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 n = 4, r = 3 cloudy days, and p = 0.5.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
Each day is independently cloudy with probability 0.5 and sunny with probability 0.5, since a day can only be one of these two states. We are asked for the probability that out of four such independent days, exactly three turn out cloudy and exactly one turns out sunny, in any order.

Step 2: Key Formula or Approach:
Since each day is an independent trial with only two possible outcomes and a fixed probability of success, this is a binomial probability situation. The probability of getting exactly r successes, here cloudy days, out of n independent trials, here days, each with success probability p, is given by \(P(r) = \binom{n}{r} p^{r} (1-p)^{n-r}\). Here n = 4, r = 3, and p = 0.5 for a cloudy day, so 1 - p = 0.5 for a sunny day.

Step 3: Detailed Explanation:
First we find the number of ways to choose which 3 of the 4 days are cloudy, which is \(\binom{4}{3} = 4\). These 4 arrangements correspond to the sunny day falling on day 1, day 2, day 3 or day 4, with the remaining three days being cloudy in each case.

Next, since cloudy and sunny each have probability 0.5, the probability of any one specific sequence of 3 cloudy days and 1 sunny day, in a fixed order, is \(0.5^{3} \times 0.5^{1} = 0.5^{4} = \frac{1}{16}\).

Multiplying the number of favourable sequences by the probability of each individual sequence gives the total probability, \(4 \times \frac{1}{16} = \frac{4}{16} = \frac{1}{4}\).

Checking the other options, 3/4 and 2/3 do not correspond to any valid binomial term for these values of n, r and p, and 3/8 would only arise from a miscount of the number of arrangements, so these are incorrect.

Step 4: Final Answer:
The probability of getting exactly three cloudy days and one sunny day out of four independent days, each with probability 0.5, works out to 1/4.
\[ \boxed{\dfrac{1}{4}} \]