what are the chances?

I spotted another outing for the old tabloid favourite of a child being born on the same day as its parents last week - this time it was the Ballingalls of Evesham (who all celebrate a birthday on August 1st) who prompted a frenzy of uncritical parroting of nonsense from various drunken hacks, principally the claim that the odds against such an occurrence are 1 in 48 million.

Just to reiterate the point I made at greater length last time, the odds against any random 3-person group (let's say mother, father, baby for the sake of argument, as that seems to have more of a pull for the newspapers, rather than, say, three randomly-chosen people down the pub, although the maths is exactly the same) sharing a birthday are 1 in 133,225. The odds of a random 3-person group all having a birthday on a date you specify in advance are indeed in the neighbourhood of 1 in 48 million, but that's not a situation that arises in any of these tabloid stories. It's easier to see the difference if you just consider one person and reflect on the difference in the answers to the questions "what are the chances of a person being born on May 15th?" and "what are the chances of a person being born on their own birthday?". The odds, given a pair of parents who share a birthday, of their child having the same birthday are - at worst - 1 in 365.

You would expect, if the odds really were 1 in 48 million, for this to be a pretty rare event. You can work out how rare by finding out how many babies are born per day in the UK, and the answer appears to be roughly 2,000 (as compared with just over 350,000 worldwide). So you'd expect one of these events roughly every 24,000 days in the UK; that's about once every 65 years. A very quick Google for similar stories just from the UK reveals the following - there are some variants in terms of which three family members share the birthday, but it doesn't affect the probabilities, and the trigger for the story is always the arrival of a new baby:

• this story from October 2010 about three siblings sharing a birthday - amusingly, the mathematics professor they wheeled in to give an opinion about probabilities gives the right answer, which the headline writer clearly ignored;
• this one from January 2016 about a baby sharing a birthday with dad and grandmother;
• another baby/dad/grandma combo from February 2015.

So, if you include the recent one (but exclude the one in my old post, as that was actually in the USA) that's four in a little under six years, which ought to suggest to the 1 in 48 million folks that they might have made an error somewhere. 1 in 133,225 would suggest an occurrence roughly every 65 days, i.e. once every couple of months.