Question

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?

• A. 1/4
• B. 3/4
• C. 2/3
• D. 3/8

Hint:

Use the binomial probability formula \(\binom{n}{k}p^k(1-p)^{n-k}\) with \(n=4\), \(k=3\), \(p=0.5\).

Answer 1

The Correct Option is A

Since a day is equally likely to be cloudy or sunny, think of each day as a fair coin toss where cloudy is heads and sunny is tails. Over 4 independent days there are $2^4 = 16$ equally likely outcome sequences in total, since each day has 2 possibilities. We want sequences with exactly 3 heads (cloudy) and 1 tail (sunny). The number of such sequences equals the number of positions the single sunny day can occupy among the 4 days, which is $\binom{4}{1} = 4$ (equivalently $\binom{4}{3}=4$). So the required probability is the count of favorable sequences divided by total sequences: $4/16 = 1/4$. This matches option (A).