A day can only be cloudy or sunny. The probability of a day being cloudy is 0.5, independent of the condition on other days. What is the probability that in any given four days, there will be three cloudy days and one sunny day?

This problem can be modeled using a binomial distribution where the number of trials is $n = 4$.

Let success be defined as a cloudy day:

• Probability of a cloudy day, $p = 0.5 = \frac{1}{2}$

• Probability of a sunny day, $q = 1 - p = 0.5 = \frac{1}{2}$

The probability of having exactly $k = 3$ cloudy days (and therefore $1$ sunny day) in $n = 4$ independent days is:

Evaluating this:

Hence, the correct option is (A).