Odds are it won't snow like this again for, well, do the math

Tuesday's fresh blanket of snow propelled this month into fourth place on the list of all-time snowiest months in the Twin Cities. That high ranking takes a little of the sting out of all the shoveling, snowblowing and spinouts. It feels like quite an achievement; after all, snowfall data in the Twin Cities go back to the winter of 1884-85.

That makes this the fourth-snowiest winter month out of the 808 since records began (counting November through April as winter months). But just how rare is that? Making a few assumptions and doing a little bit of math, we can find an answer:

This month had less than a one-half-of-1-percent chance of landing in the top-four snowiest, and we should only expect one more month like it in the next 38 years.

We'll start by finding the probability of winding up at number 1 on the all-time snowiest month list, which would mean supplanting November 1991, of Halloween blizzard fame. Here's a problem that might seem unrelated: Suppose you have one red slip of paper, and 807 blue slips. Someone places these slips randomly in 808 boxes, lined up in a row. What's the probability that the one red slip winds up in the last box?

This question is quite similar to asking for the likelihood of breaking the all-time snowiest month record this month, in our 808th winter month of data. Consider this mathematical insight: when the future and the past have no effect on each other, there's no reason to think of them as being different. Some amount of snow fell in winter months past, and some amount has fallen this month. We can think of them all as numbers chosen at random — and chosen at the same time, then randomly assigned to a year.

Returning to our slips of paper: Each one has written on it one of the 808 randomly chosen numbers. One of these numbers will be larger than the rest; that's the slip that we color red. Our boxes have the labels of each of the 808 winter months in years since November 1884-85, with the February 2019 box being last.

To find the probability of the red slip being in that last box, we employ the classic method of probability theory: Count the number of "good" outcomes (the ones we're interested in) and divide by the total number of possible outcomes. The red slip could wind up in any one of 808 boxes, giving 808 possible outcomes, but only one of those is the "good" outcome of the red slip being in the last box. That gives a probability of 1/808.

Now let's turn to finding the probability of this month finishing in the top four, rather than breaking the record. Now there are more good outcomes to count, and more total possible outcomes, but when we divide them we get 4/808, or less than one-half of 1 percent.

Another way to quantify the unlikeliness is to ask how many new top-four snowfall months we should see, on average, in various future time frames. The mathematical concept of expected value captures this idea well. We can find the expected number after n months by adding the probabilities of a new top-four result for each month, up to the nth one.

March 2019 has a 4/809 probability of landing in the top four. April has a 4/810 probability, next November will have a 4/811 probability and so on. Adding the first 100 of these numbers, we find that on average we should see 0.47 top-four months in the next 100 winter months. In order to push this up to one new top-four month, we have to go out 230 winter months, or a little over 38 years.

If you're curious about how far out we have to go to before we expect one new, all-time monthly snowfall record, the answer is 231.5 years.

What we've done here is build a mathematical model, which always requires making assumptions. One is that the snow total for each month isn't affected by the totals from past months. We've also assumed that every winter month has an equal probability of finishing in the top 4, and that no two months have exactly the same snowfall total. The model we've constructed is simple, built directly from first principles; a richer model would take the observed data into account.

A last and most crucial assumption we've made is that, on average, the amount and distribution of snowfall has remained the same in each winter since 1884. However, there's reason to believe that the changing climate has altered our snow patterns — probably making heavier snows more likely. It's possible to build models that account for this complexity, too, but doing so requires far more sophisticated methods. Contact your local college statistics department for more information.

Rafe Jones, of Eagan, is an associate professor of mathematics at Carleton College.