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: Identify the distribution.
Each day is cloudy with probability \(p = 0.5\) and sunny with probability \(q = 1 - p = 0.5\), independent of other days.
With 4 independent days, the number of cloudy days follows a binomial distribution with \(n = 4\) and \(p = 0.5\).

Step 2: Apply the binomial probability formula.
The probability of exactly \(k\) cloudy days out of \(n\) is \(P(k) = \binom{n}{k} p^k q^{n-k}\).
For \(k = 3\) cloudy days and \(n = 4\), this gives \(P(3) = \binom{4}{3} (0.5)^3 (0.5)^1\).

Step 3: Compute the value.
\(\binom{4}{3} = 4\), and \((0.5)^3(0.5)^1 = (0.5)^4 = 1/16\).
So \(P(3) = 4 \times 1/16 = 4/16 = 1/4\).

Final Answer:
The probability of three cloudy days and one sunny day out of four is 1/4. \[ \boxed{P = \dfrac{1}{4}} \]