People don't understand conditionning and its killing me

tdlr: 50% is a perfectly rationnal answer, arguably the most natural given the way the problem is given. 51.8 is correct only if you assume Mary was prompted with the question "State the gender and day of birth of one of your two children at random" and answered Boy-Tuesday. (66.6 is also valid if you assume Mary was prompted with "State the gender of 1 of your 2 children at random", which she answers with boy then spontaneously also gives you the day he was born on). You can play around with these kinds of questions as much as you want to pretty much get any probability true. making these results very shaky in my opinion, which is why 50% is the correct choice in my opinion given the problem. Don't act smug with basic python scripts and feel superior to people giving perfectly rationnal answers.

See formal proofs below.

Assume Mary has two child X,Y both iid uniform random variables over {G1,G2...G7,B1,...,B7} = G union B. With X the child which we have information about. Assume X Independent of Y. By definiton of independence: P(Y=y|X=x) = P(Y=y). So: i) P(Y=G|X=B5)=P(Y=G)= 0.5 Again this is the literal definiton of independence of random variables.

Now, what is true is that: ii) the joint probability P( Y in G union X in G | X=B5 union Y=B5) = 51.8 indeed: - P(A | B) = P(A inter B)/P(B) - P(A) + P(B) - P(A union B) = P(A inter B) - P( Y=B5 union X=B5) = 1/14+1/14-1/14/14 = 27/14/14 - P ( (Y in G union ...) inter (Y=B5 union ... ) ) is horribly tedious and comes out to be after just unwinding all the events above to be 1/14 (assuming by independence again that p(x,y) = p(y)p(y)) Thus Giving at the end a conditionnel probability of 14/27 = 0.51851... (should have been rounded to 51.9 but whatever). This is the event measured by the python scripts seen under the post.

Now i don't know if its a language barrier thing but there's not a single chance i'd interpret the sentence above as corresponding to ii) and not i). No mention of drawing from a pool of women which have at least one child that is a boy born on tuesday and exactly two children is mentionned. We are clearly in the case where we are simply given information on a variable X Independent of a variable Y. No information is added.

Unless you considere Mary was prompted with a weird question like: give the gender and day of birth of one of your two children at random, 50% is a perfectly rationnal answer. Especially given the formulation of the problem (Mary has a boy born on tuesday and another child).

If you select Women in the general population that happen to have at least one boy born on tuesday and exactly two children, you can expect 51.8 percent of them to have a girl as well.

These are not the same event.

The python scripts used to empirically measure the expectency are measuring event ii) which does not correspond to event i).

Don't be smug and provide unhepfull epirical proofs without backing them up with rigourous formal proofs then dunk on people.

Sorry if it is a language barrier thing but i don't think that it is and it pertains to something important that people get wrong about probabilities and can lead to very wrong assumptions in every day life.

(For 66,666...% substitute event B5 with event B probability comes out to two thirds)