Annoying Puzzles (and Cognitive Reflection Problems) - Numberphile
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- čas přidán 30. 01. 2021
- This video features Tim Harford.
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Liked the video as always so decided to do my part and subscribed to Brilliant through the link, looks promising!
I immediately jumped to: Is this a measurement over unit time or number of cars or miles driven over the route and since we don't know, there isn't enough information...
I guess I'm really in the vast minority...
~65-125
I'm a traffic engineer, I literally do the first example for a living 😆. A major (non-fatal) accident is weighed about the same as 50 property-damage-only accidents, depending on jurisdiction. So the answer is 50016.
+ for experts
That sounds like it depends _heavily_ on the jurisdictions definition of "major" and "minor", because if you accept one more fatality for just 50 fenderbenders less, you're an awefull traffic engineer...
Thanks! I was wondering why he kept saying the number should be much higher than 1016 - now I know. :)
That's not much different than the price of a dent repair against a new car. Is that than the same for people?
@@QemeH If you want to get technical, there are 5 standard levels of severity; K/A/B/C/PDO. A fatality (K) is usually considered equivalent to 500-1000 PDO [Property Damage Only] accidents. Major injuries (A) are ones that require immediate hospitalization.
I just thought “2000 major and 16 minor..... looks like bad data”
...or a very bad road plan!
Yeah, kind of my thoughts as well. Regardless of the layout, minor accidents have to be more common.
@@windywednesday4166 Very bad road plan indeed!
Any accident there would be like falling from a 10 story building. Almost all people will have major injuries. A few might get lucky with minor injuries, but it's not likely.
So, maybe these roads are built on the narrow edge of a high cliff?
@@z01t4n
It may depend on the margin of error. Let's say I am climbing a cliff with no safety. Is it more likely that I fall and die or that I fall and twist my ankle?
@@HermanVonPetri the intersections have no stop signs no yield signs and you can drive either direction in both lanes...
I appreciate Tim's admission of fault at the end. Obviously a smarter man than I am, but still humble and trying to teach 👍
Thanks, Noah. This is the comment I was looking for. You did not disappoint. 🙌
I agree. I have learned that my feelings are fickle. When they align with my knowledge, great! But when they don’t, (which is most of the time) I found that I need to ignore my feelings and go with my knowledge/reason. If only it were easy to do.
Tim's great about that. He had a podcast episode about how he underestimated the Coronavirus epidemic while interviewing an expert about how we were, at the time, probably underestimating the Coronavirus epidemic.
I thought it was a show of humility, too, until I realized I was only "feeling" that. His other podcast about the Coronavirus also shows this. It had seemed more like lapses in pessimism, where there is an optimistic hopeful event, only for him to be swayed back into despair. At least, that is what an analysis of it looked like to me.
You’re just being humble. You are way smarter than him, I’m sure.
The road puzzle reminds me of this lateral thinking puzzle:
An army in World War One instituted a policy that made it mandatory for troops to wear helmets while on duty at all times. As a result of this, the number soldiers treated for head injuries skyrocketed. Why?
.
.
.
.
.
.
.
.
.
Because the soldiers who would have been killed by head wounds instead were only injured.
.
When I saw the road puzzle, I thought "At least a thousand" (still a sort of wrong answer) because knowing the helmet puzzle primed me to imagine the Major Accidents were somehow going to 'convert' to Minor Accidents.
So consequently, the number of fatalities decreased in proportional numbers to the number of head injuries... Right?
Or did you simply assume that?
Perhaps wearing a helmet made the soldiers more prone to making mistakes. Maybe they felt safer wearing a helmet and thus exercised less caution?
I don't remember the story's details but i've heard it before. And as logical as it may sound that wearing those helmets could reduce fatalities for head wounds, there's no direct correlation between the two as other factors come into play.
Or to put it another way, what if kindergarteners had to wear soldiers helmets at all times? Would head injuries increase and fatalities decrease?
Or would simply head injuries decrease and other types of injuries increase?
Or, just a crazy thought... Perhaps fatalities would increase?
There's a story about aeroplane engineers during the war that would analyse the planes that returned home from bombing runs. They noticed bullets only appearing in certain areas and quickly decided to add extra armour to those places. Someone worked out that instead, the reason they didn't see bullet holes in those places is because that would cause the plane to crash, the ones that returned home were hit in non-critical locations, so they were trying to protect the wrong parts of the planes
@@RealCadde I wasn't referencing a real world anecdote or incident, but a lateral thinking puzzle. Perhaps the puzzle was based on a real set of events, perhaps not. It's not relevant.
The point is, although lateral thinking puzzles are designed to encourage counter-intuitive solutions, this particular one had instead primed me with a different sort of intuition - one equally unsuited to the road problem in the video.
@@Pembolog According to the relevant Wikipedia article on Survivorship Bias, that "someone" was Abraham Wald, a statistician at Columbia University. From the linked citations there you can read Wald's publication, which goes into great detail analyzing the damage an airplane might receive, the likelihood of any given hit being survivable, and some differences in ammunition.
The "rejoinders" by the US Military are behind a paywall; presumably this is where they talk about the proposal to add armor to the wrong places.
The wiki article also references a story like the lateral thinking puzzle...without citation or proof that it is not apocryphal.
I think it's more a matter of "what does equivalent mean in this strange hypothetical situation"
Agreed. Personally, I took "equivalent" at first to mean "though they're differently safe, the context is equivalent; eg. it's the same road before and after upgrades", which lead to an answer of "about 1000", following the logic that the second road, *being safer,* has 1000 fewer major accidents because they're instead occurring as minor accidents, and that the original minor accidents aren't happening in the first place. It wasn't until the "many thousands" answer was brought up that it even occurred that "equivalent" was intended to mean "equally bad".
I think they accept any reasonable definition, as in a major incident is worse than a minor incident, and as long as your assumptions follows from that, the answer is valid
Yeah, I think if you anchor it by giving an indication they're both processing say 100,000 cars and you're not just evaluating those numbers in a vacuum but as _per capita_ rates there might be a way to mitigate the incorrect assumptions.
Yes. My first reaction on hearing the question was what does he mean 'equivalent'. I cosidered if he meant equivalent total but decided that would be an odd question given how stacked the 'major' incidents were, and that there would be no objective answer anyway.
