The 14/27 answer in the video is correct, incidentally.
Also, I notice you said "Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre."
Remember, though, there's no "the boy". The question "On which day of the week was the boy born? Tell me, I need to know!" does not always have a well-defined answer.
Indeed, you'd get the same 14/27 answer even if "Tuesday" in the question "What proportion of two-children families with at least one Tuesday boy have a girl?" was replaced by any other day. And if this seems paradoxically in conflict with the fact that simply asking "What proportion of two-children families with at least one boy have a girl?" has instead the answer 2/3, reflect again upon the fact that some families have two boys born on different days, so that there's no single answer to "On what day was 'the boy' born?". And then just draw out the cases and count.
(Specifically, out of the 2 * 7 * 2 * 7 equiprobable cases overall for Kid 1 and Kid 2's genders and days, there are 27 cases where there's at least one Tuesday boy, and 14 cases where there's at least one Tuesday boy and also a girl. There are 3 * 7^2 cases where there's at least one boy, and 2 * 7^2 cases where there's at least one boy and also a girl.)
Many of these questions, I think, become clearer if thought of as counting questions instead of as "probability" questions (though it's all the same; the math called "probability" is just the math of various kinds of counting (from simple counting as in this case to complexly weighted continuous measurements, but still ultimately a generalized form of counting). However, despite that equivalence, the concept "probability" has developed all these other distracting connotations, such that psychologically, there can be a useful difference in perspective in switch to explicitly thinking "counting" instead. No one would long dispute that there are 27 cases with at least one Tuesday boy, etc.).
Also, I notice you said "Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre."
Remember, though, there's no "the boy". The question "On which day of the week was the boy born? Tell me, I need to know!" does not always have a well-defined answer.
Indeed, you'd get the same 14/27 answer even if "Tuesday" in the question "What proportion of two-children families with at least one Tuesday boy have a girl?" was replaced by any other day. And if this seems paradoxically in conflict with the fact that simply asking "What proportion of two-children families with at least one boy have a girl?" has instead the answer 2/3, reflect again upon the fact that some families have two boys born on different days, so that there's no single answer to "On what day was 'the boy' born?". And then just draw out the cases and count.
(Specifically, out of the 2 * 7 * 2 * 7 equiprobable cases overall for Kid 1 and Kid 2's genders and days, there are 27 cases where there's at least one Tuesday boy, and 14 cases where there's at least one Tuesday boy and also a girl. There are 3 * 7^2 cases where there's at least one boy, and 2 * 7^2 cases where there's at least one boy and also a girl.)
Many of these questions, I think, become clearer if thought of as counting questions instead of as "probability" questions (though it's all the same; the math called "probability" is just the math of various kinds of counting (from simple counting as in this case to complexly weighted continuous measurements, but still ultimately a generalized form of counting). However, despite that equivalence, the concept "probability" has developed all these other distracting connotations, such that psychologically, there can be a useful difference in perspective in switch to explicitly thinking "counting" instead. No one would long dispute that there are 27 cases with at least one Tuesday boy, etc.).