Chapter 9.2, Problem 25E

GENETICS In a certain species of roses, a plant with genotype (genetic makeup) $AA$ has red flowers, a plant with genotype $Aa$ has pink flowers, and a plant with genotype $aa$ has white flowers, where $A$ is the dominant gene and $a$ is the recessive gene for color. If a plant with one genotype is crossed with another plant, then the color of the offspring's flowers is determined by the genotype of the parent plants. If a plant of each genotype is crossed with a pink flowered plant, then the transition matrix used to determine the color of the offspring's flowers is given by

$\begin{array}{l}\text{Parent}\\ \text{Red}\text{Pink}\text{White}\\ \text{Offspring}\begin{array}{c}\begin{array}{l}\text{Red}(AA)\\ \end{array}\\ \begin{array}{l}\text{Pink}(Aa)\text{or}(aA)\\ \end{array}\\ \text{White}(aa)\end{array}\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{4}& 0\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\ 0& \frac{1}{4}& \frac{1}{2}\end{array}\right]\end{array}$

If the offspring of each generation are crossed only with pink-flowered plants, what percentage of the plants will have red flowers in the long run? Pink flowers? White flowers?