If two dice are rolled 12 times and getting a total greater than 4 is considered as a success, then which of the following statements are correct?
(A) The probability of getting a total greater than 4 in a single throw of the pair of dice is 5/6.
(B) Mean = 10
(C) Variance = 3/5
(D) The probability of getting a total less than or equal to 4 in a single throw of the pair of dice is 1/6.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
The number of successes in 12 throws follows a binomial distribution with n = 12. A pair of dice has 36 outcomes, and a total of 4 or less comes from (1, 1), (1, 2), (2, 1), (1, 3), (3, 1) and (2, 2), that is, 6 outcomes:
\( \displaystyle P(\text{total} \le 4) = \frac{6}{36} = \frac{1}{6} \)
\( \displaystyle p = P(\text{total} \gt 4) = 1 - \frac{1}{6} = \frac{5}{6} \)
\( \displaystyle q = 1 - p = \frac{1}{6} \)
So n = 12, \(p = \frac{5}{6}\) and \(q = \frac{1}{6}\).
Checking the statements
(A) The probability of getting a total greater than 4 in a single throw of the pair of dice is 5/6. — Correct. \(p = 1 - \frac{6}{36} = \frac{5}{6}\).
(B) Mean = 10 — Correct. The mean is \(np = 12 \times \frac{5}{6} = 10\).
(C) Variance = 3/5 — Incorrect. The variance is \(npq = 12 \times \frac{5}{6} \times \frac{1}{6} = \frac{5}{3}\), not 3/5.
(D) The probability of getting a total less than or equal to 4 in a single throw of the pair of dice is 1/6. — Correct. \(q = \frac{6}{36} = \frac{1}{6}\).