and meaning are simultaneous orgasms, they come together. That is one of my most fundamental intellectual tools, so when my esteemed steemian friend @youdontsay introduced me to something called "The Monty Hall Problem" in a recent article, i began to look at the problem with that in mind.
The basic set up of the problem/solution is this: You have three doors, behind two of which are goats, behind one of which is a car. Your initial choice has a 33.333% probability of being correct. When the host reveals a goat behind one of the doors you did not pick, he then asks you if you want to change your choice to the other remaining door you did not pick. Counter intuitively, so the solution says, the remaining two doors do not have an equal probability of being the correct door because the door you picked still has a 33.333% probability of being correct. That leaves the second door with a 66.666% probability of being correct since probability can't exceed 100%, and thus you are more likely to get a car if you switch doors. One of the authors of the answer, a woman named Marilyn Mach vos Savant (very cool! @youdontsay. Also, i wish i'd been born 37 years earlier! Seems quite a lovely person.), encouraged teachers to mobilize their students in testing out the solution after receiving a great number of letters telling her she was wrong (what a cool thing to do!). Well, the data from the students proved that her solution was correct... But i still think that second door is the devil, condemning you to anguish eternally in a lake of fire (with the somewhat ameliorating benefit of roasted goat meat). "Why?", i hear you breathlessly ask. Well alright, i'll tell you.
I don't know. Just kidding... I might. First, the context has changed, therefore the meaning has changed. A solitary outcome, whether goat or car, of a choice to change doors through a single round of choosing tells you nothing about statistics since it is not a statistically significant "population" of outcomes. Correct? It is a context different from the context of "statistical significance". Conversely, averaged probabilities of two different door choices arising from a statistically significant population of outcomes through a thousand rounds of choosing tells you nothing about your three→two-door-one-choice-one-round system. It's a different context. You standing in front of two doors having to make a single choice is not in the universe of the three→two-door-one-choice-THOUSAND-ROUND system, it is in the universe of the three→two-door-one-choice-ONE-ROUND system, therefore the meaning of the thousand-round system, i.e. its statistical probabilities, give you no advice about the choice you should make within your system, because ITS probabilities are not YOUR probabilities.
Intuition, unschooled by statistics, tells you that the two remaining doors have an equal chance of being correct; that they are each in the state of a 50% probability. That is TRUE, because your context of the three→two-door-one-choice-one-round system has changed its state. It has evolved into a two-door-one-choice-one-round system, and such a system has 50% probability of any choice being correct. The context of you standing in front of two doors with one choice did not, will not, and CANNOT EVER change its state in such a way that it evolves into a thousand-round system. And since a 50% probability is equivalent to "random chance" even the correct probability tells you nothing of what you should do. Furthermore, the 50% probability of you getting a car, given any door choice, is also FALSE, because you may have a greater or lesser probability of being correct than another individual given unknown criteria specific to you.
Now, please listen carefully, I'm not saying that what was "proven" by thousands of school kids testing probabilities was wrong, i'm saying it was different than what is commonly believed to have been proven. Those kids proved that a group of people through several rounds of choices had a collective average probability of 66.666% of picking the right door if they switched doors in a thousand-round system. They did not prove that the INDIVIDUAL increased their chances of winning to 66.666% by switching doors in a two-door-one-choice-ONE-ROUND system. Therefore Ms. vos Savant was wrong in her advice to an individual, while she was right in her calculation of probabilities for groups, as well as probabilities for individuals through several rounds of choice, but only right in the context of an absence of other criteria.
What i have tried to prove is that it is a logical fallacy to apply statistical probabilities derived from certain parameters to an individual case with different parameters. That it is a conflation of contexts to do so. I'm not sure if i have proven it, but if i did, and if no one has beaten me to it, i'd like to dub it "The Single-Context Fallacy" for posterity, or possibly "The Roasted-Goat Fallacy". What do you think? (12-29-2018 Addendum: It turns out that i may have discovered a case of what is already called "The Ecological Fallacy". Defined as: Incorrectly assuming that an association on the population level reflects an association on an individual level. But my reason is strained to the max here. So anybody out there who might have a leg up, i sure could use a boost...)
Go with your gut. Statistics do not apply to decisions in an individual case because the outcome of a single decision is not probabilistic, it is actual. They are therefore not valid criteria for making a decision. That doesn't mean you should stick to your guns. Change doors if you want to, but guided by a gut uninfluenced by the tantalizing smell of roasted-goatmeat-statistics. Look to your own wits. For, "It is Our WITS that make us men!" (And please pardon the language. That is a quote from "Braveheart", a movie that quite moves me. It is OUR wits that make each of us fully HUMAN! Freeeeedoooooooom!)
