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

Step 1: Each day is independently cloudy with probability \(p = 0.5\) and sunny with probability \(1-p = 0.5\).
Step 2: We need the probability that in 4 days, exactly 3 are cloudy and 1 is sunny. This is a binomial probability problem with \(n = 4\), \(k = 3\), and \(p = 0.5\).
Step 3: The number of ways to choose which 3 of the 4 days are cloudy is \(\binom{4}{3} = 4\).
Step 4: The probability of any one specific arrangement of 3 cloudy days and 1 sunny day is \((0.5)^3 \times (0.5)^1 = (0.5)^4 = \frac{1}{16}\).
Step 5: Multiply the number of arrangements by the probability of each arrangement: \(4 \times \frac{1}{16} = \frac{4}{16} = \frac{1}{4}\).
Final Answer: \[P(\text{3 cloudy, 1 sunny}) = \boxed{\frac{1}{4}}\] which is option (A).