Discussion
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?
Answer: A
The probability is 1/4.
Each day is cloudy with probability 1/2, independently, so the number of cloudy days in four is binomial. Exactly three cloudy days can happen in C(4, 3) = 4 ways (the sunny day can be any of the four):
\[ \begin{aligned} P &= \binom{4}{3}\left(\tfrac{1}{2}\right)^{3}\left(\tfrac{1}{2}\right)^{1} \\ &= 4 \times \tfrac{1}{16} = \tfrac{1}{4} \end{aligned} \]
3/8 is the probability of exactly two cloudy days, C(4, 2)/16 = 6/16, and is the usual slip. 3/4 and 2/3 do not come from any count of the 16 equally likely weather sequences.
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