I dont think anyone that interpreted the question how it was intended would answer 8, so I dont get what is interesting about it.
@@coviantlynch6913 why do people answer 32 though? that makes no sense whatsoever to me.
I think the bigger question is how big is that lake that it has 2^48 lily pads covering it?!?
The bacterial colony ones would be better in this aspect
If each lily pad has an area of 100 square cm, then the lake is larger than Argentina.
All of his supposed "puzzle" make no sense whatsoever. Kinda like "What have I got in my pocket?" ;-)
@@harriehausenman8623 What have I got in my pocket based on the exterior shape
@@mrjaquavis6444 *naughty* boi :-)
Dept. for transport: “Road B is clearly better. I think we can assume that.”
Tim: “MAKE THEM EQUIVALENT”
The whole point of the traffic problem is that it's badly worded. It's a demonstration of how your brain will work to resolve that ambiguity for you by rewriting the question rather than sending you back for more information.
Yeah, ”make them equivalent” can easily be understood as ”make the proportions between major and minor incidents equivalent”, in which 8 is the correct answer. The missing information is not only how many minor incidents make a major, but the definition of ”equivalence”.
@@voliol8070 no, the definition of equivalence results from the question you're looking at. you want to find out *total damage*, which means a total sum, not the ratios of accidents.
The missing information is how the two types of accidents should be weighed against each other.
People look at the problem 2000/16 = 1000/8 when in fact you need to use 2000 + 16 = 1000 + ?
And if you assign weights 2000x + 16y = 1000x + ?y
@@wZem The question, as stated mathematically, didn't clearly specify that it's the same amount of traffic. I have a general problem with questions like this that state important parameters informally. It okay to give a general description of the problem for context, but when you give the formal description it must include that kind of thing. Since "equivalence" is poorly defined and purposefully misleading, either solution can be validly adopted. The problem is with the phrasing of "make the situations the same" without further constraint.
@@darkwingscooter9637 The described scenario of the traffic engineers meant to me that they are looking at two possible layouts for one stretch of road. So to me that meant the same amount of traffic. But yea I guess if people misunderstand the scenario to mean different roads with different traffic volume, it is understandable to follow down a different logic.
I don't think it is the word 'equivalent' that is confusing people, though, but rather that there are two types of accidents.
If you imagined that the traffic engineers don't distinguish between types of accidents and option A simply had 2000 and option B 1000 accidents and the question was again to make them equivalent, everybody would automatically understand that option B is missing 1000 accidents.
So it is more the way the whole problem is structured that is leading people on the wrong path of thinking 2000 and 1000 are already equivalent for some other reason and therefore the equivalent of 16 must be 8.
@@wZem Yes, the whole thing would have been cleared up by just saying "for a given volume of traffic". The Monty Hall problem is similar in that it relies on ambiguous phrasing to set you on the wrong path.
In french we say « tourner sa langue 7 fois dans sa bouche » turn your tongue 7 times in your mouth before talking (or retweeting)
Yes, I turned it 7 times before replying.
And that would be easy for non-native speakers of French, who would anyway need an extensive tongue warm-up before being able to say anything at all. Lesson: always think in a foreign language.
For anyone still wondering, the correct answer is definitely 69420. Sounds about right, doesn't it?
i think this is just a case of the human brain going "OOGA BOOGA PATTERN" and just slotting 8 in there
I was so confident in my answer as well!
I feel like your comment is correct so i'll just OOGA BOOGA like it
machine learning in a nutshell. GPU GO OOGA BOOGA!
I do blame the question somewhat, he said: make these equivalent. He didn't say: make B so the total hospital costs would be about the same.
You instinctively think road B is half as dangerous so the minor incidents will be halved as well. But to make them equivalent regarding something like hospital costs, you'd have to have many minor incidents to make up for the 1000 major incidents and 1016 is the minimum of that range.
haha 8 goes brrrr
My first thought on that first puzzle is it's terribly worded. What does "equivalent" mean? What's the relative bandwidth of cars on plans A and B? Is B 1000 major accidents because it's got a ton of traffic stops and it's only allowing 1/2 as many cars through per hour?
right. When I heard "equivalent" I took that to mean "equivalent layout/structure of highway" so would have the same proportion of major accidents to minor. So in theory, if B had the exact same structure as highway A but a fraction of the amount of drivers, then 8 would be the correct answer. I was not at all thinking "equivalent amount of property damage caused by accidents"
Yeah I thought I missed the question. I think it should have been phrased like "Which number here would make us indifferent between A and B?" maybe with an "all else being equal" thrown in.
It's really a different question: How can a cognitive researcher phrase a question so the meaning of the question is unambiguous to the people who are being asked the question? I hope that was entirely clear.
Agree - "equivalent' is open to interpretation, so could be taken as 'to have the same ratio of major:minor incidents', in which case 8 is correct. Indeed, any other interpretation of 'equivalent' cannot lead to an answer without other information, so 8 is the only defendable solution. The real problem is, in real life, no one would ever ask this question. If we were really trying to compare options, we would look at actual costs of damage, considering whether lives were lost or injuries, and how the roads affected flow of traffic (rate) which means the raw number of incidents is rather meaningless.
Exactly. All too often I see traffic engineers deem throughput irrelevant, only the accident count matters. Then they make an “improvement” costing hundreds of thousands of dollars and traffic jams get much worse and happen more hours of the day. “But traffic is smoother”, they say. A crazy definition of “smooth”, I say. I see a lot of this where I live. New left turn arrows that I have to wait for even when there is no oncoming traffic. Signals forbidding right turns so traffic backs up. Signals where only one direction at a time goes from the cross street, so the main thoroughfare is halted a long, long time.
Me not falling for the examples: *signature look of superiority*
Something to bear in mind though, is that you were slightly primed for the questions. You clicked on a video with math puzzle in the title, meaning 1, you probably enjoy puzzles, and 2, you know this is a puzzle and a puzzle's answer is rarely obvious. Given this, viewers of this video are more likely to get the correct answer than random people asked this question.
??
My answer was "What do you mean 'equivalent'? How am I supposed to know, you haven't given me a cost function!"
Yep. This video is very weird.
@@ga35am Think you missed the point
@@Stettafire I don't think so. I think you missed my point.
??
@@Triantalex !!
