Question

Let the initial probability vector in Example 3.10.6 be\(v = \left( {\frac{1}{{16}},\frac{1}{4},\frac{1}{8},\frac{1}{4},\frac{1}{4},\frac{1}{{16}}} \right)\)Find the probabilities of the six states after one generation.

Short answer

Probabilities of the six states after one generation.

Step-by-step solution

Given information

A botanist is studying a certain variety of plants that is monoecious.

She begins with two plants1 and 2 and cross-pollinates them by crossing male 1 with female 2 and female 1 with male 2. producing two offspring for the next generation.

Computing the transition probability matrix.

Suppose that gene has two alleles, A and a.

An individual's genotype will result in any of the three combinations. AA, Aa, or aa.

When a new individual takes birth, it receives one of the two alleles from the other parent. the two offspring get their genotypes independently of each other.

Let the states of this population be the set of genotypes of the two members of the current population. We will not distinguish the set\(\left\{ {AA,Aa} \right\}\).

There are six states:

\(\left\{ {AA,AA} \right\},\left\{ {AA,Aa} \right\},\left\{ {AA,aa} \right\},\left\{ {Aa,Aa} \right\},\left\{ {Aa,aa} \right\},\left\{ {aa,aa} \right\}\)

For each state, we can compute the values for the probabilities that the next generation will be in each of the six states.

Therefore computing the probability matrix as

\(TP = \left[ {\begin{array}{*{20}{c}}1&0&0&0&0&0\\{0.25}&{0.5}&0&{0.25}&0&0\\0&0&0&1&0&0\\{0.0625}&{0.25}&{0.125}&{0.25}&{0.25}&{0.0625}\\0&0&0&{0.25}&{0.5}&{0.25}\\0&0&0&0&0&1\end{array}} \right]\)

The rows and columns are represented in the order

\(\left\{ {AA,AA} \right\},\left\{ {AA,Aa} \right\},\left\{ {AA,aa} \right\},\left\{ {Aa,Aa} \right\},\left\{ {Aa,aa} \right\},\left\{ {aa,aa} \right\}\)

Computing the probabilities of the six states after one generation

The initial probability matrix be

\(v = \left( {\frac{1}{{16}},\frac{1}{4},\frac{1}{8},\frac{1}{4},\frac{1}{4},\frac{1}{{16}}} \right)\)

To calculate the probability of the six states after one generation.

This is equal to the multiplication of the matrix TP and v, as shown below:

\(\left[ {v \times TP} \right]\)

\( = \left( {\frac{1}{{16}},\frac{1}{4},\frac{1}{8},\frac{1}{4},\frac{1}{4},\frac{1}{{16}}} \right) \times \left[ {\begin{array}{*{20}{c}}1&0&0&0&0&0\\{0.25}&{0.5}&0&{0.25}&0&0\\0&0&0&1&0&0\\{0.0625}&{0.25}&{0.125}&{0.25}&{0.25}&{0.0625}\\0&0&0&{0.25}&{0.5}&{0.25}\\0&0&0&0&0&1\end{array}} \right]\)

\(\left[ {v \times TP} \right] = \left[ {0.1406\;\;0.1875\;\;0.03125\;\;0.1875\;\;0.1406} \right]\)