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Markov chains find direct applications in genetics. Here is an example. An offspring of a black dog is black with probability 0.6 and brown with probability 0.4. An offspring of a brown dog is black with probability 0.2 and brown with probability 0.8. (a) Write the transition probability matrix of this Markov chain. (b) Rex is a brown dog. Compute the probability that his grandchild is black.

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Therefore, the probability that Rex's grandchild is black is between 12% and 32%.

Given that an offspring of a black dog is black with probability 0.6 and brown with probability 0.4, and an offspring of a brown dog is black with probability 0.2 and brown with probability 0.8, knowing that Rex is a brown dog, to determine the probability that his grandchild is black the following calculation must be performed:

  • Sons of Rex =
  • 0.2 black
  • 0.8 brown
  • Rex's grandchildren
  • 0.2 x 0.6 = 0.12
  • 0.8 x 0.4 = 0.32

Therefore, the probability that Rex's grandchild is black is between 12% and 32%.

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