Most students are trained to believe all the information to answer a question is provided in the description. However, this problem is lacking a key piece of information - weighting factors for major and minor accident types. Students need to be taught to perceive when information is missing - that’s as important these days as actually solving the math.
I agree 100%. This demonstrates the failure of the education system. Students think that they are smart because they know a lot but they are just machines trained to apply solution patterns from the constrained textbook questions.
@@piotrarturklos Both of you aren't that smart.
true, academics are build to make people find THE solution to all the questions asked, but rarely that answer could be "Not enough information" or "impossible to answer".
I guess they got scared everyone would answer those when they couldn't figure it out , like every time XD.
But it's a major component for critical thinking.
@@jawstrock2215 what do you base that sweeping statement on? I was taught in engineering school to assess the available information. If there's not enough you make assumptions and state those explicitly
@@jawstrock2215 Some academics? Probably, yeah. All academics? Certainly not.
Ha jokes on you, when I was a minor my parents told me I was an “accident”
a major or a minor one?
@@ruben307 When they told him he was still a minor
How may pregnancies occur on road A, and how many on road B?
Is your dad by any chance Major B. Road
??
I feel like I want to share this video, but I'm noticing my own emotions and counting to 3
When you want to ask these sorts of problems, you need to be _very_ careful with the wording. It's very easy for a small slip of the tongue to mean you're not asking the fancy trick question you meant to be asking. And then the trick answer ends up feeling completely unearned.
So, for the road layout question, you introduce the problem with a story about trying to find which layout has a _lower_ accident rate, so you've already primed the conversation that the two layouts are _not_ the same. And then you give your numbers, and ask what the missing number is. And only then, _after_ people have started to make guesses as to what the missing number is, do you say you're trying to make them "equivalent"... in some vague and handwavy fashion. It's only much later, _after_ you've already started mocking people for giving the "wrong" answer, that you finally properly explain that the puzzle is to pick a number so that they cause an equivalent _amount of damage_.
The answers of "8" aren't wrong, it's the question that's wrong.
My hope is that he actually explained the puzzle much better on the day, and some critical piece just got lost in the edit...
I'd say if you started to answer the question before the questioner finished presenting the question you're still doing it wrong.
Being very careful about the wording of a question is a point I often bring up when there are discussions about The Monty Hall Problem. Many times, a person uses words in the question which are equivalent to Monty Fall; the difference in wording can be as small as "Monty does ..." versus "Monty must ...".
The first puzzle/problem seems intentionally misleading, "equivalent" can be interpreted in many ways
that was the point of the video
@@some_rat_ not really, they explained it as you sort of forgetting the question and just shoehorning an answer despite it actually being obvious that it's the wrong answer. For this explanation to why someone would answer 8 to be true, that person must've understood the meaning of 'equivalent' "correctly" so that was actually not the point of the video at all, if I'm not misunderstanding something
the question is which road is more dangerous. so it is clear that "equivalent" refers to level of danger and certainly not to the ratio of major and minor accidents, because that has no relevance to the question.
@@wZem He diidnt even mentionned time in the question 2000 major accidents and 16 minor accident per 2 month is exactly equivalent to 1000 major accidents and 8 minor in 1 month. Erland is perfectly right to say that the question is insufficient and misleading on purpose. But this all make this video, meta with itself. I think this video is aimed toward people who can't make this distinction.
@@hurktang oh come off it. Don’t try and claim you assumed one scheme was being measured over half the time of the other.
Just admit you instinctively answered “8” and were fooled by the problem.
Nah, most annoying puzzles are of type
1=3, 2=5, 3=?
1?
Definitely.
That's why I always come up with a stupidly large number and justify my answer with the correct interpolating polynomial.
In your case it would CLEARLY be 3 = 28374687, because it is *obvious* that
f(x) = 14187340x^2 - 42562018x + 28374681
[edit: fixed a minus sign]
and therefore we have
f(1) = 3
f(2) = 5
f(3) = 28374687
@@ThisIsACZcamsAccountAsd think you mean minus in the middle term, but I agree with your point.
3====>
@@leadnitrate2194 whoops! Yes, you are right, I made a mistake when copying
Another example from that video of the brain leaping to the wrong conclusion: assuming that Yale students must be smart ;-)
they are though
Alice has five dollars, Bob has three dollars. Alice gives one dollar to Bob, what's the mass of the Sun?
Purple, because aliens dont wear hats.
Alice has five dollars, Bob has three dollars. Alice gives one dollar to Bob, how much does @jp have?
1.989 × 10^30 kg
I'm glad to say my first thought was "several million, probably?" and then "i need more info about how we're measuring the value of accidents"
Hypothesis: on average, Numberphile viewers are more careful with reasoning about this sort of thing than Yale students are.
Exactly. But then I thought: „can’t be right, because there should be some hidden information I’m overlooking because in a puzzle it needs to be a nice number and maybe I misunderstand the term equivalent“. The puzzle is annoying because it doesn’t stick to the rules of puzzles.
Figured it could not possibly be 8
And also figured many people would pick 8.
Statistics don't lie, but people do. There's a whole book on it - "How to Lie with Statistics" 70 years old, still fresh, everyone should read it.
"Thinking Fast and Slow" also appropriate.
You may want to read Tim's book. He has a section that discusses the author of that book, and the work he did for the Tobacco industry, lying with statistics.
When I saw the question I immediately thought "but how many minor accidents are equivalent to one major accident?"
Exactly. The only annoying thing about the question is how ambiguous and incomplete the information is.
As did I, then from the information given I formed the belief that 16 minor accidents = 1 major accident and came up with the answer least 16016
@@morgandavies9685 why not 16000 ?
@@quentind1924 because of the 16 in the other minor box to make them equivalent
I’ve wondered if one could design a course to teach something like (for lack of a better name) “mental humility” and questions like this would be part of the coursework. Let students encounter then confront thinking errors like this one, but in a setting where they can learn why they make the thinking error and can hopefully learn to recognize the situations that require thinking slow not fast.
And have it as standard curriculum! Like the books The Demon-Haunted World, and The Skeptic's Guide to the Universe. My primary school curriculum included a lot on propaganda (tricks in commercials to get your money) but not really any true critical thinking.
I don't think it can be learned. You'd have to carefully analyze everything, and would get nothing done. We have to rely on scientists or journalists... or the comment section... to do their job.
