Two fair dice are thrown. If it is known that at least one of the dice shows a 5, then the probability that the sum of the numbers on the two dice is 8 is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{2}{11}\)
The 36 outcomes (first die, second die) are equally likely. The outcomes with at least one 5 are the 6 with a 5 on the first die and the 6 with a 5 on the second die, with (5, 5) counted twice:
\(6+6-1 = 11\)
So the given condition leaves a reduced sample space of 11 outcomes. Among them, the sum is 8 only when the other die shows 3:
(5, 3), (3, 5) (2 outcomes)
So
\( \displaystyle P = \frac{2}{11} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{18}\) — Incorrect. This is \(\frac{2}{36}\), the probability of a 5 and a sum of 8 together, not divided by the chance of the condition.
Option (2) \(\displaystyle\frac{5}{36}\) — Incorrect. This is the probability of a sum of 8 without the given condition.
Option (4) \(\displaystyle\frac{1}{6}\) — Incorrect. This would be the answer if the first die were known to show 5; “at least one die” gives 11 outcomes, not 6.