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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.

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You seem to have approached the question correctly, and no better method comes to mind. The Law of Total Probability seems to be the best way to treat this problem, and you have then just modelled this as a Binomial random variable with $n=5$ and $p=0.455$, which seems appropriate due the independence of the events.

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