@@Milan_Openfeint I disagree. Scientists don't unlock a latent superpower of discernment, it's a skill that takes pedagogy and practice. Scientists themselves are often credulous outside their particular discipline, and sometimes within. And a population that has discernment makes a better citizenry, makes better economic decisions, is better at recognizing expertise.
@@fowlerj111 You need some basic knowledge to be able to tell a lie. Can you tell if more people die from falls or drownings? One is 10x the other (in the USA). You can't check everything, and the fact checkers.
People should stop reading Washington Post after showing such a fraudulent graph, but you can' really expect every reader, or 50%, to study it thoroughly.
A well formed program and curriculum would teach this throughout all coursework in all subjects. At my University, these types of problems were extremely common in our coursework, and we were trained to recognize them and avoid the common traps. I think that's the ideal situation.
People think "how many minor accidents will road B have" instead of "make them equivalent"
The annoying thing about the question is the definition of the word equivalent.
Yeah, my first reaction was asking: "What do you mean equivalent? Equivalent in numbers or injuries?"
Well, equivalent isn’t that bad to define. There are a few ways to do it, but I’m sure they’re all equivale - oh, shoot
Joke's on you, I only care about the probability of being in a minor incident given that I'm in an incident. ;)
But nobody answered 16
@@Milan_Openfeint But I would rather not be in an incident at all.
That's some dark humour you got there.
The video: smart people fell for it
Me who didn't understand the question in the first place: I don't have such weaknesses
That is exactly how I feel too.
Haha!!! Got em!
Yes. The question was very poorly formulated! What does "equivalent" mean in that case? Even asking that question is meaningless because he does not have an answer himself.
@@banknote501 That's incorrect. equivalent has a very specific meaning, particularly in mathematics.
@@banknote501 It makes sense. Don't assume that because you didn't understand something, that it was bad or done wrong. It isn't meaningless. It is one of those questions where the spread of answers is the actual answer. The question is meant to invoke different ways of thinking and to test who thinks what ways. Not look for an absolute answer.
I think that we often subconciously glorify the abstraction numbers and graphs offer us. We like to believe that our scientific advancement helps us explain the messy chaos that is the world we live in, so much that we forget that science mostly just describes what we witness, and that our understanding of the world must come from fitting these descriptions into the proper context. Placing the importance instead just on the raw, abstract information makes the world seem more logical, less chaotic, so when we should be thinking "Alright, what does that data actually mean given the context?", we instead think "Sure, the math works out, I can fit that into my worldview".
Or maybe that's just me.
You just described the entire challenge of working with statistics in a scientific/research setting.
The issue with the road problem is the question itself. Classic measure and evaluation issue.
The correct answer is: “what do you mean by equivalent?”
Yeah, the first answer is really easy if you ask the question properly.
It's truly annoying to be asked a question with ill-defined terms like "equivalent". Equivalent proportion of accidents? Equivalent total health related costs? Equivalent economy impact? Equivalent death toll? Because each one of these gets a different answer.
He defined the inital problem well enough. You are ingoring the goal in favor of the math. They were comparing road designs to build right? They obviously want to know which is safest overall for the users.
So we are obviously needing to compare major to minor accidents, and determin the best road- not some wierd pointless ratio of the 2.. I would have said- 'im not sure how many minor accidents add up to 1 major 1- but lets go with 10.. so 10,016?'
@@elevown Thats still ill defined, as "safest" is a vague term. thats why we establish metrics. With a metric we agree on we can determine things. One of the major problems i see in mordern media, is that a vast majority of information is presented in those vague terms without the metrics used to dertermine the conclusion. Quite often i consider news as fake news as the metrics used are kind of screwed.
Saying "there is not enough information to solve the problem" is a valid answer if that is indeed the case.
@@elevown he definitely didn't. Equivalent in terms of what? That's really important to the question. How can you suppose 10 minor accidents amount to one major accident? That's simply wrong.
My answer to the question would be "those two designs can never be equivalent because one has a half of major accidents than the other".
@@lucaslugao It is a fun introduction to "how do we sort this out". Certainly there is not nearly enough information to make even an educated guess because you need information to be educated. 10 major accidents with Ford Focus cars may amount to a smaller insurance claim than one minor accident with a Lamborghini. However, a Ford Focus can seat 5 people, a Lamborghini 2 which means in terms of people carried one Ford Focus would balance against two and a half Lamborghini's making the Ford Focus a far more human body expensive accident potential. But then you look to Germany, and think "ok so what if we are talking the Autobahn" in which case, every major accident can conceivably have one or more fatality, vs other roads where though major only a few have fatalities. So taking a step back how about comparing the impact on the non-involved; if you look at one of my employees as an example from this week, a minor accident resulted in his being late to work 1.5 hours due the traffic slowdown. Had he been on the major highway the next day where a fatality occurred he would have missed the entire 4 hour shift as the highway was shut down for 10 hours. And that does not even get into "is road A east-west, and road B north-south" or "is road A a mountain road, and road B a plains road", or "is road A a winter access road and road B closed access in winter". Yup, fun question this...
The road layout problem asked about balancing the NUMBER of accidents (0:10). It does not ask about balancing the danger levels of the 2 layouts, thus 1016 is the correct answer.
Well no wonder there are so many accidents, everybody's driving on the wrong side of the road
as long as everybody on the road agrees to drive on the "wrong" side of the road, they're fine. It'll be only a problem when some drivers use the "wrong" side, and some others the right side.
I don't think that "getting a difficult question and substituting an easier question" is quite right here. It's more that the question isn't hard - it's ambiguous. Are the two road setups equivalent in what way? I remember even back in grade school doing "word problems" in math class and observing that often there can be more than one way to interpret a question . . . but (at least at the time) only one "correct" way the teacher expects. The REAL thing to think about is to recognize where ambiguity exists, and to not "assume in" the details. "8" *IS* a correct answer if the question is "equivalent proportion of major to minor accidents".
I think that's what they were getting at. Can't speak to the original study where he got the data, but he clearly asked about the roads' safety being equal. The "substituting with an easier question" might better be explained as "substituting with a more familiar question" instead. I can't tell you how many times I have seen that chart like that and needed to solve a proportion: probability, geometry, ratios...
