On the likelihood of having all three of your kids share a birthday

In his Guardian column yesterday, Dr Ben "Bad Science" Goldacre schooled us (and the Daily Express) on some basic statistical numeracy:

Often one data point isn't enough to spot a pattern, or even to say that an event is interesting and exceptional, because numbers are all about context and constraints. At one end there are the simple examples. "Mum beats odds of 50 million-to-one to have 3 babies on same date" is the headline for the Daily Express on Thursday. If that phenomenon was really so unlikely, then since there are less than a million births a year in the UK, this would genuinely be a very rare event.

Their number is calculated as 365 x 365 x 365 = 48,627,125. But in reality, of course, it's out by an order of magnitude: one in 50 million are the odds of someone having 3 siblings sharing one particular prespecified birth date that the editors of the Daily Express sealed in an envelope and gave to a lawyer 50 years ago. In reality there is no constraint on which day the first baby gets born on, so after that, the odds of two more babies sharing that birthday are 365×365=133,225. And they might even be a bit lower, if you two feel friskier in winter and have more babies in the autumn, for example.

Then there is the context. Living on your street, hanging out with the people from work, it's easy to miss the sheer scale of humanity on the planet. In England and Wales there were 725,440 births last year. From the ONS Statistical Bulletin "Who is having babies" 14% were third births, and another 9% were fourth or subsequent births. So there are 102,000 third children born a year, 167,000 third or more-th children, and if we include the rest of the Kingdom there are even more, so on average, three shared birthdays will happen once or twice a year in the UK (although to be written about in the Express it would need to be a birth within a marriage, making 55,000 chances a year, or once every two years)