Using Bayesian statistics and inaccurate genetics to guess the eye-colour of my next child

My next child is currently in my uterus. I’ve recently had a scan and found out a little bit about its genetic make-up – that bit which has turned it into a girl. That’s the only characteristic I’ll know about for a while, and I can only speculate on the rest. So I’ve decided to speculate on the colour of its eyes.

In GCSE Biology, which I took in 1994, I learned the following model for how genes determine the colour of eyes. As a simplification, it pretends there are only two types of eyes: brown and blue. and that the eye-colour a person has is controlled by a pair of genes – one gene inherited from their mother and the other from their father. The gene determining eye colour can come in one of two variants, a brown variant, which is labelled ‘B’, and a blue variant, labelled ‘b’.

If both genes in the pair are b (written as bb), then you get blue eyes. If they’re both B (written as BB), you get brown eyes. And the B gene always overrides the blue gene, so if you’ve got one B gene and one b gene (Bb) you get brown eyes too. Because the brown-eyed gene always gets its way whenever it’s present, it’s known as a dominant gene, while the blue-eyed gene is called recessive because it meekly submits to its partner’s wishes.

If you know the genes of a mother and father, you can work out what genes their children are likely to have. If both parents have blue eyes then they’re clearly bb, and their children will get a b from both of them, so they’ll all have blue eyes too. If the mother is Bb, and the father is bb, then half the children will get a B from the mother and a b from the father and be brown-eyed Bbs, and half the children will get a b from both parents and be blue-eyed bbs.

As I mentioned before, this model was taught to me in the nineties and is an oversimplification, as many genes have been shown to contribute to eye colour. I’m not sure that it’s still taught today: on BBC GCSE Bitesize, the less happy examples of cystic fibrosis and Huntington’s disease are used to illustrate the theory of dominant and recessive genes. Until today I believed that to claim that the blue/brown eyed model was correct would be an act of disrespect towards the British Royal Family, as I had thought that the Duke and Duchess of Cambridge had blue eyes and their son Prince George had brown eyes, but now having examined many pictures of the Duchess of Cambridge I can’t decide if her eyes are brown or blue, and on revisiting a discussion page on the subject I have realised that the image of the indisputably blue-eyed woman displayed there is not in fact of the Duchess of Cambridge but a professional look-alike named Heidi Agan. Anyway, I’m going to stick the single-gene theory because nothing in my personal experience has happened to contradict it, and it makes the maths simple.

I have brown eyes, but I don’t know whether I’m a BB or a Bb. Both my parents have brown eyes. I’m assuming that my mother is a BB, as she comes from Thailand and blue eyes are rare or absent in that population. I know that my father is a Bb, as my niece (my brother’s child) has blue eyes, so my brown-eyed brother must be a Bb. My husband also has blue eyes, so he’s a bb. In fact I am the only person in my and my husband’s family whose genome with respect to eye-colour isn’t fully known.

If I were to give birth to a blue-eyed child, then it would be clear that I was a Bb, and that the next child would have a probability of 50% of also having brown eyes. If I’d already had 20 brown-eyed children with my blue-eyed partner, then I’d be pretty confident that I was a BB, as the probability of having 20 brown-eyed children if I were Bb would be $1/2^{20}$, or about 0.000001. So I wondered what the chances of the next child also having brown eyes are, given that already have one child, whose eyes are brown.

This is one of the rare opportunities in my life to apply Bayes’ Theorem. (It’s rare not because Bayes’ Theorem isn’t useful, but because I’m not involved in statistics.) Bayes’ theorem can be written as

$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

Explanation helps, though. In statistics, the raw materials are observations, such as having a child with brown eyes, and what we want to determine is the underlying probabilities. In the formula above B represents an observation, and A is an underlying probability distribution. In the context of this post, A is the probability that I am BB (say) and B is the evidence that I have a child with brown eyes.

• P(A) is the a priori probability – the probability of being BB before the evidence of a brown-eyed child came along
• P(B) is the probability of a brown-eyed child depending on the a priori probability
• P(B|A) is the probability of a brown-eyed child assuming that A has happened – i.e. that I’m BB.
• The result, P(A|B), is the probability that I’m BB based on the evidence.

So let’s evaluate those first three points in turn.

P(A)

My parents, being BB mother and a Bb, would have expected to have children who were half BBs and half Bbs. So let’s put the a priori probability, P(A) down as $\frac{1}{2}$.

P(B)

Assuming the a priori probability, the chance of my having a brown-eyed child is

$\frac{1}{2} \times \text{probability of brown eyed child with BB} + \frac{1}{2} \times \text{probability of brown eyed child with Bb}$

because I have half a chance of being BB and half of being Bb.

