Conceptual Statistics: Framing the Question
The following post is intended to illustrate the importance of framing questions in the context of sample space and Bayesian statistics, and how delicate, and sneaky, our ordinary language can be in regard to unconsciously steering our intuitions.
The problem goes as follows:
Your cousin is visiting town, and you know she has recently had two children. She tells you ‘at least one of them is a girl.’ What is the probability that both children are girls?
This sounds like a simple enough problem, and our intuition at a quick glance is to say ‘50 percent,’ since if one of them is a girl, that leaves only two possibilities for the other one, right?
Well, the point of this post is to dive into the hidden complexity behind such a a supposedly simple problem. Ostensibly, the correct answer to this problem is 1/3, because our sample space should include all paired possibilities:
These are the four initial possibilities which, upon learning that ‘at least one of them is a girl’, we can reduce to three:
Now, the original question ‘what is the probability that both children are girls’ seems visually obvious: only one outcome contains two girls in the sample space, out of three total possible outcomes, hence the correct answer: 1/3.
In this post, I’d like to point out the source of confusion a lot of people feel, or felt, when they first encountered the ostensibly-correct answer of 1/3, especially when they so strongly felt that the answer was 1/2. After all, why is the above graphic any more compelling than the original:
What’s the problem with the above graphic? Is there a problem? Why shouldn’t we privilege this graphic as much as the other one? The follow up ‘aha’ question to clarify this is a nearly identical problem:
Your cousin is visiting town, and you know she has recently had two children. She tells you ‘the older one is a girl.’ What is the probability that both children are girls?
The correct answer in this case is in fact 50%, because the above graphic correctly represents the sample space, because the word ‘older’ is a predicate which, in English, implicitly sets an order on its associated subject, ‘one.’
And here is where I really want to emphasize the trickiness of human language in regards to statistics, because no matter how powerful your tools are, they are only as powerful as the individual ability of a statistician (or a data scientist) to frame a situation coherently. Most high level disagreements, in my experience (particularly in the history of philosophy and the philosophy of language) come from deep, unspoken commitments to invisible frames of reference which seem and feel identical.
In the above problem, we can replace ‘older’ with any kind of ordering adjective, for example if an ultrasound were performed and the doctor said ‘the one on the left is a girl,’ this would be a more explicit form of ordering, and indeed the sample space would have two outcomes again: the one on the right can only be a boy or a girl (so, a 50% chance they’re both girls). Similarly, if we flip two coins:
If I reveal one coin, the act of revealing orders the information by locking it into a space and thus removing its associated outcomes (H or T) from the sample space. Thus, the answer to ‘what is the probability that both coins are heads’ will always be 50% as long as one coin is physically revealed, and the answer is only 1/3 when the sample space is constructed in a state of complete ignorance about whether the objects in question are locked in, or part of a collective sample space which is constructed from the perspective of a naive observer. Indeed, the Monty Hall problem bears great similarity to this one, but the fact that two goats and a car are stable entities behind unchanging doors implicitly order the information: it has the same effect as saying ‘the older one’ in our gender example (in addition to some added complexity about the host’s knowledge in the Monty Hall problem, since they must always reveal a goat).
To conclude, and to emphasize a final point by revisiting the original problem, let’s look at it again:
Your cousin is visiting town, and you know she has recently had two children. She tells you ‘at least one of them is a girl.’ What is the probability that both children are girls?
After looking at the above graphics, and carefully thinking about which words implicitly order our information, which words in this problem are the culprit behind our 1/2 vs. 1/3 intuitions?
Your cousin is visiting town, and you know she has recently had two children. She tells you ‘at least one of them is a girl.’ What is the probability that both children are girls?
The answer is at least. This little phrase is used all the time in English, and it normally gives us no problem. But in this problem, which was explicitly constructed to demonstrate the importance of sample space/Bayesian probabilities, it causes a huge problem. When many, and myself, said ‘50%’ in response to this question, it was because the phrase at least has a very quick connotation of ordering about it: nobody in ordinary English conversation uses the phrase ‘at least’ to mean a bizarre set of possibilities without locked objects in a sample space, unless it’s obvious in that context. We are blazingly fast at inferring which unspoken connotations are pertinent to which situations in the language we speak, which is why, in this problem, it was so absurd for many of us to think of a cousin using the phrase ‘at least’ to suggest that one child could be a girl, but where that ‘one’ is a pronoun which could represent either child in the whole sample space, and not a specific child, a child that in some sense was already revealed with a name and a face and ordered, thus leaving a sample space of two outcomes.
This boldface text, or the sense of it, is the quicker-than-lightning inference that anyone makes who says the answer is 50% and, if you were to take the side of ordinary, everyday English-users, I believe 50% to be an answer that more aligns with our intuitions about the connotations of the English language, of which this is one tiny but powerful example.
All of this is to say: such profound complexity can be revealed in a single sentence not because a problem is difficult, but because our language makes it difficult, and as aspiring data scientists, a healthy dose of reflection on those complexities, connotations, grammars, and biases could go a long way.
