Uncertainty about Parental Genotypes

A common problem in genetics is the need to calculate the probability of progeny genotypes or phenotypes when you are not certain about the genotype of one or both parents. It is very important to learn how to deal with this sort of situation and so it is a separate topic here. It is also one of the more difficult parts of solving basic genetics problems and so the more you practice it the better.

An example is shown in the pedigree below. The trait is obviously recessive (can you see why?). What if you are interested in knowing the probability that II4 and II5 have child with this trait. Given their family history we can see that it is possible that they are both carriers. How do we answer this question when we are uncertain about their genotypes?

We will start with a simpler example. Consider the situation when you have performed the cross Aa x Aa and taken one of the progeny that has a dominant phenotype. You then test cross this - that is, cross it to an aa individual - and get one progeny. What is the probability that this progeny is aa?

The difficulty in answering this is that you don't know whether you chose an AA progeny or an Aa individual and, of course, the genotype of this individual is will affect the answer. Despite this, you can still answer the question.

You can do this by applying two things that you do know:

1 - You know the probability that the one you chose is AA, and the probability that it is Aa.

2 - You know what you would get in each case. If you chose an AA individual then you will get 100% Aa from the cross. On the other hand, if you chose Aa then you would get 50% Aa and 50% aa (and, thus one half recessive phenotype).

What we describe in (2) are the Conditional Probabilities of progeny genotypes (for example, the probability of Aa IF the parents are Aa and aa). You could leave everything as Conditional probabilities, but it is possible to combine the two items listed above to come up with overall progeny genotype probabilities that account for parental uncertainty AND conditional probabilities. This will then be your predictions for progeny genotypes. (Also, your professor might not accept just Conditional probabilities!)

The way to combine these two points is to take into account every POSSIBLE cross. You can easily calculate the probability of each cross and the probability of each progeny if that were indeed the cross. The overall progeny probabilities are then calculated by applying the OR rule to the crosses. This sounds a bit difficult and it is much easier to see if you lay them all out in a Table as follows:

The Pr(Mating) column is calculated using the AND rule. It is really the probability that the two parental genotypes, given in the first column, both occur. Since we know that the Female is aa (P = 1), this reduces to the probability of the Male genotype (e.g. 2/3 x 1 in the first row).