Is this the place where I could finally get a difinitive answer to the question in this meme? I've pasted a comment about it in the description of this post. This problem keeps popping up online and people seem to keep making it more and more complicated every time.
That's not the reason.
If you have two siblings the older one could be boy or girl, born on any of 7 days, so that gives 14 options. There are also 14 options for the younger sibling.
Visualize all 14 x 14 options laid out on a big 2D grid.
now if you say you only want families with "a boy born on a tuesday" then that means you're selecting one entire row, but also one entire column. Now that would be 28 cells, and 14 of them would have a sister, however the row and column overlap - on the cell where there's a boy born on tuesday who **also** has a brother born on tuesday.
So the 14x14 cross, that's not 28 cells, but only 27. 13 of them have a male sibling, 14 of them have a female sibling for the selected brother. So the chance of a girl being the sibling is 14/27.
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And here's a simulation in Python that gives the result. It generates 10 million families with 2 kids, randomly a boy or girl, born on random days of the week.
Now IF there is a "boy born on tuesday" (which could be either the younger or older one) it then checks whether they have a sister. You get that result 14/27 times, or about 51.8% of the time.
So it's unintuitive, but of all two-child families with at least 1 boy born on a Tuesday, 51.8% of them do in fact have a sister.
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EDIT: and here' the kicker. If you remove the restriction and allow boys born on any day to count, so we only care about families with at least one boy, the chance of having a sister doesn't in fact revert to 50%, but to 66%. So the 50% was a red herring entirely.
You can try that out in the simulation by changing this line
if (c1[0]==1 and c1[1]==1) or (c2[0]==1 and c2[1]==1):
to this
if (c1[0]==1 and c1[1] < 8) or (c2[0]==1 and c2[1] < 8):
So it now counts families with a boy born on any day of the week. The result goes up to 66.6% of the families including a sister.
The reason for this is that if you don't care about days of the week, you get 4 possible family combos (based on birth order):
Boy + Boy
Boy + Girl
Girl + Boy
Girl + Girl
If we say "only count families with at least 1 boy" then we can exclude the last two-girls option, and that leaves three equally likely families left:
Boy + Boy
Boy + Girl
Girl + Boy
The options with a boy plus a girl happen twice as frequently as the option with two boys, since there are two different ways for it to happen. It's the same as flipping two coins. You'll get two heads 1/4 of the time, two tails 1/4 of the time, and heads plus a tails 1/2 the time. So heads plus a tails is in fact more likely than 2 heads, and you are in fact more likely to have a boy and a girl than two boys if you have two kids.