Probability of warming being due to random chance

Philip Bump in If you’re 27 or younger, you’ve never experienced a colder-than-average month quotes the National Oceanic and Atmospheric Administration in October 2012 as saying:

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 20th 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 332nd consecutive month with an above-average temperature.

Bump goes on to write:

If you were born in or after April 1985, if you are right now 27 years old or younger, you have never lived through a month that was colder than average. That’s beyond astonishing.

Josh Voorhees in It Has Been Nearly 28 Years Since We Have Had a Colder-Than-Average Month quotes the above Bump article (although I couldn't find the reference to February 2012 in the article myself):

The last below-average month was February 1985. The last October with a below-average temperature was 1976.

I assume Voorhees came up with February 1985 by subtracting 332 months from October 2012 (the month referred to in the NOAA quote above. It's now March 2013. So last month makes 336 consecutive month with an above-average temperature.

What's the chance of that happening by random chance? Let's assume that there's a 50% chance that temperatures in any given month could be above average (it's actually slightly less than 50% because temperatures could be above, equal to or below the average, but it's close enough for our task and if anything will understate things). The equation is basically 2³³⁶. According to Microsoft Excel the probability is 1 in  139,984,046,386,113,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 (note Excel rounds numbers this large). The Windows calculator puts the probability at 1 in 1.3998404638611276315984014253553e+101. Either way that's a very large number.

So what does it mean? It means that it's not random chance that we have had 336 consecutive months with an above-average temperature. Something has caused it and all the evidence indicates that to a large extent it's us.