Question

1. A biologist is conducting experiments with scientific cross pollination of pink roses with white, red, and orange roses. The annual experiments show that the pink can produce 60% pink and 20% white and 20% red, red can produce 40% red, 50% pink and 10% orange, orange can produce 25% orange, 50% pink and 25% white, and white can produce 50% pink and 50% white.
   1. Express the biologist situation as a Markov chain.  
   2. If the biologist started the pollination with equal number of roses then what would be the distribution after many years?

Answer 1

Yes this is Markov chain with TPM

P =

	Red	Pink	White	Orange
Red	0.40	0.50	0.00	0.10
Pink	0.20	0.60	0.20	0.00
White	0.00	0.50	0.50	0.00
Orange	0.00	0.25	0.25	0.50

P^2 =  

	Red	Pink	White	Orange
Red	0.26	0.53	0.13	0.09
Pink	0.20	0.56	0.22	0.02
White	0.10	0.55	0.35	0.00
Orange	0.05	0.40	0.30	0.25

P^3 =

	Red	Pink	White	Orange
Red	0.21	0.53	0.19	0.07
Pink	0.19	0.55	0.23	0.03
White	0.15	0.56	0.29	0.01
Orange	0.10	0.48	0.29	0.13

P^4 =

	Red	Pink	White	Orange
Red	0.19	0.54	0.22	0.06
Pink	0.19	0.55	0.23	0.03
White	0.17	0.55	0.26	0.02
Orange	0.14	0.52	0.27	0.08

P^5 =

	Red	Pink	White	Orange
Red	0.18	0.54	0.23	0.05
Pink	0.18	0.55	0.23	0.04
White	0.18	0.55	0.24	0.03
Orange	0.16	0.53	0.26	0.05

P^6 =

	Red	Pink	White	Orange
Red	0.18	0.54	0.23	0.04
Pink	0.18	0.55	0.24	0.04
White	0.18	0.55	0.24	0.03
Orange	0.17	0.54	0.25	0.04

P^7 =

	Red	Pink	White	Orange
Red	0.18	0.54	0.24	0.04
Pink	0.18	0.55	0.24	0.04
White	0.18	0.55	0.24	0.03
Orange	0.18	0.54	0.24	0.04

P^8 =

	Red	Pink	White	Orange
Red	0.18	0.54	0.24	0.04
Pink	0.18	0.55	0.24	0.04
White	0.18	0.55	0.24	0.04
Orange	0.18	0.54	0.24	0.04

P^9 =

	Red	Pink	White	Orange
Red	0.18	0.55	0.24	0.04
Pink	0.18	0.55	0.24	0.04
White	0.18	0.55	0.24	0.04
Orange	0.18	0.55	0.24	0.04

P^10 =

	Red	Pink	White	Orange
Red	0.18	0.55	0.24	0.04
Pink	0.18	0.55	0.24	0.04
White	0.18	0.55	0.24	0.04
Orange	0.18	0.55	0.24	0.04

If the biologist started the pollination with equal number of roses then what would be the distribution after many years?

Red = 18%

Pink = 55%.

White = 24%

Orange = 4%