Recently a weather website meteorologist noticed that in the United States that for 13 straight months in a row, the average temperature was in the upper third of all temperatures in the historical record. There was a heat wave. Heat waves are news worthy, so the meteorologist determine the odds of that happening.
There is one chance in three of the temperature being in the upper third and 13 months of 1 in 3 chances would result in 1/3 raised to the 13th power. That is a big number, like 1 in 1.6 million. Oh my God! That is a huge long shot that just came in! What can it mean?
Well, any month has a 50-50 change of being above average, 13 months of 50% chance is 1/2 raised to the 13th power or 1 in 8196, that is now were near as exceptional as 1 in 1.6million, but those are pretty long odds at first glance.
What are the odds that one year will be warmer than the average? 1 in 2, there is a 50-50 chance of one year being warmer than another. What are the odds of one year being the warmest in 100 years? 1 in 100.
If the instrumental record is 100 years long and this past 13 months, which is just long than one year, the odds are about 1 in 100 that this year may be the warmest of the entire temperature record.
So how did a 1 in 100 shot get boosted to 1 in 1.6million? Think about the above average year. Average is a set without any elements. If you can't be average, you have to be above or below average. The upper third is a set than can contain elements. If you give average a range instead of a hard limit, you get a more realistic probability. If average is a range of plus or minus one third of the range, then being in the upper third of the range would nearly the same as being above average. You have one chance in three each of being below average, average or above average. One chance in three but now the three is different. The closer the value in the upper third is to the "average" boundary the less likely it is to be exceptional. Since you have 1 in 100 chance of the year being exceptional, there are greater odds that each month of that year would be exceptional if the year is exceptional.
So you can consider that in a warm year, the odds of any one month being in the lower 1 third are smaller that the odds of it being in the upper one third. Once you remove the likelihood of record cold in a record warm year, you approach the month being either above the warm year average or below, 1 in 8196 if the year were a 1 in 100. If not, for a 100 year record there are 12 months per year resulting in 1 chance in 1200 of that month being a record month and 1200/3 or 1 chance in 400 of any month being in the upper one third of the record. 1/100 time 1/400 results on 1 in 40,000 for any one year. 13 times 40,000 equals 520,000 or one in about a half million of any string of 13 months being in the upper third of a 100 year record if the year where not exceptional, but an exceptional year or an exception month in that year increases the odds of other months adjacent to that month also being exceptional. So a simple approximation would be 520000 for a purely random occurance divided by 100 or 1 chance in 52,000, that 13 consecutive months in a record year would be in the upper one third of the entire record if the year where a record.
You can fine tune that, but one chance in 52,000 for 13 consecutive months being in the upper one third during a record year for a record 100 years long is close enough for government work. The one in 100 is the more important statistic anyway. Now if you where to pick the 13 months before hand, the odds of your winning ticket would be 1 in 1.6 million :)