This equals

$\frac{1}{2} \times 1 + \frac{1}{2} \times \frac{1}{2}$

which equals $\frac{3}{4}$

P(B|A)

This is, assuming I’m BB, what is the probability of having a brown-eyed child? It’s 1, because all offspring will be Bb.

So, plugging these three values back into the Bayes’ theorem formula:

$P(A|B) = \frac{1 \times \frac{1}{2}}{\frac{3}{4}} = \frac{2}{3}$

So the evidence of having a single child with brown eyes with a blue-eyed partner has made the probability of my having BB rather than Bb jump from $\frac{1}{2}$ to $\frac{2}{3}$.

Now I know the probability of being BB, I can work out the probability of having a brown-eyed child: it’s

$\frac{2}{3} \times 1 + \frac{1}{3} \times \frac {1}{2} = \frac{5}{6}$

I find it interesting to think about how much probability is sometimes just a question of perspective. Both my genome and my child’s are already established – we just haven’t examined our DNA to find out about it – so in a way there’s no probability about it. I said earlier that my parents, as a BB/Bb couple, would have expected to have half BB and half Bb children, but in fact their genomes were not revealed until it became clear that the eyes of their first child would never change from their original baby-blue.

Seven Types of Ambiguity – in Adverts

I am not very good at reading poetry, but I suspect it says important things in a way that could not otherwise be said, and that’s why I’m attracted to books which contain a small amount of poetry alongside a lot of prose telling me what to think about it. One such book is Seven Types of Ambiguity, published in 1930, when its author, William Empson, was 24. (Which is the age I learned how to use a semi-colon.) Empson studied two years of a maths degree before switching to English Literature, so I regard the book as a primer in ‘English for Mathematicians’, especially as he peppers his writing with maths-friendly phrases such as  ‘one must assume that n+1 is more valuable than n for any but the most evasively mystical theory of value.’

Empson’s thesis is that because poetry is concise and aims to say a lot of things, it tends to contain phrases which can be interpreted in several ways: i.e. ambiguous phrases. (Another category which has the same constraints and goals is names for pop bands. Consider The Wanted: a dangerous band of outlaws or a group of desirable young men? I presume, without having done much research into the matter, that both implications are relevant to the impression the band intends to create.) His book identifies seven sub-categories of ambiguity, and lists examples drawn from poetry under each.

I thought it might be interesting to take Empson’s original seven types, and see if I could find examples drawn from the field of advertising slogans to fill them. Like poetry, advertising slogans aim to say a lot of things using few words. To adapt a quote from a book:

The demands of metre catchiness and the cost of air time allow the poet copywriter to say something which is not normal colloquial English, so that the reader thinks of the various colloquial forms which are near to it, and puts them together, weighting their probabilities in proportion to their nearness. It is for such reasons as this that poetry advertising slogans can be more compact, while seeming to be less precise, than prose.

As it turned out, I couldn’t think of an example for each type. But here are the list of types, together with examples where I could find them.

Type 1 – Details are effective in several ways (comparisons with several points of likeness, antitheses with several points of difference)

I didn’t find rich pickings in this space. But here’s an antithesis with several points of difference:

Between love and madness lies Obsession

Calvin Klein

Love (assuming sexual love is meant) is sexually exciting, self-denying, and personal; madness is usually impersonal and involves one’s perceptions being in disagreement with the norm. The audience is free to choose which of these properties obsession has: if you take all the properties of sexual love together with the warped perception of madness, you get stalking; or, taking both properties of madness and only the self-denying property of love, you get something more like academic fervour – an interest in insects so strong that you forget to eat, say. Although intense sexual desire is probably the image that the marketers would most like to purvey, the vagueness of the definition given for obsession may mean the audience identifying with the slogan, and therefore interested in the product, is wider than it would be if more precision were used.

I should say that the definition of this type is wider than the truncated version I have described here. There may be more examples than I have the power to categorise. Perhaps You’re worth it (L’Oréal) qualifies because it leaves open exactly what worth and it are.

Type 2 – two or more alternative meanings are fully resolved into one

I am going to argue that my favourite advertising slogan of all time, Zanussi’s The Appliance of Science [video link], falls into this category. Again, there aren’t many examples of this type – it seems that copywriters want to use their words to say multiple things in one sentence, rather than the same thing twice. There are two ways that The Appliance of Science can be parsed:

1. Interpreting Appliance as a piece of kitchen white-ware, the slogan means ‘We create good fridges, dishwashers and cookers using the results of scientific investigations’
2. Interpreting Appliance as equivalent to ‘application’, the slogan means ‘We apply the results of scientific investigations.’

