Question

A certain city prides itself on having sunny days. If it rains one day, there is a 95% chance that it will be sunny the next day. If it is sunny one day, there is a 20% chance that it will rain the following day. (Assume that there are only sunny or rainy days.) 1. (4 pts) Give the stochastic matrix. 2. (4 pts) If today is sunny, what is the chance that it will be sunny the day after tomorrow? Show all your work. 3. (2 pts) In the long run, what fraction of all days is sunny? Show all your work.

Answer 1

1) According to the given probabilities:
P(RS) = 0.95, therefore P(RR) = 1 - 0.95 = 0.05,

P(SR) = 0.2, therefore P(SS) = 1 - 0.2 = 0.8

Therefore the stochastic matrix here is given as:

$P = \begin{pmatrix} 0.05 &0.95 \\ 0.2 & 0.8 \end{pmatrix}$

2) Given that it is sunny today, probability that it will be sunny the day after tomorrow is computed here as:

= P(SSS) + P(SRS)

= P(SS)P(SS) + P(SR)P(RS)

= 0.8*0.8 + 0.2*0.95

= 0.83

Therefore 0.83 is the required probability here.

3) Let the fraction of days that are sunny in the long run be X, then the fraction of days in which it rains in long run would be 1 - X.

From second column, we have here:
X = 0.95(1 - X) + 0.8X
0.2X + 0.95X = 0.95
X = 0.95 / 1.15 = 0.8261

Therefore 0.8261 is the required fraction of days that would be sunny here.