A pair of fair dice is thrown 180 times. Getting a doublet (the same number on both dice) is considered a success. The standard deviation of the number of successes is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 5
The doublets are (1, 1), (2, 2), …, (6, 6), which is 6 outcomes out of 36. The throws are independent, so the number of successes follows a binomial distribution with
\( \displaystyle n = 180,\quad p = \frac{6}{36} = \frac{1}{6},\quad q = \frac{5}{6} \)
\( \displaystyle \text{Variance} = npq = 180 \times \frac{1}{6} \times \frac{5}{6} = 25 \)
The standard deviation is the square root of the variance:
\(\sigma = \sqrt{25} = 5\)
Why the other options are incorrect
Option (1) 25 — Incorrect. 25 is the variance; the standard deviation is its square root.
Option (2) 30 — Incorrect. 30 is the mean np = 180 × 1/6.
Option (3) \(\sqrt{30}\) — Incorrect. This is the square root of the mean; the variance npq = 25 has to be used.