Given that the two numbers appearing on throwing 2 dice are different, then the probability of the event " sum of number on the dice is 4 " is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{1}{15} \)
This is a conditional probability. Let F be "the two numbers are different" and E be "the sum is 4".
Of the 36 outcomes, 6 have equal numbers, so F has 30 outcomes. The outcomes with sum 4 are (1, 3), (2, 2), (3, 1); of these, (1, 3) and (3, 1) are in F.
\( \displaystyle P(E|F)=\frac{P(E\cap F)}{P(F)} \)
\( \displaystyle =\frac{2/36}{30/36} \)
\( \displaystyle =\frac{2}{30}=\frac{1}{15} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{2}{15} \) — Incorrect. \( \frac{2}{15}=\frac{4}{30} \), but only 2 of the 30 outcomes, (1, 3) and (3, 1), have sum 4 with different numbers.
Option (3) \( \displaystyle\frac{1}{12} \) — Incorrect. \( \frac{1}{12}=\frac{3}{36} \) is the unconditional probability of sum 4, and it also includes (2, 2), which is ruled out by the condition.
Option (4) \( \displaystyle\frac{1}{18} \) — Incorrect. \( \frac{1}{18}=\frac{2}{36} \) is \( P(E\cap F) \); it still has to be divided by \( P(F)=\frac{30}{36} \).
NCERT (2026-27 edition):
Conditional Probability
