Why is the answer to "the most annoying math puzzle" not 8?

I’ve been watching a lot of the excellent maths YouTube channel numberphile, but this one has me confused:

In this video Tim Hartford poses this problem as an example of a cognitive reflection problem (as in, there is an answer that feels obvious right, but is wrong)

Traffic junctions are judged based on how how many major accidents happen at them (which cause serious injury or death) and how many minor accidents (that cause only minor injuries or none at all).

There is a junction A where 2000 major accidents occur and 16 minor accidents. Another junction B has 1000 major accidents occur. How many minor accidents should occur at B for it to be the equivalent of A?.

The trick here is the number of major and minor accidents is the opposite of what you’d expect (many more major than minor). As result:

while there is no ‘right’ answer the apparently obvious answer, of 8, cannot possibly be correct, it must much higher than that, to have the “equivalent” number of minor accidents it must have many thousands.

But that’s not true is it? Clearly junction A has some very strange dangerous attribute that means 99.2% of accidents result in serious injuries or death (maybe it’s perched above a sheer cliff face with no barrier, or 99.2% of the vehicles are carrying incredibly dangerous unstable explosives). Whatever it is reasonable to assume that another junction like junction B that is “equivalent” but has half as many major accidents, has that same attribute, but just 50% few accidents of all kinds happen there. In which case 8 minor accidents the right answer.

That seems at least as “right” as the answer that there must be 1000s of minor accidents to be equivalent.

What am I missing here? Or is there something missing from the description that should have ruled out 8 as an answer?

He means: for B to be as bad as A — to the point where there’d be no reason to choose one over the other, because the harm would be equivalent.

This question seems akin to asking: “all my customers pay in a combination of dollars and pennies; customer A gave me a thousand dollars and 8 nickels, while customer B gave me 500 dollars- how many nickels does B have to give me to be equivalent to A?”

It takes a lot of minor accidents to be as bad as one major accident, just as it takes a lot of nickels to be equivalent to one dollar.

It’s not an annoying math puzzle… it’s unclear language being used as a gotcha.

I think this is impossible to answer without really subjective nitpickery about how minor the minor accidents are. Even 20 mild fender-scrapings aren’t equivalent to one fatal crash.

But this isn’t a math problem any longer. It becomes semantic mumbo-jumbo about minor vs. major.

I like Tim Harford, I strongly recommend those who are interested in this stuff to check out “more or less” which is his BBC radio show on the numbers in the news.

For this problem, this is the way I am thinking of it.

We are asked to consider two potential junction layouts on the same road.
That is important because it immediately means we can assume that traffic volume and types are likely to be the same.
They want to know which layout to build and the number of accidents caused by each is an important criteria in making that choice.
So each type is modelled and we are given the numbers that you described. Type A is 2000/16 major/minor. Type B is 1000/? major/minor.

So which to build?

Imagine they were only interested in major accidents, the answer is simple. You build type B.

But the question being asked is really “how many minor accidents would have to happen on a type B junction for it to become the worse option?”

When thought of like that it is clear that “8” is not the answer.

Tim mentions that it is not a clear answer because we don’t have the full info. His camerman mentions “worth” and that’s a critical point because if we know that the average financial cost of a major accident is 100k and minor is 5k then that might give us a way to calculate an answer.
Or perhaps a major closes the road for 6 hours on average and a minor for 30 minutes.
Or perhaps it is a combination of the two or some other factor completely.

It is more an insight into how people reason in sub-optimal ways in these circumstances.

Or rather how people may be misled by a leading formulation of the question that tends to deliver wrong answers.

Whatever may be going on in that question, math is not part of it.

The question as asked is unanswerable.

First of all, it is so poorly phrased that it’s not clear what the question is. People who say 8 are not understanding the question. What number goes in the gap? If you halve the number of major accident, you will also halve the number of minor accidents, so 8.

If you phrase the question more clearly as “how many minor accidents are equal to 1000 major accidents?” it’s still impossible to answer. How many oranges equal 1000 apples?

You would need a clear definition of what is ‘major’ and what is ‘minor’, and even then the answer would be subjective. Which is worse, 1 person loses an eye, or 100 people get a broken arm? What if it were 200 broken arms? How many broken arms equals one eye?

