How to prove that if you take any 400 consecutive Octobers then exactly 14 % of those years have five Fridays and Sundays?
To have five Fridays and Sundays, October must start on Friday. A span of 400 years has 146097 days in it-remember the years that are multiples of 100 but not 400 are not leap years. This is a multiple of 7. So the pattern repeats. Count how many of the 400 start October with Thursday or Friday. You should find 56.
Corrected-used to say start on Thursday or Friday.
As people have said, the best way to do this is brute force, once you know that the calendar repeats with period 400 years.
I'd note that to have five Fridays and five Sundays, October must start on a Friday (like it does this year), and therefore run from Friday, October 1 to Sunday, October 31.
If it feels like cheating to use date rules that somebody has already computed, you could (for example) note that:
Finally, it's easy to see that the answer should be "approximately 1/7" because the calendar rules don't make reference to any particular days of the week, although of course this is not a proof. This is why the claim that went around the Internet earlier this month that an October like the current one only occurs every 823 years was so laughable; see here for a debunking.