This argument has some merit to it, but I think it misses the real issue. I suspect that, had you asked the students who answered '8' to explain what the intent of the question was, most of them would have probably gotten it right. The question is technically ambiguous, but in truth it's not really that hard to grasp what's being asked for. Rather, people don't take their time to consider what it actually is that they're being asked to do, and go for the most intuitively obvious answer.
@@BL3446 The roads' safety cannot be equal, if one road has 1000 major accidents and the other has 2000 major accidents. Thousands of bruises do not add up to death.
The problem with the traffic question is that it's ill-posed, since "equivalent" has not been defined. The correct answer is the question: what do you mean by "equivalent"?
It doesn't matter if the question was ill-posed : the point of the survey wasn't to determine the correct answer but to see how students thought.
i dont think theres a definition for "equivalent" where one has twice the rate of the other.
@@badmanjones179 When he said "what goes here to make these two equivalents?" I had assumed "proportionally equivalent",
like 2000 / 16 = 1000 / x
And the answer to that equivalence is x = 8
@@teo.reinehr i just think thats a bad assumption to make considering those are rates, and youre turning "make these two equivalent" into "make the proportion between their individual rates be equivalent" which is a bigger leap than the simple graph would lead us to realize. proportion never had anything to do with the question and yet we insert it because it makes the question easier, even though "equivalent" has a totally reasonable application to the problem on its own. thats my take at least
@@badmanjones179 That could very well be what goes inside my mind (or ours as humans). My brain wanting a solution and extrapolating the question's meaning.
It is always nice to try and challenge our cognitive bias.
In any case, I still can see the answer being 8, as well as any other number. And understand there is not enough information (I was actually thinking about the lack of it after the initial guess).
Edit:
Just to complement a little bit, for the answer to be 8 you have to assume that the rate between each layout is equal.
And for the answer to be at least 1016 you have to assume that the total number of accidents in layout B is *equal or greater* than in layout A.
Also in the question it states that Layout A is *different* to Layout B, but apparently my brain missed it to have 8 be right hahaha
This reminds me of the French/Chinese boat captain school question. There was a question that was originally part of a French study, but later ended up on a school test for Chinese children (or so the claim goes). It seemed like a perfectly standard maths word problem about the captain of a ship transporting livestock. It went into detail about how many of each species were on the boat, and how much they all weighed, and the capacity of the boat, etc. Then it got to the actual question: "How old is the captain?" Most students took the numbers they were given and symbolically manipulated them in various ways, adding or multiplying or dividing different combinations to arrive at a numerical answer. Even some adults who were shown the problem went on to try and find an answer by using external information, averages, and assumptions (such as "well, with this many sheep and this many cows, the ship must be at least this big, and in China to captain a boat that size requires a certain license which you have to be at least X years old to obtain, so the captain is X years old").
The correct answer is, of course, "there isn't enough information to accurately answer the question". But the majority of people who see it assume there must be a concrete answer and find one where it doesn't exist. Funnily enough, this kind of thing overlaps with artificial intelligence as well. One of the latest and most exciting AI's is called GPT-3 (publicly revealed in May 2020), and while it's effectively a fancy autocomplete, it makes connections so well that it's the closest thing humanity has ever invented to generalized intelligence (not quite there yet, but close). One person reviewing it pointed out a major flaw: while it's GREAT at answering common-sense, factual, and mathematical questions (among many others), if you give it nonsensical or impossible questions, it spits out a nonsensical answer. "A human would recognize that the question is nonsense and say that," they argued, "but this AI isn't quite smart enough to do so." Except... the boat problem, and other problems like the ones mentioned in this video, prove that humans are NOT, in general, smart enough to call out when a question can't be answered.
What's really cool is that someone else who read that article was curious about what would happen if he told GPT-3 that it should respond to nonsense questions with the answer "yo, be real" -- and it worked. It continued answering reasonable questions well, but whenever it got a nonsense question, it didn't spit out any nonsense answers: it just said, as requested, "yo, be real". And I think that's possibly one of the most humanoid behaviors it's ever shown, because like Tim said in this video, if you tell a person they're allowed to say "I don't know/there's no answer", or if you give them a moment to reflect, they'll make the right decision, but if you don't, they'll just spit out an answer that's as nonsensical as the question.
Finished Tim’s book a few weeks ago and absolutely loved it! I’d strongly recommend!
I chose 8 because, it wasn't immediately clear to me we should assume the two highways had the same amount of traffic. If we take major accidents to be a sort of control group, then yeah 8 makes sense. But that was me adding my own assumption that highway A had twice as much traffic as highway B.
That is a flawed assumption, since it was stated from the start that there was two possible road layouts. You are not looking at two roads, but two ways to build a road. Therefore we can assume the same amount of traffic, because it will be the same commuters that would use it.
it's a pretty decent assumption. I also had it. That is why it is important to spend some time discussing assumptions and context when discussing complex topics
@@Arthur0000100 Not to be rude, but what makes it a decent assumption just because you also had it? I agree with you that it is important to discuss assumptions, but I don't think this one is decent, because if you listened closely you could hear that it wasn't the case.
A decent assumption would be any assumption on the conversion factor between major and minor accidents, because that is not something that is be given.
@UCdVxrv8Q8ulRwhd4wJ6hQCg But if it is POSSIBLE road layouts of one road (he even says the road layout, singulat). It is weird to assume different volumes of traffic, since it would be expected that the same need is present in both cases since it is essentially the same road. So again I don't find it to be decent, because you have either intentionally or by accident ignored one of information given to you.
If you don't agree, then please tell me why should we assume different amounts of traffic on essentially the same road, where only the layout differs?
Its not 2 roads but 1. it will have the same traffic obviously.
I feel like part of the problem with the first puzzle is the term "equivalent". What does it mean for two road schemes to be equivalent? Later in the video, the term equivalent is expanded upon further with the whole discussion involving the "exchange" rate between major and minor accidents. However, I feel this definition of equivalence was not communicated properly when the question was asked. If a question contains ambiguous terms, I feel that many people (myself included) will fill in whatever definition they want to try to make sense of the ambiguous question. Therefore, I don't think it would be correct to say that 8, 32 or 1016 is wrong since the definition of equivalent road schemes was not properly communicated in the phrasing of the question.
