The Probabilistic Paradox
Let’s start simple.
A family has two children. One of them is male. What’s the likelihood that the other child is also male?
I found it while watching shorts. It’s one of those social media math problems, the kind of thing titled “99% get it wrong”, followed by clickbate images.
How stupid; there’s an equal chance of having males and females, unless it was supposed to be a biology problem. I scroll through the comments:
Answer: C. 1/3.
Wait, what? I felt stumped. Then I realized how my mind fell into a trap, a trap of lazy thinking, of shallow understanding, of false intuition.
I wasn’t the only one stumped. The problem, being far from stupid, was on the Chinese National Examination. Something brutal and many times more difficult than the infamous GaoKao.
Out of all the extended essays, logic problems, math puzzles and challenging decision-making questions, that problem had one of the lowest accuracy rates.
It’s difficult to imagine how I got it wrong, and even harder to think how the doctorates and policymakers got it wrong.
They studied years of higher mathematics, only to stumble and fall on such a humble-looking problem.
But it surely reveals something: people are bad at probability. It is counterintuitive and goes against the human nature.
Perhaps they don’t have a deep understanding of the fundamental ideas, or maybe it’s just because their intuition strongly contradicts with the logic, concealing the hidden truths.
Perhaps they will never realize their crucial mistake if the problem was given in a real-world context.
Why Not 1/2?
Those who have studied probability may already know the answer.
I’ll make an attempt so that you intuitively feel the answer isn’t 1/2.
What happened?
Something that you should keep in mind is: the genders of both children have been decided before you came, and will always stay the same. The other child isn’t being randomly “generated” as you decide its gender.
To apply probability correctly, we have to back go a time where nothing has been decided: when both children are being born. If we apply probability to this scenario, the collection of possible events is all possible pairs of genders, (boy-boy, boy-girl, girl-boy, girl-girl).
With this consideration in mind, there are a total of 4 possible events.
Out of these 4 possible events, the conditions eliminate one scenario (girl-girl), leaving us with three scenarios. Only one of these scenarios work: boy-boy.
A Problem of Math and Biology
Credits to Greg Herlihy for pointing this out. If we changed the wording of the problem slightly so it becomes
A family gave birth to a pair of twins. One of them is male. What’s the likelihood that the other twin is also male?
then the solution changes drastically. The key idea here is biological: there are two types of twins, either identical or fraternal; the latter is more common, with fraternal twins occurring twice as often as identical twins, since fraternal twins can be (boy-boy, boy-girl, girl-boy, girl-girl) while identical twins can only be (boy-boy, girl-girl).
There are 6 possible events, namely (boy-boy, girl-girl, boy-boy, girl-boy, boy-girl, girl-girl). The events occurring in both identical and fraternal twins are repeated.
Now, our given condition eliminates two of the scenarios, namely the scenarios where both twins are girls. This leaves us with a list of four possible events: (boy-boy, boy-boy, girl-boy, boy-girl).
Out of these four events, the two boy-boy scenarios satisfy our requirements. Hence, the desired probability is 2/4 = 1/2.
Further Comments, and the famous Bayes’ rule
The famous Bayes’ rule tells us to remember our priors. It is helpful for such problems, but the key reason for failing this particular problem is choosing an incorrect set of possibilities. Even if Bayes’ rule was applied in this context, the answer would still be the incorrect 1/2, because the set of probabilities we chose was (boy, girl).
However, a key benefit of Bayesian thinking is to make us consciously aware of the set of possibilities. We are advised to start with the broadest set of possibilities, then eliminate them as conditions are added. This conscious thinking helps to prevent incorrect intuition, allowing us to have a correct start.
For this particular problem, the key idea is that what happens already happened. That sounds awkward, but it basically means that the possibilities have been decided beforehand. This causes the gender of the other child to be dependent on the given information.
You may ask, why are boy-girl and girl-boy two separate scenarios? This is because order matters. It’s twice as likely to have a mixed gender combination than a same gender combination.
On the other hand, here’s another problem with a completely different answer:
A mother gives birth to a boy. What’s the probability that, assuming she’ll have another child, they’ll be a girl?
Now, this is asking about something about to happen in the future, and is independent of what has already happened. The probability in this case, is the intuitive 1/2.
