Q: In a given town, $270$ days of the year have a winter-like weather and the other $95$ days have a summer-like weather. On a winter-like day the probability of a sunny day is $0.3$. On a summer-like day the probability of a sunny day is $0.9$. Out of the $365$ days of the year you pick $5$ days with replacements at random (uniformly). Calculate the probability that exactly $3$ out of the ﬁve days are sunny.
My Approach: P(Sunny|Winter) = 0.30 P(Sunny|Summer) = 0.90 Therefore using Bernoulli's trials where Success = Sunny P(Success) = P(Sunny|Winter)P(Winter) + P(Sunny|Summer)P(Summer) P(Success) = (0.3)(270/365) + P(0.9)(95/365) = 0.455 Therefore P(3 days out of 5 Sunny) = 5C3(0.455)^3(0.545)^2 = 0.279
Was wondering if I am correct or not? And is there any other way to approach the problem.