Two dice are thrown together. If the sum of the numbers on the two dice is at least 10, then the probability that both dice show the same number is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{1}{3}\)
Since the sum is known to be at least 10, only those outcomes are possible. This reduced sample space is
Sum 10: (4, 6), (5, 5), (6, 4)
Sum 11: (5, 6), (6, 5)
Sum 12: (6, 6)
That is 6 equally likely outcomes. Among them, both dice show the same number in (5, 5) and (6, 6), which is 2 outcomes.
\( \displaystyle P(\text{same number} \mid \text{sum} \ge 10) = \frac{2}{6} \)
\( \displaystyle = \frac{1}{3} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{6}\) — Incorrect. This is the probability of the same number on both dice without the given condition (6 outcomes out of 36).
Option (2) \(\displaystyle\frac{1}{18}\) — Incorrect. This is \(\frac{2}{36}\), the probability that both happen together, not divided by \(P(\text{sum} \ge 10) = \frac{6}{36}\).
Option (4) \(\displaystyle\frac{1}{2}\) — Incorrect. Only 2 of the 6 possible outcomes show the same number, not 3.