I need to rethink my response!
The coat of a certain breed of cat can be either black, white or grey.
When a black-coated male is crossed with a white-coated female, the offspring kitten has a 50% chance of having a black coat, and a 50% chance of having a white coat.
When a white-coated male is crossed with a white-coated female, the offspring kitten has a 20% chance of having a black coat, a 60% chance of having a white coat and a 20% chance of haing a grey coat.
When a grey-coated male is crossed with a white-coated female, the offspring kitten has a 10% chance of having a black coat, a 45% chance of having a white coat and a 45% chance of having a grey coat.
(a) Assume that each successive generation of cats is produced by crossing all cats with a white-coated female. Write down the transition matrix for this Markov process.
(b) If initially there are equal numbers of black-coated and white-coated cats, and no grey-coated cats, what percentage of the next generation will be grey-coated?
(c) In the long run, what percentage of cats will have black, white and grey coats?
there should be a vector that multipies with the transition matrix
P . x1 = x2
but i wasn't sure what that vector should be.
if in the case of your matrix,
i assume the vector equation in question (b) is
0.5 ? as there are equal numbers of black and white cats ? correct me if i am wrong
0.5
0
Either way .. I'm not too sure about b, but, Long term transition probabilities I'm fairly sure simply involve taking your matrix P to a high power... I think you'll find that for B, W, and G respectively you'll get something like 25.7% for B, 54.45% for W, and 19.8 % for G
There's a bit of error in these percentages because all should add up to exactly 1 due to the stochastic nature of the matrix P