332 straight months of above-average temperatures

I ran across this on Facebook today.  It turns out that I have never, in my whole life, experienced a month with below-average temperatures:

The average temperature across land and ocean surfaces during October was 14.63°C (58.23°F). This is 0.63°C (1.13°F) above the 20^th century average and ties with 2008 as the fifth warmest October on record. The record warmest October occurred in 2003 and the record coldest October occurred in 1912. This is the 332^nd consecutive month with an above-average temperature.

In other words, for 332 straight months, the average temperature has been above the 20th century average temperature.

This got me thinking about just what the chances of that are.  I want to make one simple assumption and then test the hypothesis that global warming is not occurring.

Temperature Deviations

Let's look at temperature deviations.  Call the average temperature over the century \(T\).  We should think of this as the long-haul temperature.  If global warming isn't happening, then this is the "baseline temperature" of the Earth.

Each month, the average temperature doesn't necessarily need to be the same as the long-haul temperature.  It might be above or below, depending on whether there's something like El Nino going on, or if the sun is extra-bright, and so on. Either the temperature in that month is above or below the long-run temperature\(T\).  Whether or not the temperature is above or below -- one of those two outcomes -- is what we'll focus on.

Here's our one simplifying assumption.  The probability of monthly temperature being above or below \(T\) in one month does not depend on whether it was above or below \(T\) in previous months.

This lets us treat monthly temperature as a Bernoulli process.  This is a standard piece of finite mathematics which I'll proceed to explain.

Bernoulli Processes

(If you want to get to the meat, skip this section and go to the next one.)

A Bernoulli process is an experiment with the same two outcomes ("success" and "failure") and the same two probabilities (of success, of failure) repeated over and over again.  The probabilities don't change from experiment to experiment.

The standard example of a Bernoulli process is flipping a coin over and over again.  There are two outcomes each time (heads and tails), and each time the probabilities are the same (\(0.5\)) each).

The usual question in a Bernoulli process is, "What's the probability of getting this many successes in that many trials?"  For example, we might ask, "what's the probability of getting \(2\) heads if we flip a coin \(3\) times?"  Here's a quick explanation of where the answer comes from.  It requires two "black box facts."

We did not specify the order of the heads.  So, for example, the outcomes HHT, HTH, THH all qualify as "three heads."  Each of these outcomes has the same probability: since the probabilities don't change from experiment to experiment, the experiments are independent.  Black-box fact one: The probability of a string of independent outcomes is the product of each outcome's probability.  That is, the probability of HHT is the probability of H times the probability of H times the probability of T.

Now to get the probability of two heads (instead of HHT, say) we just have to add up the probabilities of all the different ways we can have two heads in three flips.  Each has the same probability, so we just need to multiply that probability by the number of different ways we can get two heads in three flips.  Black-box fact two: the number of ways of choosing two of the three flips to be heads (order doesn't matter) is the number \(C(3,2) = 3\).

So the probability of flipping two heads in three tosses is \(3(0.5)^3\).

More generally, the probability of getting \(k\) successes in \(n\) trials (if \(p\) is the probability of success and \(q\) is the probability of failure) is
$$C(n,k)p^kq^{n-k}.$$

Climate as Bernoulli Process




Our climate model is basically a series of coin flips.  If the long-haul temperature average isn't changing, then each month, the probability of above-average temperatures is just the same as the probability of below-average temperatures.  That is, there's a \(50\%\) chance that either occurs.  We want to know the probability of \(332\) above-average temperatures in as many trials.

This just like treating each month as a coin flip.  If you flip a coin \(332\) times, what's the chance that you get \(332\) heads in a row?

$$C(332,332)\bigg(\frac{1}{2}\bigg)^{332}\bigg(\frac{1}{2}\bigg)^0 = \frac{1}{2^{332}}.$$

That is an incredibly small, tiny, vanishing, eensy-weensy, little number.  It's on the order of \(10^{-100}\), that is, one in \(10^{100}\).  By comparison, there are roughly \(10^{80}\) atoms in the observable universe.  So if we gathered all of those atoms into one box, colored one blue and all the rest white, and grabbed one of the atoms while blindfolded, we're billions of times more likely to pick the blue atom than we are to have \(332\) straight months of above-average temperatures.

The tl;dr?  If global warming is not happening, and if the temperature each month is independent of the temperature the previous month, then the probability of \(332\) straight positive deviations is a billion trillion times less likely than reaching into a bin of all the atoms in the observable universe and randomly picking the one blue atom.  So, global warming is almost certainly happening.