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In a recent exam our teacher gave us the following questions:

Assume that D, E, F, G, H, and I are autosomal genes on different chromosomes. From the mating (parent A) DdeeFfGGHhIi x (parent B) DdEEFFGgHhii: a. What is the probability that one of the offspring will have the genotype DdEeFFGghhIi? b. What is the probability that one of the offspring will be heterozygous for each allele?

Is there any way of answering these questions without drawing a super large Punnett square? Thanks in advance for helping out.

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You need to understand that each locus follows a law of independent assortment. This means, each probability of combinations are multiples of the probability of each element. For example, Dd x Dd goes by

DD (1/4) : Dd (2/4) : dd (1/4)

and Ff x FF goes by

Ff (2/4) : FF (2/4)

If the case is DdFf x DdFF, it is

DDFf (2/16) : DDFF (2/16) : DdFf (4/16) : DdFF (4/16) : ddFf (2/16) : ddFF (2/16).

You can verify my countings by making a Punnett square. However, notice on the first case, the probability of DDFf is basically DD * Ff itself, 1/4 * 2/4 = 2/16. So as others.

To solve your problem, use Punnett squares (or just directly doing it with no issue) for each of the locus. Like this:

  1. Dd x Dd -> DD : Dd : dd
  2. ee x EE -> Ee
  3. Ff x FF -> Ff : FF

What is the probability of having DDEeFF?

DD * Ee * FF -> 1/4 * 4/4 * 2/4 = 8/64

Try this on the question asked. Remember, solve smaller Punnett squares, and multiply for your desired genotype

Good luck!

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  • $\begingroup$ The hint that the loci independently assort comes from being on separate chromosomes. If they were close together on the same chromosome we would strongly suspect dependence. If the on different chromosomes was absent, I would suspect that the exam wants me to assume independence but I might be temped to include Fre'chet bounds anyway. $\endgroup$
    – Galen
    Commented Apr 30, 2022 at 16:26
  • $\begingroup$ I see, @AgnesianOperator. In the case of nonindependently assorting genes, we can opt to genetic linkage. If so, there would be usually two types of questions, First, calculating each probability with genetic distance information in centimorgans. Second, determining the genetic distances between loci from an observed distribution table. Fre'chet's inequalities are something interesting, but I have not been updated that it is used in genetic mapping or genetic analysis. I would search for more if I missed it. $\endgroup$
    – cc12amu
    Commented Apr 30, 2022 at 17:45
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    $\begingroup$ Yes, using centimorgans is fine if you know the conditional probability of recombination given centimorgans. Haldane's map is an example that might be suitable for a student exercise. I think we agree that in an empirical research setting the data distribution must be considered. $\endgroup$
    – Galen
    Commented Apr 30, 2022 at 18:27
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    $\begingroup$ The Fre'chet bounds work for any probability distribution, so they certainly apply whenever we are considering recombination probabilities. Their main limitation is that they can be conservative; the price of assuming very little. If you can make more assumptions about the probability function, then sometimes you can put tighter bounds on the joint distribution. $\endgroup$
    – Galen
    Commented Apr 30, 2022 at 18:29

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