Post Script: There is another mystery here. What mechanism causes cars to be placed behind unpicked doors more frequently than picked ones?
Post Post Script: i may have made some mistakes in some of my formulations. If you can show me where and how, i'd be much obliged...
This has been a public service announcement. Paid for by taxation of the very cutting edge of The Million Things' current ability to reason out correctly. Thanks for listening.
I enjoyed reading this and it's certainly well written, but I must admit I disagree with your conclusion.
"Those kids proved that a group of people through several rounds of choices had a collective average probability of 66.666% of picking the right door if they switched doors in a thousand-round system. They did not prove that the INDIVIDUAL increased their chances of winning to 66.666% by switching doors in a two-door-one-choice-ONE-ROUND system."
I guess I just don't see how you're distinguishing between the statistical probability and the in-the-moment probability. I think the confusion might be arising because the problem excludes all circumstances in which the first door opened by the host reveals the car and that we've already lost.
The effect is more obvious with more doors. Imagine its 10 million goat doors and one car door. We choose door #1 and the host opens all but one of the other ten million doors. Obviously, in most cases (99.9999998%) he's going to end up opening a door with the car behind it, revealing that we've lost. The other .0000002% of the time, he opens nothing but goat doors and leaves one door untouched. Your original door and this other door both had a .0000001% of having the car when the game started, but now we've watched ten million doors open and reveal goats, all the while ignoring one random specific door besides your own... History shows us that if we let this scenario play out millions of times, 99,999,999 out of 10,000,000 who choose to switch doors end up being correct, while 99,999,999 out of 10,000,000 who choose to keep their original door end up being wrong-- so it would seem to me wise to go with the 99.9999998% success rate of switching doors, or likewise, the 66% success rate of switching doors in the original formulation of the problem.
Glad you disagree! Feels like the left and right sides of my brain (so to speak) are having a debate, and i can't tell which is which!
i'm pretty sure i follow your logic... Which has an answer to the "mystery" i mentioned: What mechanism causes cars to be placed behind unpicked doors more frequently than picked ones? Your answer, if i understand, would be something like, "The mechanism is that there are more unpicked doors than picked ones. It's simple and straight-forward. Cars are scarce man, there ain't enough to go around. And goats? Well, goats are abundant, but nobody wants a goat do they? It's the system man... Fight the power!" And i have a special place - either in the cockles of my heart, or in the left or right side of my brain, though possibly in my corpus callosum - for that answer... And i did see an explanation of that thousand door variant.
"How do you distinguish between the statistical probability and the in-the-moment probability?" i actually can't tell if the in-the-moment probability is 50%, 66.thedevil%, or indeterminate. What distinguishes them, i think, is that the in-the-moment case, the real case, has no "population" of statistical significance. Neither a population of choices, nor a population of rounds, nor population of people, which are the same thing really. The population in-the-moment is just one. And since it's just one, i think that it is a case of the "Ecological Fallacy", or a "Fallacy of Averages" or "The Statistical Fallacy" if there is such a thing as either of those, to apply probabilities derived from a different population size.
Doesn't some other factor in-the-moment have to determine the probability... Er, is "probability" per se even determinable in a single case with a population of one? Put it this way, if i run the game a dozen times the potential probability that i get a DOZEN correct answers improves if i do NOT follow the rule of always switching doors, since the more times i run the scenario while always switching doors actually destines me to ONLY being correct two-thirds of the time as the round count approaches infinity. Therefore, many of my teachers found me rather annoying back in school.
Now, i might not actually be doing statistics when i reason this way. But are we doing statistics when we say, "Hey Bob, switch doors and your chances improve." if he only has one chance? i'm not sure if all of my reasoning is sound, but i believe my conclusions -1. That there is a fallacy. 2. What the kids actually proved - are roughly correct nevertheless.
But i'm no mathematician! i just don't believe anything; in the sense that if i can't reason something out for myself than what i've got is a hypothesis, not a certainty, and if i can reason it out for myself then i still have a hypothesis, since new data might change things.
i maintain that: If "Bob" gets a goat because he switched doors, we erred in advising him to switch. It's starting to seem like a pedantically trivial point...
(It's occurring to me that an unknown ancestor somewhere in my family tree may have been a troll... The horror! The shame!)
...But i think it has something to do with the creative power and existence of an individual, which any "authority of concepts" can't touch, whether mathematical or otherwise. i think it's because concepts aren't true, they're just useful for some purposes sometimes.
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