A single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 5 the second time.
To find the probability of two independent events occurring, we multiply the individual probabilities.
The probability of rolling an even number on a fair die is $\frac{3}{6}=\frac{1}{2}$, since there are three even numbers (2, 4, and 6) out of a total of six possible outcomes.
The probability of rolling a number greater than 5 on a fair die is $\frac{1}{6}$, since there is one number (6) greater than 5 out of a total of six possible outcomes.
Therefore, the probability of rolling an even number the first time and a number greater than 5 the second time is $\frac{1}{2}\cdot\frac{1}{6}=\boxed{\frac{1}{12}}$.