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 with p = 0.5 and n = 4.

Answer 1

The Correct Option is A

Step 1: Count the ways to choose which day is sunny.
Out of 4 days, exactly one day is sunny and the other three are cloudy.
There are $\binom{4}{1} = 4$ ways to pick which single day is the sunny one.

Step 2: Find the probability of one such arrangement.
Cloudy and sunny each have probability $0.5$, and the days are independent.
Any one specific sequence of 3 cloudy days and 1 sunny day has probability $(0.5)^3 \times (0.5)^1 = (0.5)^4 = 1/16$.

Step 3: Add up all the arrangements.
Each of the 4 possible positions for the sunny day gives this same probability $1/16$, and these outcomes do not overlap.
So the total probability is $4 \times 1/16 = 1/4$.

Final Answer:
Three cloudy days and one sunny day out of four occur with probability 1/4, so option (A) is correct. \[ \boxed{P = \dfrac{1}{4}} \]