Yeah exactly, the question he tried to ask was "how many minor accidents does B have to have to make the two layouts have an 'equivalent' amount of damage so the Department would not have a numerical basis to decide between them", but what he asked was just "what goes here to make these two schemes equivalent", which i interpreted as "they're trying to figure out which layout to use, and that decision is helped by measuring the accidents; how many minor accidents would B have if the two schemes are equivalently built". And since I've been given minimal information, I have to go off the fact B has fewer major accidents, so clearly it's safer and there should be fewer minor accidents too
Tim Harford is always such a treat
About that lily pad problem: It is quite interesting to note what size the population covers after 48 days actually.
Assuming the exponential doubling holds on every day and that a lily pad has the size of about 1 dm^2, the lily pads cover an area of about 2,814,749 square kilometers. That area is larger than the Mediterranean Sea!
And 2000 or even just 1000 major accidents on a single road segment both sounds like an awful lot to me...
No time frame was given. 1000 every year is a lot. 1000 over the lifetime of the road is not that much.
"Cognitive Reflection Problems" or "How do I make it sound like the person being asked is the problem and not the way the question is phrased"
my thought, exactly. But still, the idea that our brain substitue hard question for easier one is interresting. It happend to me and to other countless time. Although, I still feel that a well phrased question would solve that issue.
I agree, even more so about the problem that is the deliberately misleading phrasings of sensationalist news articles and badly-formatted statistics.
And exactly here is also where I see a problem of the road scenario: We normally don't expect to be tricked, especially if the asking party has no reason to trick us. Surely there is no gain for a road designer to deliberately ask us misleading questions. It's not only asking a misleading question, it's asking a misleading question in a misleading scenario.
Tim Harford! I finished reading his book a few weeks ago.
How’d you like it? Thinking of ordering it.
@@ramsesabreu1870 I liked it a lot. It's got lots of examples of real incidents. It's not very technical. Has kind of a "what to look for to understand data" approach. I'd say it has significantly changed the way I see statistics and what questions I ask. Written in a conversational style. I enjoyed it a lot. That being said, everybody is different and there is a slight chance it might not fit your preferences, but that is the case with every book, you only know after you've read it and this one is definitely worth a read.
Tim Harford is awesome!
For the first example, the way I thought about it was that, if you assign some "badness" weight `a` to a major accident, and some "badness" weight `b` to a minor accident, then the problem is "Solve for x in 2000a + 16b = 1000a + x*b", so x = 1000a/b + 16. To "solve" the problem, you have to invent the ratio of major badness to minor badness, then the answer is that a thousand-fold, plus sixteen.
Not surprised on the Washington Post story; that should not be a go to for you; NYT might be OK, WSJ as well, but unfortunately if it is politically related, that editorial staff is not trustworthy
Yeah, pretty much all news is very politically biased and literally worships a certain party. And it also gets worse when super-partisan people own fact-checker sites, because they can hold a "claim" on the truth. Would you really trust fact-checkers? Seriously, think about it. They make money to tell the masses if certain claims are true or not. How do you know that the fact-checker isn't biased?
equivalently dangerous vs equivalent proportions of minor to major
Answering questions with incomplete factors is my normal type of work: I'm an engineer!
Same here, for some reason, I haven't been tricked by those and I felt smug, and now that you point this out, I realise that whenever I get some new requirements from the management, my first move is to figure out what information did they forget to provide. Now I don't feel smug anymore and I have bit more compassion for the management.
I’m convinced Tim’s point on us all needing to be aware of our own biases and susceptibilities to misinformation is like the #1 thing this world needs right now
Low-key wouldn't mind you starting an economics channel just as an excuse to get more Tim Harford.
For once I felt "smart" watching a Numberphile video!
My first reaction to the puzzle was "well I don't know... How many minor accidents do I need to compensate for 1 major accident?". Had I had been forced to say an answer it definitely would have been in the 10s of thousands.
Yeay, I shall now rule the universe.
Love these videos :)
Love Tim Harford and More or Less. One of the best shows on the radio.
I like that the title primed me to think carefully about the road accident puzzle.
If the title was "Ratios on the Road" or something I would definitely have got it wrong!
I wish the puzzle was explained more clearly before giving away the answer. "Equivalent" in what way?
I thought that was the whole point, though.
Exactly. It's perfectly reasonable to interpret the question as "What's the number that makes the roads have equivalent accident distributions" rather than "What's the number that makes the roads be equivalently dangerous?"
@@romanski5811 if that was the point he could have said "I'm not going to define equivalent, think about that for yourself. Now pause the video if you want to think about it." But they gave the answer while I was still waiting for the actual puzzle.
Exactly my thought. didn't want to comment on it but my first reaction was equivalent on what ? cause they are proportional if the answer is 8 ...
Well of course equivalent in a sens of a "cost function', and they have to have the same value in order for them to be indistinguishable
Am I the only one who the whole time was thinking "1000 x some number + 16, at least 1016"? And then when he said it looked like it should be 8 I started questioning my sanity?
Nope, same here. I was thinking “something big...” and when he said 8, I thought “Wait what was the question? Did I mis-hear it?”
Please, can you explain why is 1000 x some number + 16? Because I don't really get how that makes sense. I don't know if I correctly understood the question in the first place.
Edit: Nevermind, I get it know. I didn't understand it at first because it doesn't really makes sense to have so many major accidents and only 16 minor accidents.
Thank you for tying in mathematics with psychology. Fascinating.
The problem with the question is "equivalent" being an extremely ambiguous term.
"As costly for society" would be a better expression. Unless "equivalent" was referring to sum of delays to others in the traffic grid...
Yeah it seems like it could mean, road A has a 2000/16 ratio between major and minor, how many minor does road B need to make its ratio the same. The answer to that question is definitely 8. Quite ambiguous.
You say this as if they don't talk about it in the video but they do, around the halfway mark
I disagree. The framing of the question in the context of the city trying to decide which road is safer should make it plenty obvious what's being asked.
@@Nomen_Latinum In this context they would presumably know the actual numbers though, from their modelling? It doesn't make sense that only the number of major accidents would be known, and if it was unknown you still wouldn't frame the comparison this way - you'd collect data on the actual number of minor accidents and then compare them, in which case I'd expect people to arrive immediately at the 'what is the exchange rate' conclusion.
