More on birthday coincidences

My daughter's birthday was last week (12 January), so I had a personal interest in today's  Telegraph article about a family with 4 children all having the same birthday - 12 January. 

Family with 4 children - all born on 12 January

Anybody who has read our book or seen our Probability Puzzles page will be familiar with the problem of 'coincidences' being routinely exaggerated (by which I mean probabilities of apparently very unlikely events are not as low as people assume). There is the classic birthdays problem that fits into this category (in a class of 23 children the probability that at least two will share the same birthday is actually better than 50%); but of more concern is that national newspapers routinely print ludicrously exaggerated figures for 'incredible events'*. 

So when I saw the story in today's Telegraph I did what I always do in such cases - work out how wrong the stated odds are. Fortunately, in this case the Telegraph gets it spot on: for a family with 4 children, two of whom are twins, the probability that all 4 have the same birthday is approximately 1 in 133,225. Why? because it is simply the probability that the twins (who we can assume must be born on the same day) have the same birthday as the first child times the probability that the youngest child has the same birthday as the first child. That is 1/365 times 1/365 which is 1/133225. It is the same, of course, as the chance of a family of three children (none of whom are twins or triplets) each having the same birthday. The Telegraph also did not make the common mistake of stating/suggesting that the 1 in 133,225 figure was the probability of this happening in the whole of the UK. In fact, since there are about 800,000 families in the UK with 4 children and since about one in every 100 births are twins, we can assume there are about 8,000 families in the UK with 4 children including a pair of twins. The chances of at least one such family having all children with the same birthday are about 1 in 17.



*Our book gives many examples and also explains why the newspapers routinely make the same types of errors in their calculations. For example (Chapter 4) the Sun published a story in which a mother had just given birth to her 8th child -  all of whom were boys; it claimed the chance of this happening were 'less then 1 in a billion'.  In fact, in any family of 8 children there is a 1 in 256 probability that all 8 will be boys. So, assuming that approximately 1000 women in the UK every year give birth to their 8th child it follows that there is about a 98% chance that in any given year in the UK a mother would give birth to an 8th child all of whom were boys.

Sally Clark revisited: another key statistical oversight

The Sally Clark case was notorious for the prosecution’s misuse of statistics in respect of Sudden Infant Death Syndrome (SIDS). In particular, the claim made by Roy Meadows at the original trial – that there was “only a 1 in 73 million chance of both children being SIDS victims” – has been thoroughly, and rightly, discredited.

However, as made clear by probability experts who analysed the case, the key statistical error made was to consider the (prior) probability of SIDS without comparing it to the (prior) probability of murder of a child by a parent. The experts correctly focused on the critical need for this comparison. However, there is an oversight in the way the experts built their arguments. Specifically, the prior probability of the ‘double SIDS’ hypothesis (which we can think of as the ‘defence’ hypothesis) has been compared with the prior probability of the ‘double murder’ hypothesis (which we can think of as the ‘prosecution’ hypothesis’). But, since it would have been sufficient for the prosecution to establish just one murder, the correct hypothesis to compare to ‘double SIDS’ is not ‘double murder’ but rather ‘at least one murder’. The difference can be very important. For example, based on the same assumptions used by one of the probability experts who examined the case, the prior odds in favour of the defence hypothesis over the prosecution are not 30 to 1 but rather more like 5 to 2. After medical and other evidence is taken into account this difference can be critical. The case demonstrates that, in order to use probabilities in legal arguments effectively, it is crucial to identify appropriate hypotheses.

My published article on this is here.