So to give one answer to my calendar puzzle, which you may recall as this: for any given month and year, we know with certainty whether there’s a Friday the 13th in it. And yet, we can say that “Friday the 13ths are more likely than any other day of the week”, and mean something by it, and even mean something true by it. Thanks to the patterns of the Gregorian calendar we are more likely to see a Friday the 13th than we are a Thursday the 13th, or Tuesday the 13th, or so on. (We’re also more likely to see a Saturday the 14th than the 14th being any other day of the week, but somehow that’s not so interesting.)
Here’s one way to look at it. In December 2011 there’s zero chance of encountering a Friday the 13th. As it happens, 2011 has only one month with a Friday the 13th in it, the lowest case which happens. In January 2012 there’s a probability of one of encountering a Friday the 13th; it’s right there on the schedule. There’ll also be Fridays the 13th in April and July of 2012. For the other months of 2012, there’s zero probability of encountering a Friday the 13th.
Imagine that I pick one of the months in either 2011 or 2012. What is the chance that it has a Friday the 13th? If I tell you which month it is, you know right away the chance is zero or one; or, at least, you can tell as soon as you find a calendar. Or you might work out from various formulas what day of the week the 13th of that month should be, but you’re more likely to find a calendar before you are to find that formula, much less work it out.
If I don’t tell you what month it is, though, you have to resort to an empirical probability: there are four months in 2011 or 2012 which have a Friday the 13th; there are twenty-four months I might have chosen from. So if I wasn’t biased towards or away from any particular month, then, I had a four in twenty-four chance of picking a Friday the 13th-equipped month. The 13th of the selected month, in 2011 and 2012, we might say has a 1/6 chance of having been on Friday.
If I pick some month in 2010, 2011, or 2012 — only August had a Friday the 13th in 2010 — then there were five chances in thirty-sex of picking a Friday the 13th. Picking a 13th in this three-year span, at random, would give us a 5/36 chance of picking one of those 13ths which was a Friday. Adding in the two Fridays the 13th of 2013, we find seven chances in forty-eight of a Friday the 13th between 2010 and 2013.
And this seems to get at what we mean by saying that an arbitrarily picked month has some chance of having a Friday the 13th. It’s not that there’s any uncertainty in what the days of the week of the month are, if we know the month and year. (Even if we know the month and year, we might not know or have easily available the knowledge of where it falls on the calendar, so we can treat this as being like not knowing which particular month and year are being discussed.) What the uncertainty is, I suggest, is that we don’t know whether we’ve selected a month with a Friday the 13th or not.
In the 48-month stretch between 2010 and 2013, there are seven months with a Friday the 13th. In the 60-month stretch from 2010 to 2014, there are eight months; in the 72-month stretch from 2010 to 2015, we find eleven months. The probability that an arbitrarily picked month was one of the Friday the 13th months is fairly obviously fluctuating. We’d naturally expect that as the length of the interval we consider, the number of months we examine, grows that the probability should settle down to whatever the true number should be. Happily this is what it does.
How long an interval do we have to look at, if the ratio of Friday the 13th months to total months keeps changing? How do we know when to stop? The Gregorian calendar offers a logically compelling interval to examine: 400 years. The Gregorian calendar repeats its pattern of days-of-the-week every 400 years, and not over any shorter period, so, 4800 months it is. And in those 4800 consecutive months, there will turn up 688 Fridays the 13; 687 Sundays and 687 Wednesdays the 13th; 685 Mondays and 685 Tuesdays the 13th; and 684 Thursdays and 684 Saturdays the 13th.
So if we picked a month, without knowing which one it was, or without checking its days-of-the-week pattern, we would have 688 chances in 4800 of picking one with a Friday the 13th, better than our chances of picking one with any specific other day of the week. And this interpretation seems to match both what we mean by “the 13th is more likely to be Friday than any other day” and with our knowledge that there’s no month for which we actually don’t know whether the 13th is a Friday or not.