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 idea: count the ways to place the single sunny day among four days, then multiply by the probability of each fixed pattern.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Each day is independent, and cloudy and sunny are equally likely, each with chance $0.5$. We want exactly one sunny day out of four, with the other three cloudy.

Step 2: Key Formula or Approach:
List the four days as slots. The single sunny day can sit in any one of the four slots, and each specific pattern of three cloudy plus one sunny has the same probability, since every day's chance is $0.5$ regardless of position.

Step 3: Detailed Explanation:
The probability of any one fixed pattern, say cloudy, cloudy, cloudy, sunny in that order, is
\[ 0.5\times0.5\times0.5\times0.5=(0.5)^4=\frac{1}{16} \]
Since the sunny day could be the first, second, third, or fourth day, there are $4$ equally likely patterns that all give three cloudy days and one sunny day. Adding their probabilities,
\[ 4\times\frac{1}{16}=\frac{4}{16}=\frac{1}{4} \]

Step 4: Final Answer:
The probability of exactly three cloudy days out of four is $\dfrac{1}{4}$, so the correct choice is option (A).