2. Every year, during the March through September growing season, a gardener use

Question

2. Every year, during the March through September growing season, a gardener uses a chemical test to check soil condition. Depending on the outcome of the test, productivity for the new season can be one of three good, fair, and poor. Over the years, the gardener has observed that the last year's soil condition impacts current year's productivity and that the situation can be described by the following Markov chain Good Fair Poor Good 0.3 0.1 Fair 0.6 0.6 0.4 Poor 0.1 0.3 0.55 0.05 a. Determine the probability of the condition of the soil to be poor next year, it b. Determine the probability of the condition of the soil to be good in the next c. Determine the probability of the condition of the soil to fair in the next two now is fair. two years if now the condition of the soil is poor. years, if now the condition of the soil is fair.

Explanation / Answer

a) From the table, probability thta the soil would be poor next year, if now it is fair is 0.3

b) When soil is poor, the probability of it being good,fair and poor in the next year are 0.05,0.4 and 0.55 respectively.

The probability that soil is good after two years = 0.05*0.3+0.4*0.1+0.55*0.05 = 0.0825

c) When soil is fair, the probability of it being good , fair and poor in the next year are 0.1,0.6 and 0.3 respectively

The probability that soil is fair after two years = 0.1*0.6+0.6*0.6+0.3*0.4 = 0.54

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