My point was that when you are dealing with a couple who has two children of the same sex (gender, whatever), there are 13 combinations, because them both being born on Thursday is just one, but with a mixed pair, you have the boy born on Thurs, and the girl born on another day, or the girl born on Thurs, and the boy born another day. This is why it’s 13/27, and not 14/28 (which would of course, be 1/2).
Specifying that there is one girl who was born on a Thursday is what keeps it from being 1/2. If the question were simply that “there is one girl, what are the odds, that both children are girls,” then it would be 1/2. The additional information of Thursday changes it to 13/27.
ETA: I think, additionally, if BOTH children’s days were specified, then it would change to 13/26, or again, 1/2. I haven’t checked that out thoroughly, though.
The rarer the second condition, the closer it comes to 1/2. (Or further conditions applied.)
So, if I picked out of a large crowd someone who has two children, one of which is a girl born on February 17th, then you would have 729/1459 chance that the other child is a girl.
No, that’s what keeps if from being 1/3. If I pick out of a large crowd someone who has two children, one of which is a girl, then the chances that that person has two girls is 1/3.
I don’t follow you here. If we just know that one of the children is a girl, then the the probability that the other child is a girl is 1/3, not 1/2. Providing more information about the one child we know is a girl brings the results closer to 1/2. If I do the calculations correctly (and we allow for the assumptions made above), when we know that the girl was born in January the results are 23/48. When we know that the girl is born on Jan 1, the results are 729/1459. The more precisely you can identify the girl that we know the closer we get to a 50% result for the remaining child.
(Assuming 12 equal length months and no leap years)
The nice thing about days of the week is that you can assume an even distribution.
In order to make a determination on the odds when the additional information is that one of them is 5’7", I’d first have to know the distribution of heights of girls.
As we are culling the respondents with more precise identifiers we impact the likelihood of the different parent groups responding “yes”.
We have 4 equally sized groups of parents with two children BB, BG, GB, GG. When you ask if they have “a girl” the last three groups all respond Yes, because they each have 100% chance of having “a girl”. So the chance of that subgroup having a second daughter is 1/3.
When you add specificity to the question like “born on an odd numbered day”, the GG group is more likely than either BG or GB to have a girl with that characteristic, so it skews the percentages of having a second daughter.
If the specificity is absolute uniqueness, such as “with Social Security number XXX” the GG group is exactly twice as likely as either BG or GB to have a girl with that number, getting you to a 50% chance of having the second daughter.
And I don’t know about “but only one has”, it should be, “and one has”, as high enough values of “a lot” to approach the 13/27 range would take enough that there should be multiple that meet such criteria.
So, "I know a few thousand families with two kids, and I chose one from the group that has at least 1 girl born on a Thursday. What’s the chance their other child is also a girl?”
That requires a slightly less absurd assumption, in that I would know that he will always tell me the gender and day of birth of his friend’s kids, but would only tell me about his own if they happened to match.
I remember arguing about the simpler two boys vs boy and girl version on the James Randi forums 20 years ago, after someone posed a similarly ambiguous question. Nowadays I could just point to Wikipedia to show I was right.
I’m pretty sure it’s 1/2, not 1/3, because girl/boy, and boy/girl are separate outcomes.
There is a 1/4 chance of girl/boy, a 1/4 chance of boy/girl, a 1/4 chance of boy/boy, and a 1/4 chance of girl/girl. Since girl/boy and boy/girl are considered the same, there is a 1/2 chance of this outcome.
You seem to be describing the situation where @Biotop has randomly volunteered a sex, which would not change the odds from 1/2. But the same principle applies to him randomly volunteering a birth-day-of-week. If he has two kids born on Thursday, there is 100% chance he will pick Thursday. If they are born on different days, there is 50% chance. This brings the odds of two girls back to 1/2 again.
I think, if I’m doing the math right, then if instead of Thursday, the selected set is one with a girl that was born in the AM, then that changes the probability to 3/7 chance that both are girls.
That second piece of information about the set that you are choosing for makes a difference in the probability, as there is more overlap, a higher chance that two have the same criteria, the chance approaches 1/3, as the second condition becomes more rare, and less chance of overlap, the chance approaches 1/2.
Okay, so how does it work out if my second criteria for the set is that she was born on a weekday(5 of 7), rather than a particular day of the week?