12.19 Assume it is known that the color of eyes of humans is determined by a particular gene. We assume that this gene has three alleles B,

b, and g, where B is dominant and both b and g are recessive. A person with genotype containing the dominant allele B has brown eyes. A person with both recessive alleles containing allele b (bb or bg) has blue eyes. Finally, a person with genotype gg has green eyes.

Every child takes one allele at random from each of the parents.

Suppose, in a remote village where all the villagers have blue eyes (genotype bb), one day an escaped fugitive with brown eyes (genotype BB) takes refuge. He marries one of the villagers and they have one child. This child marries further in the village and the story continues every time having exactly one descendant. Write down the transition probability matrix for the successive generations of children. Calculate the expected number of generations until a descendant with blue eyes appears.