Solve the problem. 25) Suppose that the weather in a certain city is either sunn

Question

Solve the problem. 25) Suppose that the weather in a certain city is either sunny, cdoudy, or rainy on a given day, and consi 25) following: 1) If it is sunny today there is a 70% chance it will be sunny tomorrow and a 30% chance that it will cloudy 2) If it is cloudy today thereis a 40%chance it will be sunny tomorrow, a 40% chance that it will be c and a 20% chance that it will be rainy. 3) If it is rainy today there is a 40% chance it will be sunny tomorrow, a 30% chance that it will be cic and a 30% chance that it will be rainy. In the long run, how likely is it that the weather will be rainy on a given day? Hint steady state vector is needed. Use Linear algebra and show work on a separate paper) A) 10.6% B) 9.5% C) 12.1% D) 11.3%

Explanation / Answer

Here lets say probability of being a sunny, cloudy and rainy day are s, c & r respectively

so,

s + c + r = 1

so by given conditions

s = 0.7s + 0.4 c + 0.4 r

0.3s = 0.4 c + 0.4 r.

3s = 4c + 4r ..(i)

c = 0.3s + 0.4c + 0.3 r

0.6c = 0.3s + 0.3 r

2c = s + r.....(ii)

r = 0.2c + 0.3 r

0.7r = 0.2 c

7r = 2c ..(iii)

c = 7r/2

putting it in equation (i) and (ii) and s + c + r = 1

we get following results

s = 4/7 (sunny)

y = 1/3 (cloudy )

z = 2/21 (rainy)

so the weather will be rainy on given day with 2/21 percentage = 9.5% (option B s correct)

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