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: Identify the type of probability problem.
Each day is an independent trial with exactly two outcomes, cloudy with probability \(p=0.5\) or sunny with probability \(q=1-p=0.5\). Over four days we want the chance of getting exactly three cloudy days and one sunny day, in any order. This is a binomial probability situation.

Step 2: Write down the binomial probability formula.
For \(n\) independent trials, each with success probability \(p\), the probability of getting exactly \(k\) successes is
\[ P(k) = \binom{n}{k} p^{k} (1-p)^{n-k} \]
Here a success is a cloudy day, so \(n=4\), \(k=3\), and \(p=0.5\).

Step 3: Count the number of ways to choose which three days are cloudy.
\[ \binom{4}{3} = 4 \]
This counts the four possible arrangements: the sunny day could be day 1, day 2, day 3, or day 4, with the other three days cloudy.

Step 4: Multiply by the probability of one such arrangement.
For any one arrangement, the probability of exactly that sequence of cloudy and sunny days is
\[ (0.5)^3 \times (0.5)^1 = (0.5)^4 = \frac{1}{16} \]

Step 5: Combine the count and the probability.
\[ P = 4 \times \frac{1}{16} = \frac{4}{16} = \frac{1}{4} \]

Step 6: Final answer.
The probability that exactly three of the four days are cloudy is
\[ \boxed{\frac{1}{4}} \]
Hence, the correct option is (A).