Single DES and Double Yolks

It was reported in the Daily Mail this week that a woman bought a carton of half a dozen eggs, which she later found to be all double-yolked, as shown below

Since the chances of getting a single double-yolk egg are  around 1-in-1000, then it appears that we have witnessed an extremely rare event, in fact one that is a practical impossibility. If we assume that the likelihood of each egg being double-yolked is independent, then the picture above is conclusive evidence of a  1-in-10^{18} event manifesting. This is a bit less likely than guessing a DES key at random at 1-in-10^{17}. The Daily Mail article goes on to give reasons why this event is not as unlikely as it seems, because on face value, the event is so unlikely that we would never expect to witness it over the lifetime of all eggs that have ever been produced.

Apparently the eggs are all likely to come from hens in the same flock and of the same age which reduces the likelihood to “only” 1-in-729 million. And the occurrence becomes even more likely (or less unlikely – take your pick) when we account for eggs of a similar weight being sorted into the same boxes.

A bit more detail is given over at the wonderful Understanding Uncertainty blog. If the 1-in-10^{18}  odds were correct then given the number of eggs consumed in Britain each year, we are looking at waiting 500 years to see the photo above, so the independence assumption is not plausible. Factoring in that eggs coming from the same group (who may have a propensity for double yolks), packing by weight, noting that some supermarkets can detect and sell double yolked eggs, then the event seems less impressive. But impressive nonetheless!