It's an unlikely scenario I know, but if the players on the AFL Seniors lists ever got to talking about shared birthdays I'd wager they'd find themselves perplexed.

As chestnuts go, the Birthday problem is about as hoary as they come. It's about the probability that two randomly selected people share a birthday and its longevity is due to the amazement most people express on discovering that you need just 23 randomly selected people to make it more likely than not that two or more of them will share a birthday. I'll venture that few if any of the 634 players on the current Seniors lists know that but, even if any of them did, they'd probably still be startled by what I'll call the AFL Birthday phenomenon.

In the standard Birthday problem, analysis focusses on the probability of getting a single birthday match, but we could well ask another shared birthday related question, namely, given X randomly selected people, how many pairs of shared birthdays should we expect to find amongst them? Now the Seniors Lists of the 16 AFL clubs each contain about 40 players - a few more in Geelong's case - so let's make X=40 in my question. Then, cranking the analysis we find that the answer is 2 - in other words, we should expect a typical AFL club to have about 2 pairs of shared birthdays amongst the players in its Seniors list.

Well the actual average for the current lists is about 2.3 per club, which is the first hint that something interesting is afoot. Further, as you can see from the following table, 4 clubs have 4 shared birthdays within their Senior player ranks. There's only about a 12% chance of getting this many or more matches amongst a randomly selected group of 40 people, so getting 4 of these in 16 clubs is only about a 9/1 shot. So, why are there so many teams with so many shared birthdays?

The simple answer is that the birthdays of AFL players are not random - they're skewed towards the first half of the year, so much so that almost 60% of them occur in the months January through June as you can see in the following chart.

This bias, it's postulated, is at partly due to the practice of age-based team selection in the junior football ranks, with January 1st as the cutoff date. So, for example, a kid born in January and playing under 13s has almost a full year's extra development on an opposition kid born in December of the same year. As a consequence, the January-born kid will tend to excel throughout his junior years and so will be more likely to enjoy the sport and continue playing. Conversely, the December-born kid will get heartily sick of manning up against faster, stronger and hairier opponents and so will give the game away.

Clumping of birthdays leads inevitably to clumping of shared birthdays, and so we find that, of the 37 shared birthdays amongst AFL Seniors, 21 of them (57%) occur in the first half of the year.

With all this talk of birthdays, I'm sure at least some of you are curious about the number of players who share your birthday, so here's the data: 

So, if you're born March 15th, May 1st, June 7th or June 26th, you share your birthday with 6 AFL Seniors. Again, note that these days are all from the first half of the year.

I'll leave you with one last chart to ponder:

Why so few births on Saturday through Monday and so many on Wednesday and, especially, on Thursday?

This trend has been noted before and the shift away from weekends partially attributed to increased rates of ceasarian section and inductions. That doesn't explain the uneven spread of births from Monday through Friday though. There's another interesting paper on the effects of the Baby Bonus on the timing of births around its implementation date, which makes reference (in footnote 23) to the skew towards Thursday births, but I haven't been able to find any reason posited for the existence of this skew. Answers on the back of a stamped, self-addressed envelope please. 

(Thanks to Jevon, one of our blog readers, for providing the idea for this piece by sending me through details of this recent WSJ article, which alludes to the an analysis of the 2009 AFL Seniors list player birthdays that is the same as I've done here for the 2010 list.)