Disclaimer: Rowling owns everything except the calculator that I did this math on.
Arthur had always wanted a girl. When convincing Molly to have another child so soon after Bill he used this argument:
"We had a ½ chance of having a boy before. Now we have another ½ chance. But since 1/2 x 1/2 is ¼, we really have only a ¼ chance of having a boy, making very likely that we'll get a girl next, dear. And then we can stop."
The same when convincing her to have what turned out to be Percy:
"The chances are 1/2 x 1/4 x 1/2 now dear. If we did this 16 times (Molly gulped) 15 of those times we would have a girl."
And the same for Fred & George:
"1/256, Mollywobbles! There's only a 1 in 256 chance that it'll be a boy!" Instead they had gotten two boys.
Yet again with Ron:
"Literally a 99.99% chance of having a girl, love! Let's just give it one more go."
After that they had given up. Apparently it wasn't in the cards for them to have a girl. Besides they had six beautiful boys, how could they complain?
Then, one day Molly went to the healer feeling sick and came back with news of a baby. She and Arthur both felt safe telling the boys to expect a new brother.
So when, nine months later, Arthur heard the cry 'It's a girl' it is understandable why he fainted dead away.
The Weasley couple had not needed a ¾ chance, a 15/16 chance, a 255/256 chance, or even a 99.99% chance. All they had needed was 1 in 7.
A/N: Okay, so it's really short, but so cute I couldn't resist! I'm pretty sure I didn't apply statistics correctly or the math is incorrect (big surprise, a writer who can't do math) so feel free to tell me what I did wrong.