p(both are girls | you're told at least one is a girl)
= p(you're told at least one is a girl | both are girls) * p(both are girls) / (
p(you're told at least one is a girl | both are girls) * p(both are girls)
+
p(you're told at least one is a girl | they aren't both girls) * p(they aren't both girls)
)
The solution there assumes that p(you're told at least one is a girl | both are girls) = p(you're told at least one is a girl | they aren't both girls).
Pretty much all of them are like this and it's actually a terrible article.
For those thinking about how can the likelihood of them being a girl can change consider this;
>A family has two children. You're told that at least one of them is a girl. What's the probability both are girls?
"The question writer took all sets of two child families and ruled out the bb case. Then they asked the exact question above" This is 1/3 chance - select gg from [gg,bg,gb]'
vs
"The question writer came across a girl from a two child family, then they asked the exact question above". This is 1/2 chance - select gg from [gg, gg, bg, gb] with gg listed twice since there's two ways to select a girl from that set; ie. coming across a girl is twice as likely to occur from the gg case than it is either gb or bg.
I think that's the clearest wording to get the message across. Either way it's the exact same question but it reasonably has a completely different answer. There's no way to resolve this ambiguity with the question as written. Pretty much all questions here are like this.
Amusingly the Monty Hall problem in it's original form was written to avoid this ambiguity. There's a comment thread above on this. But you know what the author of the article linked here did? They reworded it and added ambiguity similar to the above. The Monty Hall problem needs to specifically state "the host opens a door that he knows does not contain the prize" or it's unanswerable. The author of the article removed this statement without awareness of its importance!
This sounds mean but honestly I really really think this article was written by someone with no education or knowledge on statistics. As in they broke perfectly reasonable questions and added ambiguity to the point they are not answerable as written in an article trying to demonstrate how easy stats is.
I did not mention a lot of assumptions. For example my problem statement also does not answer the question if Monty always gives you a choice to switch or only when you picked in a certain way. There are quite a few possible variations, all of which change the answer: https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...
In the derivation of the likelihood the assumptions are clearly stated.
While my knowledge on statistics is certainly unsatisfactory, your assessment of the education I had is quite wrong. My final exam in order to obtain a Master's degree in mathematics is next week. One of the subjects I chose is Bayesian Statistics, so wish me luck.
I do wish you luck. Fwiw I'd always work forward from the question and talk to the ambiguities present especially when people may come across this and not realize. Stating the Bayesian probabilities of one possible interpretation and having that as your way of stating assumptions is a bit like working backwards from one possible answer.
As an example of how presenting this way can hurt; The link you had for the sisters paradox doesn't talk to the ambiguities of a question that is well known to have no answer - https://en.wikipedia.org/wiki/Boy_or_girl_paradox . This information spreads and people see it restate it. This leads to a false belief that the sisters paradox has no ambiguities and that the answer is clearly 1/3 (the wikipedia page on this has the correct statement that it's ambiguous and the answer can be 1/3 or 1/2 depending on one of two reasonable interpretations).
I think a far better article would talk to ambiguities of each of these. Not subtly stating assumptions. I'll also point out that these types of ambiguous questions are commonly used in DS interviews (I've worked in big tech for many years now). The expected response of a strong candidate is a discussion on the ambiguities. You'll usually be prompted "is there any other interpretation of this question that leads to a different result?". If people read this as their guide rather than the more detailed wikipedia articles etc they may be misled which is why i'm strongly negative on this article as written. I'd hope no one reads and doesn't realize the assumptions here since they are critical.
This is not about someone choosing to lie, it’s about someone choosing what true thing to say.
If “you're told that at least one of them is a girl” was intended to mean “you ask whether at least one of them is a girl and you’re told that at least one of them is a girl” it was pretty easy to make it clear.
In that case it’s straightforward that p(you're told at least one is a girl | both are girls) = p(you're told at least one is a girl | only one is a girl) = 1 (with just the mild and reasonable assumption that you’re not being lied to.)
Edit: also the reasonable assumption that your question is unrelated to the number of girls - it’s easy to imagine settings where you asked because of something which may be correlated with the number of girls.
> Likelihood of at least one girl
What the “mechanism” requires is “likelihood of being told that at least one of them is a girl”.
Use Bayes rule to correctly solve probability riddles:
https://news.ycombinator.com/item?id=45056790
The solution there assumes that p(you're told at least one is a girl | both are girls) = p(you're told at least one is a girl | they aren't both girls).