It's a trick question, or a poorly framed one. Though I do find it interesting that some people naturally arrived at the lower bound as their answer. When confronted with questions like this where there's clearly an expected answer but not an objectively correct one I usually just throw up my hands and look confused and rant to my friends later about how poor the question was.
The comment is right that "equivalent" is not defined. You can observe that every answer has its own invariant (proportion, product, sum, weighted sum) and define an equivalence relation mathematically (even though it may make no sense in real world). I think what the video wants to say is that people tend to be overconfident about their answer before slowly checking it with logic, purely a psychological topic.
What's interesting to me is I felt that the right metric would be getting the inverted ratio of major to minor accidents, like: 2000/16 = x/1000, solve for x. That gives you 125,000, which I immediately felt must be wrong because it's so big. But it turns out my intuition when playing with the numbers was on the right track.
I didn't even notice this was Tim Harford until the end of the video. He's such a great guy!
I feel the problem is the semantics of the word "equivalent" that is not well defined for the question. What if equivalent means "has the same ratio of major vs minor incidents" and not "has the same (ill defined) `cost` " or "Has the same number of total incidents"? I hate those kind of voluntary poorly defined problems. It is hard only because it is not well defined, just to trick you. The trick is only in the way the question is asked, if asked correctly everyone would have answered a reasonable answer, im pretty sure.
Exactly. This example is a semantics problem, not a maths problem.
If the question is asked in the abstract then this is certainly a fair point. However I think with the given context of a government trying to decide between two different road layouts there is enough information for someone to take a second and realise what "equivalent" means in this situation. The fact that what is being asked is not straightforward is a large part of the reason that people tend to internally replace the question with something simpler, particularly when what is actually being asked doesn't have a definitive answer.
@@Willd2p2 But then the numbers presented don't make sense. As soon as I saw that there were 100 times more major accidents than minor ones on road A (that never happens in real life) I knew that it was not a real world problem and we are not supposed to find reasonable answers.
@@banknote501 Sure, but if anything that just further emphasises the point being made in the video. Despite the problem being strange, people still default to interpreting it in a simple way despite that interpretation not really making sense in the given context, without taking the time to consider what is actually being asked.
This sort of reaction would only be amplified in a situation that is more "intuitive", which really just means the reaction is even more easily prejudiced by pre--existing bias.
@@Willd2p2 If you would ask me that question in a real world scenario, I would request all the information I need and then seek an answer. If you ask me that in a mathematical (or psychological) quiz setting, I give the most obvious answer-
here's how the question was posed: We've got these things. "they're trying to figure out which one will cause the fewest accidents". blah blah filler data that we assume is useful but is not. "fill in the gap to make A and B equivalent".
The reason the brain gets tricked isn't because the question is complicated and we want an easy answer. It's because your setup and final questions are unrelated. Making B match A doesn't tell you which layout will cause less accidents, it lets you solve a pointless math problem with no implication on the real problem we were supposed to be solving.
This reminds me of an old "joke"/riddle I heard several times as a kid that was about a bus, the person asking the riddle would give out a bunch of information about the number of passengers and what not, and at the end they'd ask "so what color are the eyes of the driver?". What's happening is misdirection/trolling, you assume you're gonna get a reasonable question with some logic but instead get the rug swept from under you
The data provided is actively worked against you. The 2000 and 1000 seem correlated when in the problem they are supposed to be completely independent major accident tallies for two different road layouts. The 16 is overflowing with factors of 2 begging to be divided or multiplied by 2.
I think the way mathematics is taught in school, especially with word problems, conditions people to expect certain intentional types of math solutions.
Yep, I agree. The issue is that you're given the premises of the problem, then asked a (in my opinion) very vague question that can be interpreted differently.
"fill in the gap to make A and B equivalent", equivalent in what way?
Equivalent in the amount of total damage done? In that case, sure, B minor accidents would be a stupidly high number. But the whole story before hand leads you to want to make A and B equivalent in ratio of major to minor accidents, hence 8 being the most common answer.
When you find that delightful content and it jiggers your soul, stop and treat with extreme suspicion
I actually thought immediately: How much worse is a major accident than a minor?
For the car, total loss against a scrape and dent has a monetary ratio, order of 20..50.
But for humans?
Well, for the first question, since an exchange rate isn't given, the correct answer is minor accidents = 16 + 1000x, where x is the exchange rate.
Ah ha! Yes! Best answer.
fwiw a Traffic engineer responded and said x=50 so the correct answer is 50016
Why 16+something? Why not just something?
Why would it even have more major than minor incidents in the first place?
Maybe given the circumstances if something went wrong, they were only likely to go horribly wrong. With only very few who were lucky.
Maybe like an icy road that has a blind crest, with a turn, leading onto a narrow bridge, over a rushing waterway.
Maybe the design of the road layout is so bad that it turns accidents which would have been minor into major ones.
It's a fair point. Any road that has more major accidents than minor ones is a badly designed road.
But, it could be that the data is referring only to accidents that get reported. Obviously almost every major accident will get reported because otherwise the ambulance won't come. But minor accidents might not get reported so the data shows a much lower number than the true reality
That could be a thing.. imagine some super sketchy cutting machine with big flying blades that has no safey guards or features at all.. that regularly jams and needs the operator to clear.. you'd have WAY more major accidents than minor ones! Sure the example is contrived- but my point being, depending on what we are talking about- major accidents can easily be common than minor ones.
Maybe because one road involves a jump?
This reminds me of Simpson’s paradox in probability. Depending on how you “average” the differences the answer may change. There is also a very simple geometric intuition behind, concerning areas of parallelograms (no traffic of psychology here though).
This is what I subscribed for
"So what goes here to make these equivelent?"
Well... If a minor accident is half as bad as a major one then a ton right? like 4000
"It feels like 8"
*pardon?*
The reason people say 8 in response to the traffic puzzle is "when something looks like a nail, you grab your hammer." The puzzle superficially looks like the typical proportionality problems we are usually presented, so we assume it is one. Only when we think about what is actually being asked do we realize it is not.
Right. It looks obvious but if you think twice you realize that is a grade school level problem... and if you have any awareness you realize you are at Yale University... poor little freshmen.
In a world full of ambiguity, making emotional decision can be time saving.
You're forgetting to factor for the time wasted during recovering from the wrong decision, and after you do that, you still have to spend the same amount of time to research the correct decision properly.
