Empiricism is the belief that knowledge derives from sensory experience, or, by some definitions, experiment. However, empiricism itself can show that it is not the best tool for a given scenario. This article provides an example of when this happened to me.

The Two-Child Puzzle is a logic problem. The version I first heard goes like this:

I know my co-worker has exactly two kids that each have an independent 50/50 chance of being a boy or a girl. I ask her if at least one of her kids is a boy. She tells me that is correct. The question is: From my perspective, what is the probability that the other one is a boy as well?

The answer, I immediately supposed, was 1/3, while the trick was that most readers would guess 1/2. I posed the question to a friend and asked him what he thought. He thought it was clearly 1/2, and said multiple times throughout our ensuing discussion that he was very confident. I tried a variety of tactics to persuade him, initially using pure logic. However, the complexity of language and my inexperience as a teacher made us run in circles, as he would continue to state something like "but what we know about one of the kids can't change the probability of the other" or "if she brought a boy to work one day, instead of telling you she had a boy, you would think that the other kid had a 1/2 chance of being a boy instead of 1/3".

I attempted to use more empirical tactics. First, I used coins to demonstrate the puzzle. They happened to come up with the "girl" answer the first three times, which is more likely if the probability is 1/3 of being a boy than 1/2 of being a boy, yet the method seemed unreliable and my friend was highly skeptical.

Then, I created a spreadsheet with 1000 rows using random values, showing different averages like "both girls", "both boys", "one or more boys", "kid 1 is a boy", and "kid 2 is a boy". I showed that when all results of "both girls" were factored out, the spreadsheet was repeatedly displaying a 1/3 chance of "both boys". Yet my friend continued to state that one kid being a boy was distinct from kid 1 being a boy, and that therefore the spreadsheet was irrelevant (The spreadsheet was still somewhat useful, as I learned that the chance for kid 1 being a boy was 2/3).

My friend called someone who he described as smart and math-oriented and we explained the puzzle. The math-oriented friend created diagrams to attempt to understand the puzzle better but came down on the side that the probability was 1/2, though he sometimes agreed that the probability of a given kid being a boy was 2/3.

All this time, we had avoided looking at the puzzle's provided solution. We did this now, and it stated clearly that 1/3 was the answer. However, my friend did not believe that this could be true and found other versions of the puzzle that had 1/2 as the correct answer and that stated that either answer was a valid solution since the question was ambiguous.

I would say that these four methods (coins, spreadsheet, math-oriented friend, and official answer) were all, to some degree, empirical methods for determining the truth, as they were all observations of the outside world. However, none of them had swayed my friend much. Given that I later convinced my friend easily by using non-empirical means, was empiricism simply not effective in this case?

I would say: first-level empiricism was not effective, while second-level empiricism was. I had used four ways to test if the answer was correct. However, I had forgotten the second level: to test which method was best at convincing my friend. When I began to watch him more closely, I noticed that empirical evidence held little sway over him while puzzling through with logic had sometimes made him doubt his position. I honed in on a specific area where he seemed the least sure and where I was the most confident. He had continued to state that when the coworker states that at least one of her kids is a boy, this set that kid apart and the two kids were no longer interchangeable. So I rephrased the puzzle:

I know my co-worker has exactly two kids that each have an independent 50/50 chance of being a boy or a girl. I ask her if both of her kids are girls. She tells me that is false. The question is: From my perspective, what is the probability that both kids are boys?

He agreed after very little effort on my part that this was equivalent to the other puzzle. He then also agreed that the matrix [GG GB BG BB] was correct (he had previously stated that the GB cell should be crossed out as well). And he then agreed that the answer to the original puzzle was 1/3. I asked two other people what they thought of this reframing of the question, and one responded positively, the other very positively.

Looking back on the solution later, I can see that I should have used second-level empiricism to recognize that my first-level empiricism was not working. Just as a Utilitarian could knowingly take a Deontology pill that would cause her to change moral systems if the expected Utility would be higher, second-level empiricism can prove that first-level empiricism is not useful in a given situation.

Cover photo by Michal Janek on Unsplash.

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5 comments, sorted by Click to highlight new comments since: Today at 9:33 PM

I'm curious if people have the same reaction of disbelief to an analogous situation: Punnett squares where two heterozygous parents have kids, but being homozygous recessive is fatal very early in pregnancy. What fraction of the resulting offspring in that case will be homozygous dominant? In that case, the fact that an offspring survives at all tells us they are not homozygous recessive, which in the gender example is analogous to ruling out two girls. But the 1:2:1 pattern in biology is extremely robust and was known all the way back to Mendel - green and yellow peas are still how we teach genetics in high school.

I guess what confuses some people is the phrase "the other one" which sounds like denoting a specific (in terms of SSN) child while it's not at all clear what that could even mean in case of two boys. I think step one when being confused is to keep rephrasing the puzzle until everything is well defined/clear. For me it would be something like:

My friend has two kids, and I don't initially know anything about their sex beyond nation level stats which are fifty-fifty. She says something which makes it clear she has at least one boy, but in such a way that it just prohibits having two girls without hinting at all if these are two boys or one, perhaps something like "i have to pick up Johny from kindergarten". How much should I bet she actually has two boys vs a boy and a girl?

The first sentence of that phrasing is great! It makes things much more clear. But:

"i have to pick up Johny from kindergarten"

actually would give the probability of the other kid being a boy a fifty-fifty chance still, I believe. I still think the clearest way to phrase that part of the puzzle is for the narrator to ask the woman "is at least one of your kids a boy?".

I agree that my phrasing was still problematic, mostly because it seems to matter if she said something spontaneously or as a response to a specific question. In the first case, one has to consider how often people feel compelled to say some utterance in various life scenarios. So for example in case one has two boys the utterance "i have to pick up Johny from kindergarten" might have to compete with "i have to pick up Robert from kindergarten" and might be strange/rare if both are in similar age and thus both should be picked up etc. Still, I think that without knowing much about how people organize their daily routines, my best bet for the question "does she have two boys?" would be 33%.

It's get funnier with "i have to pick up my younger one, John from kindergarten" :)