Chapter 10.2, Problem 28E

Consider beginning with an individual of known type and mating it with a hybrid, then mating an offspring of this mating with a hybrid, and so on. At each step, an offspring is mated with a hybrid. The type of the offspring can be modeled by a Markov chain with states AA. Aa, and aa.

a. Find the transition matrix for this Markov chain.

b. If the mating process in Exercise 27 is continued for an extended period of time, what percent of the offspring will be of each type?

27. A genetic trait is often governed by a pair of genes, one inherited from each parent. The genes may be of two types often labeled A and a. An individual then may have three different pairs: AA. Aa (which is the same as aA), or aa. In many cases the AA and Aa individuals cannot be otherwise distinguished: in these cases gene A is dominant and gene a is recessive. Likewise, an AA individual is called dominant and an aa individual is called recessive. An Aa individual is called a hybrid.

1. a. Show that if a dominant individual is mated with a hybrid, the probability of an offspring being dominant is 1/2 and the probability of an offspring being a hybrid is 1/2.
2. b. Show that if a recessive individual is mated with a hybrid, the probability of an offspring being recessive is 1/2 and the probability of an offspring being a hybrid is 1/2.
3. c. Show that if a hybrid individual is mated with another hybrid, the probability of an offspring being dominant is 1/4, the probability of an offspring being recessive is 1/4, and the probability of an offspring being a hybrid is 1/2.