I\'ve been working a little with M. G. Bulmer\'s Principles of Statistics (Dover

Question

I've been working a little with M. G. Bulmer's Principles of Statistics (Dover, 1979) and cannot see how to properly compute half of one question. This should be a basic probability computation, but I am not getting the same answer as the book's suggestion.

Here is the question (Chapter 2, Problem 2.4): If the three genotypes (i.e., AA, Aa, aa) are all distinguishable, find the probabilities that a) a pair of siblings and b) a pair of unrelated individuals will appear the same when p=q=.5.

The book offers .59 for a) and .38 for b). I can get b correctly, but I don't understand how to properly compute part a).

This does not seem like a complicated exercise!

Explanation / Answer

I got .59. In response to WYSIWYG:

(Note: I'm using A and a for the two alleles. My copy of Bulmer uses B for the second, which makes a little more sense if the alleles are codominant.)

a.) The AA x aa cross should be twice as common as the AA x AA and aa x aa crosses, just as HT is more common than HH or TT when we toss a pair of identical coins. In particular, this cross should occur with frequency 2*(.25*.25)= .125. Same goes for AA x Aa, AA x aa, and Aa x aa.

b.) In an Aa x Aa cross, there is a 3/8 probability that two offspring will have the same genotype. I reasoned as follows:

There is a 0.5 probability that the first offspring will be heterozygous (Aa). In that case, there is a 0.5 probability that the second offspring will also be heterozygous.

There is a 0.25 probability that the first offspring will be homozygous AA: In that case there is a 0.25 probability that the second offspring will also be AA. Likewise for aa.

Summing these: (1/2)^2 + (1/4)^2 +(1/4)^2 = 1/4 + 2/16 = 3/8.

I believe that resolves the issue. Apologies if I've missed something.

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