This came up frequently in my time as a mathematical educator. Far too many "word problems"(*) are written in ways that require some words and phrases in the problem to be interpreted in their everyday sense, and others to be strictly interpreted mathematically even when this directly contradicts their usual meanings. Learning which are in each category often turns out to be equivalent to "guessing the password", often without even the benefit of instructional material or consistency between problems.
In my experience, problems in probability or statistics are by far the worst of this type.
(*) in the pedagogical sense, not the one that means testing identity of semigroup elements.
This is an incredibly insightful comment. You've perfectly described why so many people find word problems, especially in probability, so frustrating. The "guessing the password" analogy is spot on.
It's fascinating to see how that same challenge—interpreting ambiguous information to figure out probabilities—applies in completely different fields, like strategic gaming.
For instance, 'Big Brother: The Game' is essentially one massive, real-time probability problem. Players are constantly trying to calculate the odds of eviction, the chances of an alliance holding true, and interpret intentionally misleading "data" from other players. It's a surprisingly complex system.
I was so absorbed by this aspect that I actually built a solver tool to help track these variables and calculate the odds for different scenarios. It's called BB Last Solver: https://bblastsolver.com/
It's just funny how a tool for a game ends up tackling the same core issues of interpretation and probability you mentioned in math education.
An excellent post. (I make a related point in this old comment about the famous “three mathematicians walk into a bar” joke.)
EDIT: Also, it’s worth noting that another way to put the point mentioned in this paragraph—
Now, let’s assume that this really happens and you are trying to figure out the meaning of your friend’s words. The first thing is that you would be really surprised: this is not how humans normally convey information.
—is that, in order to interpret the speaker’s words in the way that is required in order to get the “right” answer to the puzzle, one must ignore the Gricean maxims (i.e., assume that the speaker is not following the “cooperative principle”). But doing so in ordinary human communication is, of course, irrational…
Nassim Taleb called this The Ludic Fallacy
"the misuse of games to model real-life situations". Taleb explains the fallacy as "basing studies of chance on the narrow world of games and dice"
He talks about it more in the context of people treating probability like it was about "games and dice", but it easily applies to examples like yours too. And it's indeed something to be careful of. Maybe it should be a tag..
Interesting post, and illustrative of a more broad-reaching issue (I believe exists) with academia in the past century. Too much of our understanding is based on experiments conducted in a lab, so we underestimate the complexity and connectiveness of the outside world. We generalize results from sanitized vials and beakers into the dirty, chaotic world and then wonder why we get so many things wrong.
Cheers
What further confounds the problem is that even if a smart researcher observes a quirky human behavior that doesn't align with homo economus and doesn't in any way say it's irrational many online news writers and people that cite research articles will start citing the researcher's paper and explaining how it means humans are irrational.
Separately from my other comment—this seems as good a time as any to note that much of behavioral economics may be nonsense to begin with, for the simple reason that some of its most important (alleged!) findings are the result of scientific misconduct (read: fraud). (See also.) And one need hardly mention the replication crisis, which has hit this field hard…
A very well known puzzle in probabilities says:
Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
There is much discussion about this puzzle because depending on how you find out that one of the kids is a boy, the probability changes. I understand this, but this is not what I wanted to discuss. If you want to know more about this, you can read this Wikipedia page.
What I wanted to discuss is that for me to give the answer that I am expected, I need to be aware that the person posing the question is expecting me to write a tree with all the possible combinations etc. The tacit assumption about the problem is that it is a puzzle, i.e a problem of a small world with very clear rules. There are many puzzles such as this one that are used to illustrate things such as: humans behave irrationally, humans are very bad at judging probabilities, humans lack a basic understanding of the most fundamentals concepts in economics, etc. I feel that many of these puzzles rush too quickly into those conclusions and we should always treat them with a pinch of salt.
A small world is an artificial environment created with well-defined rules and probabilities. Usually, small worlds are inhabited by agents that are uniform (meaning that all of them behave the same under the same circumstances), have perfect knowledge, lack any feelings and are perfect reasoners; if the problem has a solution (in their small world environment) they will reach the solution.
We don't live in a small world. We don't live in a world where most problems can be assigned a clear probability such as the ones you can compute when flipping a coin.
In the last two hundred years, a large part of economic thinking, now considered to be the orthodoxy, has used simplified models of the world to understand the complex system of economics. In these models, every agent of the system has perfect information about the system, takes decisions in a way that maximizes a well-defined utility function and every agent is equivalent to every other agent.
In the 60s, the American thinker Herbert Simon conceived the idea of Bounded rationality, in which humans are not perfect thinkers but often take sub-optimal decisions, as a response to the Homo economics model of humans. A few years later, Kahneman and Tversky started studying how human choices depart systematically from rational choice, founding the field of Behavioral economics. Many of the results in this field are extremely interesting and relevant, but there is one particular thing I feel a bit uncomfortable with: the definition of rationality is based precisely on the Homo economicus model, an agent that lives in a world that does not exist.
For instance, there is an experiment where one person is given 100 dollars, and then it has to split the money with a second person, giving him whatever quantity he wants. The only rule is that the second person can decide to reject the money; if this happens, no one gets anything. So what does classic economics say about this? That you give just one cent to the other person (or the minimum divisible amount). The other person will accept it because as a rational entity, he is better off with that cent than without it.
What happens in the real world if you make the experiment with real people?
Surprise surprise, people offered a cent reject the money. And people tend to split the money more or less 50/50, though there are important departures from this. Does this show that humans behave irrationally? Only if you define rational as only caring about maximizing a utility function called Money. So one of the problems of Behavioural economics is that the null hypothesis, the rational agent, is defined in a very narrow way. Human beings are more rational than they are credited sometimes, especially if you compare them with the environments they used to face when they evolved. It should not be so surprising that humans give weird answers in weird scenarios that exist only in a simulated unreal world.
Imagine that you are presented this problem in this situation:
You bump into a friend after 20 years without seeing each other. He says: I have two kids, and at least one of them is a boy.
Now, let's assume that this really happens and you are trying to figure out the meaning of your friend's words. The first thing is that you would be really surprised: this is not how humans normally convey information. A sane person would say something like I have one boy and a girl, or I have two girls. But if he said that, with those words, for starters, I would assign a pretty large probability to the other kid being transexual and not really sure about their gender (why on Earth would he be using those words otherwise?). This is not a sign of me being an imperfect reasoner, is me being able to integrate more information about the word than only the one in the message, namely, the rules of normal communication between humans.
When people answer 50% to the question, this is very often the way they have to say: I have no clue. The only way we can provide the (normally) expected answer of 1/3 is by assuming that every boy/girl is a coin flip and that the speaker is purely stating a fact about probabilities.
I am not advocating for not using probabilities, ignoring mathematical models or thinking that humans are in fact, perfect reasoners (not at all). If this post had to be summarized in a couple of sentences it would be:
When using puzzles to illustrate a point, don't forget most puzzles take place in a small world. Don't expect either that the people listening to the puzzle are aware of all the rules and assumption surrounding that small world and don't consider them irrational if they fail to behave according to the predictions of how an agent in a small world should behave.
The world is a large world: the agents in the large world are not uniform, do have feelings and their reasonings are adapted to the large world they inhabit (or better said, to the large world where they evolved and not so adapted to the strange new world they created). The large world is full of uncertainties that are not easily measurable. The large world is not a small world.