Two fair dice are thrown. If it is known that the numbers on the two dice are different, then the probability that the sum of the numbers is 6 is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{2}{15}\)
Of the 36 outcomes, 6 show the same number on both dice, so the given condition leaves a reduced sample space of
\(36-6 = 30\) outcomes
The outcomes with sum 6 are (1, 5), (2, 4), (3, 3), (4, 2), (5, 1). The pair (3, 3) has equal numbers, so only 4 of them are in the reduced space:
\( \displaystyle P = \frac{4}{30} = \frac{2}{15} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{5}{36}\) — Incorrect. This is the probability of sum 6 without the given condition.
Option (3) \(\displaystyle\frac{1}{9}\) — Incorrect. This is \(\frac{4}{36}\); it removes (3, 3) but still divides by all 36 outcomes.
Option (4) \(\displaystyle\frac{1}{6}\) — Incorrect. This is \(\frac{5}{30}\); it wrongly keeps (3, 3), which does not have different numbers.