I am a traffic engineer and i do these estimates for a living - if example A has 2000 major incidents and only 5% minot - then its just a bad project, bad example - it cannot be that way at any crossroad. It is just BAAD! and A should be banned from even thinking about it. The ratio of major/minor is statistically prevailed and given that the first example was calculated correctly, some bad speed limits, then the second should just be ~10 so 8 for example - if it was the same persona that created plan A and B :)
this is because they drive in the wrong directions :)
it's a British CZcams channel ;-)
@@tobyk.4911 yeah, I know :)
@@tobyk.4911 oh, he's drunk, how would he know where they are going.
English translation:
It's because they drive on the wrong side of the road. _boom tish_
_No need to thank me you're welcome._
@@BlackDuke235 I don't drink, you understood me well :)
The question is annoying because it's so poorly worded.
David Pemberton exactly
Not at all. It was actually very simple. The question was: "Which one will cause the fewest accidents?"
You just missed the point of the video that your answer should have been: "I do not know. Because I have not enough information about these two traffic situations."
Think of it like this: "Do I have enough information about plan A and B, that I can fill in this table with numbers?"
Answer: "No"
Bat one is defintely but you have to deatch from it and its simple
I worked as a Claims Adjuster for one of the biggest auto insurance companies in the US for years.
My first reaction was "well, about 10,000 because minor accidents nowadays usually cost around $5k, and most people in the US carry the minimum BI and PD required by law, which tends to be around $50k total, and that very often gets maxed out in a serious accident. So x10 it is."
This was an incredibly interesting video
"Stuff on the Internet should be taken with a grain of salt, " according to a video on the Internet.
I believe everything on the internet except this video.
File it under "takes one to know one"
Nothing exceptional about the internet in that regard
The point of the video is not that "stuff on the internet should be taken with a grain of salt." It's that we should use proper judgement to determine what is and what isn't worth a grain of salt.
That's what Isaac Newton said
My answer would have been a lot, it depends on the weight of major and minor accidents
I think that kind of thinking is what also ensures an answer of 32, which is what I chose, because of how they would be weighted. Choosing 1016 would just be if each death was weighted as the same.
@@nickfifteen if I were left to guess, minor accidents (small injuries and material damage) would be weighted less than major accidents (serious injuries or death)
So I would hope to see a number higher than 1016
"The pub" is becoming one of those "I'm this old" memes...
An example of this happened where I live. Years ago the state that I live in (Illinois) decided to raise some tax, I can't remember exactly which, but it was going from 3% to 5%. They tried to sell it by telling everyone that it was only a 2% increase. People weren't fooled and realized that in fact it was a 66% increase.
“Gordon Pennycook”
I’m gonna take a wild guess and assume this guy is British....
So the solution to the first math puzzle is, that it is no math puzzle🤔
It's doesn't even qualify as a puzzle. More like pick-a-random-OEIS-sequence :-)
@@harriehausenman8623 Indeed. It's literally nonsense. The alternate layout was chosen to reduce accidents, so what does it even mean to make them "equivalent"? 🤨 It smacks of those "viral math problems" on social-media. 😒
@@user-vn7ce5ig1z It's actually kind of ironic. Although combined with his questionable motivation, Ii's more dangerous than ironic.
@@user-vn7ce5ig1z He's in serious need for some category theory.
@@user-vn7ce5ig1z It wasn't chosen. You are evaluating alternatives. And you can definitely rule some values out, even though you can't give a precise answer without more information.
When he asked "how to make these equivalent" I thought he was still setting up the question and wanted the same proportion. I wish it were more clearly worded so I'd know I was wrong for sure, but I will make it obvious what "equivalent" means when I show this to people.
I couldn't understand what was happening with the cupboard in the background, until I realised the bottom right hand side was the the cushion of the sofa. It looks like a crazy Escher picture :D
32? I hoped for a question to 42. :(
@Butterfly Sword Wrong base mate
When the question was presented I asked, "How many minor accidents equal a major?" And then my immediate next thought was, "Well, if it's on Numberphile there must be a clever, statistical answer regardless. 1,016?"
I also assumed that majors must convert to minors. And 1016 sounds like one of those funky numberphile numbers that get to be a video title.
Big props to the notion of checking the data even if the presentation agrees with your biases.
One of the issues I have with this problem is the way it was laid it out on the paper. At first glance it looked like a ratio problem, so most likely, anyone who passed grade school math would cross multiply and divide. Had this problem been laid out differently on the paper (for example, all on one line) I think the results would be much different.
I am just here to say that it is a play in words. The word equivalent is hardly precise enough to fault someone for saying 8. Equivalent in what?
The premise literally just isn't explained properly
If the most common answers to the first problem was nonsense like 8 or 32, that makes me suspect the question was just ill formed and it wasn't clear what was actually being asked.
My initial objection to the first puzzle was the notion of a road layout causing 'accidents'. That's not how it works.
loved the video :)
Tim makes a mistake here that is quite common: students at Yale are not smarter than the average person, they're RICHER than the average person.
wrong, they are indeed also smarter in general
@@tronalddump2267 [citation needed]
How do you measure smarts?
How do you even define "smarter"?
I guarantee you, people who know more about psychology than you have tried to answers these questions, and come up empty-handed.
@@DontMockMySmock know more, think better
@@tronalddump2267 What do you mean by "think better"? How are you intending to measure that?
@@DontMockMySmock IQ tests?
Well there's your problem, everyone's driving on the wrong side of the road
I am not claiming to be smart. But for someone whose second language is english the qustion wasn't as snappy and intuitive, because the brain slapped meaning onto words, so the hard question couldn't become an easier one.
Oh, and my answer if i was being asked on the go would've been "32? 200? i don't know"
Just because there's less major accidents... doesn't mean there's less minor ones too. Those major ones go somewhere. Not sure why everyone thinks 8, I could see why I guess, but I instantly thought it would be a large number. I'm a roadgeek so I guess I'm used to seeing these numbers.
It's like when a 4-way intersection with traffic lights is compared to alternatives (like a roundabout). The alternatives may have more conflict points, but they're usually less deadly as well, so it's still seen as an improvement. It's better to have two cars brush together, than to t-bone or head-on collide. If one fatality can be avoided, then it's worth the extra minor stuff.