And that’s why the answer isn’t 8.

yes, as is said in the video this about behavioural psychology, not maths.

For you specifically you are missing the fact that this is a question about the modelling of potential junction layouts on the same road.

Starting the video, and watching the initial presentation where he tells us that it “feels like 8” is the right answer, and it’s even more clear that this is a case of:

  1. Using very imprecise language (bordering on incorrect) to state the objective.

  2. Deciding that one possible interpretation of the language is the correct one.

  3. Declaring the problem really tricky because you might not interpret the objective as intended.

Again, not a puzzling problem. Just a failure to ask the question clearly.

The very next example in the video is “if a bat and ball cost $1.10, and the bat costs $1 more than the ball, what do they each cost? Many people incorrectly jump to the answer: ‘$1 and $0.10’”.

There is a question where the math and objective is clear and unambiguous, yet people apparently jump to the wrong solution. The “make these ratios equivalent” question in the OP is broken to begin with.

To calculate the answer using maths, one would have to apportion a value to major and minor accidents. This value would be arrived at by calculating all the costs associated with each.

That calculation would be complicated by considerations such as: Whose costs - the council who have to repair street damage; Insurance companies, who cover repairs to vehicles; the people involved who bear any uninsured costs in addition to the almost unquantifiable emotional cost.

One could allocate some arbitrary figure like ‘10’ for a major and ‘1’ for a minor, but that would only be guesswork.

One could simply respond “greater than 16” and be correct. Presumably, one could probably even answer “greater than 1016” and be even more correct, since it seems fair to assume a major accident should be considered at the very least not less than a minor accident.

But then it’s not really a cognitive reflection problem. When you actually explain what you are trying to answer then 8 is no longer the obvious answer it’s clearly (16+1000.x) where x is the number of minor accidents you consider equivalent to a major accident. But that’s not how the question was defined.

The “a bat and ball cost $1.10, and the bat costs a dollar more than the ball. how much does the bat and ball cost individually?” Question is much better.

Even backing out of the math and trying to apply the problem to the real world comparison it tries to address, there seems to be no reason why “how many minor accidents would it take on road B to reach an equivalent amount of harm as road A” is a more correct interpretation than “road A causes this many major and minor accidents. In comparison, road B causes X major accidents. How many minor accidents are likely to occur, given what you know about the relationship between major and minor accidents on road A?”

In fact, I’d say the first interpretation is a wild stretch compared to the second, without more context given at the outset.

I agree with this in certain circumstances, with the caveat that the exact number of minor accidents would not be very meaningful.

If I were a traffic engineer with a fixed budget, I would probably rather focus my money on improving the sites that have very few minor accidents but lots of major accidents, since there are probably easy mitigation strategies to them, like signage, guard rails, or traffic lights, rather than the ones that also have lots of minor accidents, since the easiest and cheapest mitigations are probably already in place.

“>1016” is what I immediately said. But even folks who are freaking data scientists were arguing with me.

But then I’ve long since realized that successful Data Scientists often operate in these four steps:

  • Identify who is the most important person in the process
  • Find out what conclusion they want to reach
  • Figure out a mathy sounding justification for reaching that conclusion
  • Profit/Promotion

The video, and the question as posed, is a really good example of real-world situations that come up time and time again. The point of the video is not to purposefully trick anyone but to shine a light on how we fool ourselves, jump to conclusions, fail to clarify the question etc.

You mention using imprecise language. Well sort of, but all the info you need to know it is not 8, is actually there in the set up. However, that (and the impossibility of giving an accurate answer) can be obscured by the big glaring “obvious” answer that we all tend to grab onto as soon as it presents itself.

Sure it could be made clearer and that is where self-awareness of this potential trap-door is important because it would lead you to writing down, in clear language that makes sense to you, what is it that you are being asked? what are the facts that you know? what information is missing? What are the limitations of your analysis?

But, as I say, given a lovely big obvious answer that comports with our initial preconceptions we tend to throw all that rigour out of the window.

No. There is a fixed, known and objective ratio of nickels to dollars. That question is answerable.

The ratio of minor to major accidents is arbitrary and subjective. It can’t be answered “correctly”.