Here’s a little puzzle in probability which, in a slightly different form, I gave to my students to work out. I get the papers back tomorrow. To brace myself against that I’m curious what my readers here would make of it.

Possibly you’ve encountered a bit of calendrical folklore which says that Friday the 13ths are more likely than any other day of the week’s 13th. That’s not that there are more Fridays the 13th than all the other days of the week combined, but rather that a Friday the 13th is more likely to happen than a Thursday the 13th, or a Sunday, or what have you. And this is true; one is slightly more likely to see a Friday the 13th than any other specific day of the week being that 13.

And yet … there’s a problem in talking about the probability of any month having a Friday the 13th. Arguably, no month has any probability of holding a Friday the 13th. Consider.

Is there a Friday the 13th this month? For the month of this writing, December 2011, the answer is no; the 13th is a Tuesday; the Fridays are the 2nd, 9th, 16th, 23rd, and 30th. But were this January 2012, the answer would be yes. For February 2012, the answer is no again, as the 13th comes on a Monday. But altogether, every month has a Friday the 13th or it hasn’t. Technically, we might say that a month which definitely has a Friday the 13th has a probability of 1, or 100%; and a month which definitely doesn’t has a probability of 0, or 0%, but we tend to think of those as chances in the same way we think of white or black as colors, mostly when we want to divert an argument into nitpicking over definitions.

Most years have 365 days, that is, one day more than 52 weeks exactly. There are many results from this; among them is that if the 1st of January this year was a Saturday (as it was), then the 1st of January next year will be one day ahead in the week cycle, a Sunday. The other days of the week advance accordingly. However, every four years there’s a leap year, with 366 days, or two days more than 52 weeks exactly. And then, going over this, the day-of-the-week for any given date advances by two days of the week rather than one. The first of March, 2011, was Tuesday; the first of March, 2012, is a Thursday.

Leap years ordinarily happen every fourth year, so that the first of January progresses, starting in 2011, from Saturday, to Sunday, to Tuesday, to Wednesday, to Thursday, to Friday, skip a day to Sunday, to Monday, and so on. The start of January, or any month, progresses through the week in a pattern which would eventually repeat back after 28 years. Except …

The Gregorian Calendar, which tries to make the match between the calendar year and the seasonal year a bit better, drops three leap days out of every 400. These are the leap days that happen in century years not divisible by 400: 2100, 2200, and 2300 will not be leap years. 2400 will be again. So the 28-year period of the starts of Januaries breaks down. There is a 400-year cycle, though. Your calendar for the year 2011 will work correctly again, apart from movable feasts, for 2411, and 2012 will impersonate 2412 perfectly, and 2013 matches 2413, and so on.

So here’s the thing: there isn’t any “probability” about any month having a Friday the 13th. It either has or it hasn’t, and whether it has is defined by a rule laid down in the 16th century, and which we can project for every year imaginable indefinitely far into the future, and, for that matter, which we can proleptically project back to the Big Bang, or even farther back if we want to say something about whether 2,038,679,230,082,266,003 BC would have been a leap year. Arguing about the probability is like arguing about the probability that Franklin Roosevelt won the Presidential Election of 1936.

But we definitely mean *something* by saying that a month has a probability of having a Friday the 13th. More specifically, a month has a probability of ^{688}/_{4800} of having a Friday the 13th. And yet, no month has a probability, unless it’s either 1 or 0, of having a Friday the 13th.

Where does the *chance*, the probability that’s between zero and one, come from?

It comes from the unspoken formulation “if a month is picked at random, what is the probability that it has a Friday the 13th.”

By the way, I’m sure there’s a fringe out there that would argue that FDR did *not* win the election of 1936 – it was stolen.

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Yes, that’s the interpretation that makes most compelling sense to me. But I do find it interesting when there are such divergences between what we say and what we mean; for example, if we ask about the chance that July 2065 has a Friday the 13th, we can say it’s a little over 1/7th and everyone understands that, even though the month actually is either certain to or certain not to have one.

I don’t doubt there’s a crazy enough set that would claim Roosevelt stole the election of 1936, or any election, although they seem to have avoided my mailbox, which is fine. Anyway we can be certain he won to the limits of being able to know any historical fact (down to knowing the result of the coin you just tossed, which you might after all be perceiving incorrectly, by mistake or by conspiracy).

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