If you accept that the subject in interpretation [2] is implicitly kitchen white-ware, both parsings amount to the same thing. And the slogan rhymes! Amazing.

Type 3 – Two apparently unconnected meanings are given simultaneously

This type, which allows you to say two relevant things using one set of words, is very well represented in the advertising world. Almost all punning advertising slogans fall into this category. I’ll start with my second favourite slogan of all time, which is sadly under-exposed, being a private employee induction slogan for my current employer, Tesco.

A chance to get on

Tesco

This five word sentence contains an impressive three meanings.

1. Taking get on to mean ‘ascend’, it is a chance to get on the vehicle that is employment at Tesco (whether that’s a carriage to the stars or a bus to a business park in Welwyn Garden City.)
2. Taking get on to mean ‘associate amicably with people’, it is a chance to work in a constructive manner alongside colleagues. (Tesco prides itself, in my view with justification, on the supportive environment it offers employees.)
3. Taking get on to mean ‘achieve career success’, it is a chance to, well, achieve career success.

The next example is another favourite of mine because it brazenly addresses the squeamish subject of what the product actually does.

Simplicity press-on towels (1989) [video link]

From the context of the advert, which features women rollerskating and cycling, it seems clear that the primary meaning of the slogan derives from its usual idiomatic meaning – to take something in your stride is to experience something negative it without it affecting you. Which is to say that if you use Simplicity press-on towels, you will feel able to get on with life when you have your period. But the more literal meaning of ‘stride’ is to do with the separation of your legs – which is, of course, where the press-on towel goes.

Type 4 – Alternative meanings combine to make clear a complicated state of mind in the author

I haven’t been able to think of any good examples for this type, and I think it’s because adverts aren’t usually trying to convey a complicated state of mind. I thought I might be able to mine something from charity or public health slogans, which aim to deter bad things rather than promote good things, but the ones I can think of are pretty unambiguous e.g. Oxfam’s Make Poverty History, Save the Children’s No child born to die. Perhaps the best example is the below, which I think I’ve seen on a British advert from the government-sponsored Think! road safety campaign but can’t find anywhere.

Here, the D, the I and the E of ‘Drive’ in the slogan ‘Don’t drink and Drive’ are highlighted so that it simultaneously says ‘Don’t drink and DIE’. This could be said to represent a complex state of mind in the author, because drink-drive adverts are expected to emphasise the harm that can be done to passengers and other road users, but this one also points out that drunk drivers also risk harming themselves.

The example is a bit of a cheat, though, because it relies on colouring in certain letters, so it’s really two slogans instead of one.

Type 5 – Author is discovering their idea in the act of writing

I also haven’t been able to think of any examples of this type of ambiguity. The problem of type 4, which is that copywriters aren’t usually trying to convey a complicated state of mind, also affects this type, because this type suggests at least two states – the state of not having the idea, and the state of having it. But in type 5 the problem is compounded by the lack of space – you need more words than slogans afford to go from one state to another.

Type 6 – Forcing the reader to invent interpretations

In this type a statement says nothing, so the reader is forced to invent interpretations for themselves. There are quite a lot of examples of this type in advertising.

The claim is clearly not intended to be believed. However, it does allow the viewer to imagine strength and Scottishness diffusing out of a mouthful of Irn-Bru into their body.

Impossible is nothing

This slogan qualifies on the grounds of grammatical delinquency. The viewer may correct it to ‘Nothing is impossible’, or may turn Impossible into a noun, meaning either ‘the set of things which are impossible’ or impossibility itself.

I’m not sure I understand this type, so I’m going to quote its definition in full:

An example of the seventh type of ambiguity … occurs when the two meanings of the word, the two values of the ambiguity, are the two opposite meanings defined by the context, so that the total effect is to show a fundamental division in the writer’s mind.

Taking this definition on its own, I would claim that

Tax doesn’t have to be taxing

qualifies. Taking taxing literally, as in ‘pay money’, tax, obviously, is taxing, so the statement is a contradiction, but interpreting taxing to mean ‘be hard work’ the statement means that ‘it’s not much administrative effort to pay tax’, which is the intent of the slogan.

If this were the extent of this type then the examples in the seventh chapter of Seven Types of Ambiguity would all be the type of puns that Idlewild are fond of making, such as I think you’re young without youth (American English) and And did I hear you sing / That we exist without existing (I understand it). However, the examples in the chapter are statements which say the opposite of what they mean. The true meaning of the statements is has to be deduced from the context. The nearest things to this I can think of in adverts are those adverts which don’t explicitly endorse the products that they’re promoting:

You either love it or hate it