headline of the day

I suppose it is a bit like harpooning a walrus in a bathtub, spearing an elephant seal in a wash-basin, torpedoing a manatee in a paddling pool, or any other even easier variant of shooting fish in a barrel, but I offer for your enjoyment anyway this splendid example of tabloid innumeracy - here's a heart-warming story in the Daily Mail about a couple who share a birthday (July 19th, since you ask) who have now had a son who - wait for it - was born on the same date! Now I'll grant you this is fairly unusual and a nice little conversation piece at dinner parties before the wine kicks in and someone comes out with something toe-curlingly racist, but perhaps not as noteworthy as the Mail headline suggests:

A quick moment of reflection on the number of days there are in a typical year, and the obligation for any baby, I don't care who your parents are, to be born on one of them should prompt the amateur mathematician to smell a bit of a rat here. You can see where the Mail have got that number from - all three of them have the same birthday, there are 365 days in a year (let's forget February 29th for the moment, just to keep things simple), therefore the probability of all three having the same birthday is one in (365 x 365 x 365 = 48,627,125). Right? Well, no.

For starters, the odds of two people sharing a birthday are one in 365, not one in (365 x 365 = 133,225) - the point being it doesn't matter which day the first person's birthday is on, the second one just has to match it, and has a one in 365 chance of doing so. Add a third person into the mix and the odds of them having the same birthday (and therefore that date being shared by all three) just adds another factor of 365 to the mix, i.e. the probability of the whole thing - boy meets girl, discovers they share a birthday, impregnates her, baby is born on their mutual birthday - is one in 133,225. No millions involved, still less 48 of them.

However, we're not interested in the probability of the whole shooting match, just the baby bit, since the parents getting together and sharing a birthday bit has already happened. The odds of this are going to be at best (or worst, depending how you look at it) one in 365, and they are only this until you know you're pregnant - once you know this the range of available dates for the baby's birthday narrows considerably. Let's assume that we require the baby to be born healthy and normal in all the usual respects - if you're in a first world country with decent medical care this probably means a window of 24-40 weeks after conception as possible birth dates (and the lower end of that could be touch and go survival-wise). That's 16 weeks, i.e. 112 days. So, unless you already know you're out of the game (if the Parkers had conceived in August, for instance, there's absolutely no way the baby could be born on July 19th), you're actually only looking at odds of one in 112 or thereabouts. Less of a snappy headline for a news story, I'll grant you.

Probability is inherently counter-intuitive and hard to grasp, though, even for quite clever people, never mind Daily Mail readers. The notion that throwing ten heads in a row doesn't make a tail more likely next time is a hard one for some people, people whose heads would probably explode if confronted with either the birthday problem or the Monty Hall problem, both seemingly simple situations where the obvious "common sense" answer is wrong.

Interestingly, this article applies the birthday problem's conclusion (i.e. that you only need a group of 23 people to make it more likely than not that two of them will share a birthday) to the handy existing data set of all the US Presidents (of which there have been 43, making it over 90% likely that there'll be a match) - sure enough James Polk and Warren Harding share a birthday, November 2nd. QED!