Two fair dice are rolled and it is known that at least one of them shows a 4. What is the probability that the sum of the two numbers is 7?

The condition shrinks the sample space: outcomes with at least one 4 number 6 + 6 − 1 = 11. Among them the sum is 7 only for (3,4) and (4,3), so the conditional probability is 2/11. (The unconditional answer 6/36 = 1/6 ignores the given information.)