7.1; Some means and variances of counts

For pea-plants, the alleles at a single genetic locus determine the flower color. We will label the alleles R and r, and suppose that plants with two R alleles (genotype RR) have red flowers, those with two r alleles (genotype rr) have white flowers, and those with one of each (genotype Rr) have pink flowers. In this case, if two pink-flowered pea plants are mated ("crossed" is a better word than "mated" for plants), the offspring plants are independent of each other, and each has red flowers with probability 1/4, pink flowers with probability 1/2, and white flowers with probability 1/4. If a plant with pink flowers is crossed with a plant with white flowers, again the flower color types of the offspring are independent of each other, and in this case each offspring plant has pink flowers with probability 1/2, and white flowers with probability 1/2.

Another trait of pea plants which Mendel considered is whether the seeds are wrinkled or round. Again, this is determined by the alleles a plant has at just one genetic locus. In this case, the allele for wrinkled seeds (W) is dominant to the allele for round seeds (w). This means that plants with both genotypes WW and Ww have wrinkled seeds, and only those with genotype ww have round seeds. If two plants of genotype Ww are crossed, the probability an offspring has wrinkled seeds is 3/4, and the probability it has round seeds is 1/4. Again, the characteristics of different offspring are independent of each other.

NOW YOU CAN DO QUESTIONS 1,2,3 in the next section

Now let's assume what is probably not true -- that is the the genetic locus determining flower color is on the same pair of chromosomes as the one determining seed shape, and in fact fairly close.

First, we are first going to cross a plant from a variety which has, for generations, had red flowers and wrinkled seeds. We know it has genotype RR for flower color, and can be pretty certain it has genotype WW for seed shape. (Let's assume that.) We are going to cross it with a plant with white flowers and round seeds (genotype ww,rr). The offspring plant must have genotype Rr for flower color (pink flowers) and Ww for seed shape (wrinkled). In fact we know more: we know the R and W came from one parent, so are on the same chromosome, and also the r and w on the other chromosome, from the other parent. We write this RW/rw.

Now we cross this plant to another plant with white flowers and round seeds, genotype rw/rw. This other plant will always hand on rw to its offspring, so we don't have to bother about that. Our RW/rw plant can pass on any of the four combinations RW, Rw, rW, rw. If there is no recombination between the two loci, it will hand on RW or rw, each with equal probability. If the is recombination, it will hand on Rw or rW, each with equal probability. Let's suppose the recombination probability between these two loci is 0.2. Then the offspring of our cross are independent of each other, and the probabilities of their types are as follows:

Pink flowers,   wrinkled seeds:  probability (1/2)(1-0.2) = 0.4
Pink flowers,   round seeds:     probability (1/2)(0.2)   = 0.1
White flowers,  wrinkled seeds:  probability (1/2)(0.2)   = 0.1
White flowers,  round seeds:     probability (1/2)(1-0.2) = 0.4