Unlikely != Impossible

My sister is currently in law school, and earlier today she sent our father, a former NOAA weather observer, the following question:

I am pondering the nature of black ice. In tort liability cases, a person who slips and falls on business property cannot recover for his injuries if "a reasonable person" should have discovered and realized the danger. For example, if I go to Starbucks and see the person in front of me slip on some ice in front of the door, and then I slip on the ice and injure myself, I "accepted the risk" of slipping, and I can't sue Starbucks, even if Starbucks negligently maintained their property. Knowledge is an essential component of risk acceptance. I cannot accept a risk if I have no knowledge of its existence.

Now to my question: the nature of black ice is that it is nearly impossible to see. A person of ordinary, average intelligence knows that he should "be careful of black ice," but he doesn't have actual knowledge of any specific patch of black ice until it's too late.  Do you agree? Is it possible to "know" black ice?

Here was his response:

Black ice forms from either rain falling on a freezing surface or when moist air comes in contact with the freezing road surface. This can be in the air or can come from car exhaust. I think your question is can one know that there is black ice, right?

I say you can. You can, should know when the temperature is nearing freezing. Most people don't but ignorance is no excuse.

He likes to tease me for my scientific precision, and as we were discussing this he joked that he knew there was no black ice on the road today, with temperatures in the 80s, but that I might allow for the possibility.  I thought it might be fun to figure out exactly what the probability of black ice today would be, to show that there is some finite possibility.

In order to get ice, we'll need temperatures around freezing.  Temperature describes the average speed molecules are moving in a material, but not every molecule will be going at this speed.  We could imagine a small part of a material having molecules that are mostly going a below-average speed.

In an ideal gas (air is pretty close to one), the probability of a molecule going a specific speed is given by the Maxwell Speed Distribution:

where m is the mass of the molecule, k is Boltzmann's constant, and T is the temperature.  It's possible to get an average speed as a function of temperature from this:

Using these two, we can find the probability of finding a patch of freezing air.

According to this, the average speed of nitrogen (the main component of air) at 32°F will be 453 m/s.  Plugging this into the Maxwell Speed Distribution at a temperature of 80°F gives a probability of 0.2%.  This may seem shockingly high, but remember that this is only for a single molecule.  To find the probability of a patch of cool air, we'll need the number of molecules in the patch.

Let's suppose we're looking for a block of freezing air 25 cm by 25 cm by 1 cm.  The density of air is approximately 1.2 kg/m^3.  Multiplying by the volume of our block, and converting to the number of molecules we find 1.6*10^22.  Now, to find the probability of all the molecules moving slowly enough, we raise the single molecule probability to this power.  Unfortunately, there's no easy way to express the tininess of this number, but please note that it is not zero.

This talk of exceedingly unlikely events reminded me of a design I made one year for Swarthmore's annual Physics Department T-Shirt Contest.  I forget whether I ever submitted it (it didn't win in any case), but here it is (click to enlarge):