Probability: Tricky Questions Solved (13)

Question

Two dice are thrown. Given that the sum is $7$, find the probability that one of them shows $2$.

Solution — Step by Step

The required probability is $\dfrac{1}{3}$.

Why This Works

Conditional probability restricts the sample space. Once we condition on “sum is $7$”, we’re no longer working with all $36$ pairs but only the six pairs that sum to $7$. Within this restricted set, count the favourable cases as a fraction.

The formula $P(A|B) = P(A\cap B)/P(B)$ also works: $P(A\cap B) = 2/36$, $P(B) = 6/36$, ratio $= 1/3$. Same answer.

Alternative Method

Use the formal formula. $P(A\cap B) =$ probability of (one die = 2 AND sum = 7) $= 2/36$. $P(B) = 6/36$. Ratio $= 1/3$.

Common Mistake

Computing $P(\text{sum 7 AND one die }=2)/P(\text{sum 7})$ but forgetting to count both $(2,5)$ and $(5,2)$. Students often answer $1/6$. JEE Main 2023 had this exact problem with sum $= 8$ and one die $= 3$ — same trap.