Markov chains find direct applications in genetics Here is a

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. Rex is a brown dog.

(a) Compute the probability that his grandchild is black.

(b) In a long run, what portion of the offsprings is black?

Solution

A) HERE WE WILL HAVE CASES

CASE 1) CHILD OF REX IS BROWN

0.8*0.2 = 0.16(PROBABILITY THAT CHILD WILL BE BROWN = 0.8 AND OFFSPRING OF CHILD OF BROWN = O.2)

CASE 2 WHEN CHILD IS BLACK

0.2*0.6 = 0.12( PROBABILITY THAT CHILD OF BROWN IS BLACK = 0.8 AND HIS CHILD IS BLACK IS 0.6)

WE CAN ADD BOTH HENCE IT WILL BE 0.28

B) TAKING THE PROBABILITY IN BOTH CASES

THE BROWN WILL BE 1.2

AND THE BLACK WILL BE 0.8

HENCE 0.8/0.12 = 2/3