The probability of rolling an even number followed by an odd number is $\frac{1}{4}$, or $25\%$.

When rolling a standard six-sided die, there are three even numbers: $2$, $4$, and $6$, as well as three odd numbers: $1$, $3$, and $5$. Consequently, the probability of rolling an even number on the first roll is calculated as follows:

Similarly, the probability of rolling an odd number on the second roll is also:

To determine the combined probability of both events occurring in sequence—rolling an even number first and then an odd number—you multiply the probabilities of each individual event. Therefore, the calculation is:

Thus, the probability of rolling an even number followed by an odd number is $\frac{1}{4}$, which is equivalent to $25\%$. This means that if you were to roll a die twice repeatedly, you would expect this specific sequence to occur approximately $25\%$ of the time.