https://media.cheggcdn.com/media%2Fee6%2Fee6b1ed3-aa7f-405b-9043-bb0c783e1e08%2FphprZjMP4.png

a)it' sunny today. So the probability that all four days are sunny = 0.8^4 = 0.4096

b) it's sunny today. Probability of rain on the last day can be in following ways = sunny*sunny*sunny*rainy + sunny*sunny*rainy*rainy + sunny*rainy*rainy*rainy + sunny*rainy*sunny*rainy= 0.8^3*0.2 + 0.8^2*0.2*0.6+ 0.8*0.2*0.6^2 + 0.8"0.2*0.4*0.2= 0.2624

If it's raining today, probability of rain on the fourth day = rain*rain*rain*rain + rain*sunny*sunny*rainy + rain*sunny*rainy*rainy + rainy*rainy*sunny*rainy = 0.6^4 + 0.6*0.4*0.8*0.2+ 0.6*0.4*0.2*0.6+ 0.6^2*0.4*0.2= 0.2256

Thus total probability of rain on the last day = 0.2624+0.2256= 0.488

c) Here since 365 days is a long period of time. We will consider the long term probability. And we find that probability of raining out is 0.33

https://media.cheggcdn.com/media%2F8b8%2F8b88e855-e232-483e-a29d-f15cff02c881%2FphpHAeeBi.png