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. \(\dfrac{1}{4}\)
• B. \(\dfrac{3}{4}\)
• C. \(\dfrac{2}{3}\)
• D. \(\dfrac{3}{8}\)

Hint:

Count how many of the four days can be the single sunny day, then multiply by the chance of one fixed order of three cloudy and one sunny day.

Answer 1

The Correct Option is A

Step 1: List the possible orders.
We want three cloudy (C) days and one sunny (S) day among four days. The sunny day could fall on day 1, day 2, day 3 or day 4, giving the four orders CCCS, CCSC, CSCC and SCCC.

Step 2: Find the probability of one such order.
Since each day is independent and cloudy or sunny each has probability $0.5$, any single fixed order of three C's and one S has probability
\[ (0.5)\times(0.5)\times(0.5)\times(0.5) = (0.5)^4 = \frac{1}{16} \]

Step 3: Add up all four orders.
Since these four orders cannot happen at the same time, we add their probabilities:
\[ 4\times\frac{1}{16} = \frac{4}{16} = \frac{1}{4} \]

Step 4: Check this against the options.
This value of $\frac{1}{4}$ matches option (A) exactly. The other fractions, $\frac{3}{4}$, $\frac{2}{3}$ and $\frac{3}{8}$, do not come from counting the four valid orders of three cloudy days and one sunny day, so they can be ruled out.

Step 5: Conclude.
\[ \boxed{\dfrac{1}{4}} \]