Posted on December 5, 2012.

Unless you've been living under a rock recently, you are well aware that Kate Middleton is pregnant and suffering from a severe form of morning sickness called hyperemesis gravidarum. Hereafter, for simplicity, I'm just going to refer to this as HG. This has been widely reported in the press as being linked to having twins, which is true. But since then, the odds at Paddy Power of the pregnancy being twins has shrunk from 50-1 to a mere 8-1. This didn't seem right to me, so I've had some fun with Bayes' theorem to see what the numbers should be.

## Bayes' Theorem

$$ P\left(T|HG\right) = \frac{P\left(HG|T\right)P\left(T\right)}{P\left(HG\right)} $$

This subtle theorem tells us how we can incorporate all the knowledge we have in order to answer a question. A full discussion of how what Bayes' Theorem is and how we might arrive at the above expression is over at Wikipedia, but I'm going to forgo the tradition explanation of 'Priors' and 'Evidence' here - it's there if you're interested!

The question in this case is 'What is the probability (P) that Kate Middleton is having twins, given that she has hyperemesis gravidarum', and this is written more formally on the left hand side of the above expression. The syntax I've used here is 'T' for 'Twins' and 'HG' for hyperemesis gravidarum. The pipe is standard notation for 'given that'. Now that we know how to read these probabilities, we can easily understand the right hand side:

- \(P\left(HG|T\right)\) - the probability that one had HG, given that you had twins.
- \(P\left(T\right)\) - the probability that one had twins.
- \(P\left(HG\right)\) - the probaility that one had HG.

So 'all' we need to do is come up with these probabilities. Fortunately, we have the entire history of human scientific endeavour to draw on, and so these numbers are reasonably straightforward to locate (at least, if you have an academic internet connection with access to journals...)

## \(P\left(HG|T\right)\)

What I thought was going to be the hardest number to find actually turned out to be fairly straightfoward. The best paper I could find about this is Sex Ratio and twinning in women with hyperemesis or pre-eclampsia, which is a Danish study from 2001. They conducted a study on all births in Denmark between 1980 and 1994. They found 143 births of twins to those with HG, and 9140 sets of twins to those without. They also found 700 births of twins to those with pre-eclampsia. We therefore conclude $$P\left(HG|T\right) = \frac{143}{9140+143+700} = 0.0143. $$

## \(P\left(T\right)\)

This number is actually highly determined by both family history, and age. Fortunately, neither Will or Kate have a significant history of twins in the family (my source for this is currently a Daily Mail article, who I don't want to link; a more reputable source would be welcome), so we can discount that effect.

As the age of the mother increases, so does the incidence of twins. The best (quantitative) reference for this I could find was Association of Maternal Age and Parity with Birth Weight, Sex Ratio, Stillbirths and Multiple Births, which gives a figure for the chance of twinning in a mother aged 30-34 of \(P\left(T\right) = 0.033\).

*It is sad to note here that the chance of stillbirths is actually higher still, at over 5%.
I had no idea that number was so high. As a male in my mid-twenties with no intention to have
a child anytime soon, there's clearly a lot about the process I'm totally unaware of.*

## \(P\left(HG\right)\)

We return to the first paper here, and easily extract the relevant numbers: 768040 births from healthy mothers, 6084 births with mothers suffering hyperemesis, and 24065 suffering pre-eclampsia. This gives us

$$P\left(HG\right) = \frac{6084}{768040+24065+6084} = 0.0076.$$

## Conclusion

So, how likely is a set of royal twins? Substituing in the above numbers to Bayes' theorem:

$$ P\left(T|HG\right) = \frac{0.0143\times0.033}{0.0076}$$

which gives us a probability of 5.6%, or about 1-in-18. This is nearly twice the probability of her having twins had she not suffered from HG, but is a long way from the 1-in-8 you can get from the bookies.

Of course, the bookmakers will take other factors into account - most notably rumour and hearsay - which could influence the odds, and they will certainly want to err on the side of them making money. The maths for the moment, however, says hold on to your cash.