Question

If the probability of rain on any given day in city X is 50%, what is the probability that it rains on exactly 3 days in a 5-day period?

• A. \(\frac8125\)
• B. \(\frac225\)
• C. \(\frac516\)
• D. \(\frac825\)

Hint:

For “exactly k successes” problems, always use the binomial probability formula and remember \(p + q = 1\).

Answer 1

The Correct Option is C

We have \(n = 5\) days, and probability of rain on any day is \(p = 0.5\), probability of no rain is \(q = 0.5\).
We need the probability of exactly \(k = 3\) rainy days. This follows a binomial distribution:
\[ P(X = k) = \binomnk p^k q^n-k \] Substitute values:
\[ P(X = 3) = \binom53 (0.5)^3 (0.5)^2 \] \(\binom53 = 10\), and \((0.5)^5 = \frac132\).
Thus:
\[ P(X = 3) = 10 \times \frac132 = \frac1032 = \frac516 \] Hence, the probability is \(\frac516\).