# Colour blindness and Bayes’ theorem

Following from my last post, there’s more to say about colour blindness in my family. This time we’ll look at my side of the family.  My maternal grandfather was red-green colour blind. That means my mother is a carrier of red-green colour blindness, with a “faulty” (well, “alternative” would be better) X-chromosome.  And as a male, it meant I had a 50:50 chance of being red-green colour blind, as I have only one ‘X’, inherited from Mum.  As it turns out, I got the non-alternative version, and so see colours “normally”.

But what about my two sisters?  They both have one ‘X’ inherited from Mum, and one from Dad. But which of Mum’s X-chromosomes did they get? Since Dad wasn’t red-green colour blind, we know they would have got a normal version from him, and so, if one or both sisters inherited Mum’s alternative-X, it wouldn’t show. They both have a 50% chance of being a carrier of colour-blindness, but not colour-blind themselves.

Or not.  The thing is, we actually know a bit more about one of my sisters.  She has two boys, neither of whom is red-green colour blind. That changes this sister’s probability of being a carrier.

How can that be? Surely she had a 50% chance of inheriting the faulty chromosome – how can it matter what happened after her birth?  The key is that we now have more information about her that we don’t have for other sister (who does not have boys).   The fact that both are not colour blind gives a significant steer that she is not a carrier. Because, if she were a carrier, having two non-colour blind boys would be only a 25%  (2 times 50%) chance.  If that is puzzling, think about this extreme, very hypothetical scenario.  Suppose she has another 8 boys. (She shudders to think…) All of them turn out to have “normal” colour vision. Which is more likely then: 1. That they all happened to inherit a healthy version of the X-chromosome rather than an unhealthy version, or 2. That there wasn’t an unhealthy version to inherit in the first place?  The second is far the more likely. Or think about the case where one of the boys actually turned out to be colour blind. This would mean it would be guaranteed that sister was a carrier (100%). So, the fact that they are not, must give some steer the other way.

With just two boys, the chances of 2. being correct are less certain than they would be with ten normal-vision boys, but it is still better than 50:50.   In fact, using Bayes’ Theorem, we can work it out exactly. It turns out that sister is four times more likely to have two healthy X-chromosomes that having one healthy and one faulty. In other words she has a 20%, or 1 in 5 chance, of being a carrier. In  other words, if she had another boy (…she shudders to think…) the child would have just a 10% chance of being red-green colour blind. (If she were a carrier, then the chance child would inherit the faulty X would be 50%, so 50% of 20% gives us 10%).

This tells us nothing about sister two however, who does not have boys. She has a 50% chance of being a carrier.

With colour blindness, these chances are of very little consequence. However, with some of the nastier genetic conditions, these calculations really would be a big deal.

A further practical consequence of Bayes’ Theorem in science is that we always have to interpret new evidence in light of existing evidence. We can’t take one experiment, for example, that indicates that P causes Q, and take it out of the context of a thousand others that indicate that it doesn’t, as so many of our pseudoscience ‘friends’ love to do.