Question

In Markov City, a sunny day is followed by a rainy day with probability 0.82, and a rainy day is followed by a sunny day with probability 0.81. What proportion of days will be sunny in Markov City, in the long run? (Give the probability rounded to four decimal places.)

Answer 1

P(rainy day | previous day is sunny) = 0.82

P(sunny day | previous day is sunny) = 1 - 0.82 = 0.18

P(sunny day | previous day is rainy) = 0.81

P(rainy day | previous day is rainy) = 1 - 0.81 = 0.19

P(sunny day) = P(sunny day | previous day is rainy)*P(previous day is rainy day) + P(sunny day | previous day is sunny)*P(previous day is sunny day)

= P(sunny day | previous day is rainy)*P(rainy day) + P(sunny day | previous day is sunny)*P(sunny day)

= 0.81 * (1 - P(sunny day)) + 0.18 * P(sunny day)

P(sunny day) = 0.81 * (1 - P(sunny day)) + 0.18 * P(sunny day)

P(sunny day) = 0.81 - 0.81*P(sunny day)) + 0.18 * P(sunny day)

P(sunny day) * (1+0.81-0.18) = 0.81

P(sunny day) = 0.81 / (1+0.81-0.18)

P(sunny day) = 0.4969

proportion of days that will be sunny in Markov City, in the long run = 0.4969

(please UPVOTE)